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https://people.maths.bris.ac.uk/~matyd/GroupNames/448/C4sDic7s7C4.html
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## G = C4⋊Dic7⋊7C4order 448 = 26·7
### 3rd semidirect product of C4⋊Dic7 and C4 acting via C4/C2=C2
Series: Derived Chief Lower central Upper central
Derived series C1 — C2×C14 — C4⋊Dic7⋊7C4
Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C2×C4⋊Dic7 — C4⋊Dic7⋊7C4
Lower central C7 — C2×C14 — C4⋊Dic7⋊7C4
Upper central C1 — C23 — C2.C42
Generators and relations for C4⋊Dic77C4
G = < a,b,c,d | a4=b14=d4=1, c2=b7, ab=ba, cac-1=a-1, dad-1=ab7, cbc-1=b-1, bd=db, cd=dc >
Subgroups: 604 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C7, C2×C4, C2×C4, C23, C14, C42, C4⋊C4, C22×C4, C22×C4, Dic7, C28, C2×C14, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.63C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C22×Dic7, C22×C28, C14.C42, C7×C2.C42, C2×C4×Dic7, C2×Dic7⋊C4, C2×C4⋊Dic7, C4⋊Dic77C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C22×C4, C2×D4, C2×Q8, C4○D4, D14, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D7, D28, C22×D7, C23.63C23, C2×C4×D7, C2×D28, C4○D28, D42D7, Q8×D7, C4×D28, C23.11D14, C23.D14, C22.D28, Dic73Q8, Dic7.Q8, D142Q8, C4⋊Dic77C4
Smallest permutation representation of C4⋊Dic77C4
Regular action on 448 points
Generators in S448
(1 134 15 120)(2 135 16 121)(3 136 17 122)(4 137 18 123)(5 138 19 124)(6 139 20 125)(7 140 21 126)(8 127 22 113)(9 128 23 114)(10 129 24 115)(11 130 25 116)(12 131 26 117)(13 132 27 118)(14 133 28 119)(29 155 50 148)(30 156 51 149)(31 157 52 150)(32 158 53 151)(33 159 54 152)(34 160 55 153)(35 161 56 154)(36 162 43 141)(37 163 44 142)(38 164 45 143)(39 165 46 144)(40 166 47 145)(41 167 48 146)(42 168 49 147)(57 190 78 176)(58 191 79 177)(59 192 80 178)(60 193 81 179)(61 194 82 180)(62 195 83 181)(63 196 84 182)(64 183 71 169)(65 184 72 170)(66 185 73 171)(67 186 74 172)(68 187 75 173)(69 188 76 174)(70 189 77 175)(85 218 106 204)(86 219 107 205)(87 220 108 206)(88 221 109 207)(89 222 110 208)(90 223 111 209)(91 224 112 210)(92 211 99 197)(93 212 100 198)(94 213 101 199)(95 214 102 200)(96 215 103 201)(97 216 104 202)(98 217 105 203)(225 340 239 364)(226 341 240 351)(227 342 241 352)(228 343 242 353)(229 344 243 354)(230 345 244 355)(231 346 245 356)(232 347 246 357)(233 348 247 358)(234 349 248 359)(235 350 249 360)(236 337 250 361)(237 338 251 362)(238 339 252 363)(253 376 267 385)(254 377 268 386)(255 378 269 387)(256 365 270 388)(257 366 271 389)(258 367 272 390)(259 368 273 391)(260 369 274 392)(261 370 275 379)(262 371 276 380)(263 372 277 381)(264 373 278 382)(265 374 279 383)(266 375 280 384)(281 393 305 412)(282 394 306 413)(283 395 307 414)(284 396 308 415)(285 397 295 416)(286 398 296 417)(287 399 297 418)(288 400 298 419)(289 401 299 420)(290 402 300 407)(291 403 301 408)(292 404 302 409)(293 405 303 410)(294 406 304 411)(309 429 333 440)(310 430 334 441)(311 431 335 442)(312 432 336 443)(313 433 323 444)(314 434 324 445)(315 421 325 446)(316 422 326 447)(317 423 327 448)(318 424 328 435)(319 425 329 436)(320 426 330 437)(321 427 331 438)(322 428 332 439)
(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)
(1 227 8 234)(2 226 9 233)(3 225 10 232)(4 238 11 231)(5 237 12 230)(6 236 13 229)(7 235 14 228)(15 241 22 248)(16 240 23 247)(17 239 24 246)(18 252 25 245)(19 251 26 244)(20 250 27 243)(21 249 28 242)(29 263 36 256)(30 262 37 255)(31 261 38 254)(32 260 39 253)(33 259 40 266)(34 258 41 265)(35 257 42 264)(43 270 50 277)(44 269 51 276)(45 268 52 275)(46 267 53 274)(47 280 54 273)(48 279 55 272)(49 278 56 271)(57 293 64 286)(58 292 65 285)(59 291 66 284)(60 290 67 283)(61 289 68 282)(62 288 69 281)(63 287 70 294)(71 296 78 303)(72 295 79 302)(73 308 80 301)(74 307 81 300)(75 306 82 299)(76 305 83 298)(77 304 84 297)(85 320 92 313)(86 319 93 312)(87 318 94 311)(88 317 95 310)(89 316 96 309)(90 315 97 322)(91 314 98 321)(99 323 106 330)(100 336 107 329)(101 335 108 328)(102 334 109 327)(103 333 110 326)(104 332 111 325)(105 331 112 324)(113 349 120 342)(114 348 121 341)(115 347 122 340)(116 346 123 339)(117 345 124 338)(118 344 125 337)(119 343 126 350)(127 359 134 352)(128 358 135 351)(129 357 136 364)(130 356 137 363)(131 355 138 362)(132 354 139 361)(133 353 140 360)(141 365 148 372)(142 378 149 371)(143 377 150 370)(144 376 151 369)(145 375 152 368)(146 374 153 367)(147 373 154 366)(155 381 162 388)(156 380 163 387)(157 379 164 386)(158 392 165 385)(159 391 166 384)(160 390 167 383)(161 389 168 382)(169 398 176 405)(170 397 177 404)(171 396 178 403)(172 395 179 402)(173 394 180 401)(174 393 181 400)(175 406 182 399)(183 417 190 410)(184 416 191 409)(185 415 192 408)(186 414 193 407)(187 413 194 420)(188 412 195 419)(189 411 196 418)(197 433 204 426)(198 432 205 425)(199 431 206 424)(200 430 207 423)(201 429 208 422)(202 428 209 421)(203 427 210 434)(211 444 218 437)(212 443 219 436)(213 442 220 435)(214 441 221 448)(215 440 222 447)(216 439 223 446)(217 438 224 445)
(1 85 29 64)(2 86 30 65)(3 87 31 66)(4 88 32 67)(5 89 33 68)(6 90 34 69)(7 91 35 70)(8 92 36 57)(9 93 37 58)(10 94 38 59)(11 95 39 60)(12 96 40 61)(13 97 41 62)(14 98 42 63)(15 106 50 71)(16 107 51 72)(17 108 52 73)(18 109 53 74)(19 110 54 75)(20 111 55 76)(21 112 56 77)(22 99 43 78)(23 100 44 79)(24 101 45 80)(25 102 46 81)(26 103 47 82)(27 104 48 83)(28 105 49 84)(113 204 141 169)(114 205 142 170)(115 206 143 171)(116 207 144 172)(117 208 145 173)(118 209 146 174)(119 210 147 175)(120 197 148 176)(121 198 149 177)(122 199 150 178)(123 200 151 179)(124 201 152 180)(125 202 153 181)(126 203 154 182)(127 218 162 183)(128 219 163 184)(129 220 164 185)(130 221 165 186)(131 222 166 187)(132 223 167 188)(133 224 168 189)(134 211 155 190)(135 212 156 191)(136 213 157 192)(137 214 158 193)(138 215 159 194)(139 216 160 195)(140 217 161 196)(225 318 261 284)(226 319 262 285)(227 320 263 286)(228 321 264 287)(229 322 265 288)(230 309 266 289)(231 310 253 290)(232 311 254 291)(233 312 255 292)(234 313 256 293)(235 314 257 294)(236 315 258 281)(237 316 259 282)(238 317 260 283)(239 328 275 308)(240 329 276 295)(241 330 277 296)(242 331 278 297)(243 332 279 298)(244 333 280 299)(245 334 267 300)(246 335 268 301)(247 336 269 302)(248 323 270 303)(249 324 271 304)(250 325 272 305)(251 326 273 306)(252 327 274 307)(337 428 367 400)(338 429 368 401)(339 430 369 402)(340 431 370 403)(341 432 371 404)(342 433 372 405)(343 434 373 406)(344 421 374 393)(345 422 375 394)(346 423 376 395)(347 424 377 396)(348 425 378 397)(349 426 365 398)(350 427 366 399)(351 443 380 409)(352 444 381 410)(353 445 382 411)(354 446 383 412)(355 447 384 413)(356 448 385 414)(357 435 386 415)(358 436 387 416)(359 437 388 417)(360 438 389 418)(361 439 390 419)(362 440 391 420)(363 441 392 407)(364 442 379 408)
G:=sub<Sym(448)| (1,134,15,120)(2,135,16,121)(3,136,17,122)(4,137,18,123)(5,138,19,124)(6,139,20,125)(7,140,21,126)(8,127,22,113)(9,128,23,114)(10,129,24,115)(11,130,25,116)(12,131,26,117)(13,132,27,118)(14,133,28,119)(29,155,50,148)(30,156,51,149)(31,157,52,150)(32,158,53,151)(33,159,54,152)(34,160,55,153)(35,161,56,154)(36,162,43,141)(37,163,44,142)(38,164,45,143)(39,165,46,144)(40,166,47,145)(41,167,48,146)(42,168,49,147)(57,190,78,176)(58,191,79,177)(59,192,80,178)(60,193,81,179)(61,194,82,180)(62,195,83,181)(63,196,84,182)(64,183,71,169)(65,184,72,170)(66,185,73,171)(67,186,74,172)(68,187,75,173)(69,188,76,174)(70,189,77,175)(85,218,106,204)(86,219,107,205)(87,220,108,206)(88,221,109,207)(89,222,110,208)(90,223,111,209)(91,224,112,210)(92,211,99,197)(93,212,100,198)(94,213,101,199)(95,214,102,200)(96,215,103,201)(97,216,104,202)(98,217,105,203)(225,340,239,364)(226,341,240,351)(227,342,241,352)(228,343,242,353)(229,344,243,354)(230,345,244,355)(231,346,245,356)(232,347,246,357)(233,348,247,358)(234,349,248,359)(235,350,249,360)(236,337,250,361)(237,338,251,362)(238,339,252,363)(253,376,267,385)(254,377,268,386)(255,378,269,387)(256,365,270,388)(257,366,271,389)(258,367,272,390)(259,368,273,391)(260,369,274,392)(261,370,275,379)(262,371,276,380)(263,372,277,381)(264,373,278,382)(265,374,279,383)(266,375,280,384)(281,393,305,412)(282,394,306,413)(283,395,307,414)(284,396,308,415)(285,397,295,416)(286,398,296,417)(287,399,297,418)(288,400,298,419)(289,401,299,420)(290,402,300,407)(291,403,301,408)(292,404,302,409)(293,405,303,410)(294,406,304,411)(309,429,333,440)(310,430,334,441)(311,431,335,442)(312,432,336,443)(313,433,323,444)(314,434,324,445)(315,421,325,446)(316,422,326,447)(317,423,327,448)(318,424,328,435)(319,425,329,436)(320,426,330,437)(321,427,331,438)(322,428,332,439), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,241,22,248)(16,240,23,247)(17,239,24,246)(18,252,25,245)(19,251,26,244)(20,250,27,243)(21,249,28,242)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,270,50,277)(44,269,51,276)(45,268,52,275)(46,267,53,274)(47,280,54,273)(48,279,55,272)(49,278,56,271)(57,293,64,286)(58,292,65,285)(59,291,66,284)(60,290,67,283)(61,289,68,282)(62,288,69,281)(63,287,70,294)(71,296,78,303)(72,295,79,302)(73,308,80,301)(74,307,81,300)(75,306,82,299)(76,305,83,298)(77,304,84,297)(85,320,92,313)(86,319,93,312)(87,318,94,311)(88,317,95,310)(89,316,96,309)(90,315,97,322)(91,314,98,321)(99,323,106,330)(100,336,107,329)(101,335,108,328)(102,334,109,327)(103,333,110,326)(104,332,111,325)(105,331,112,324)(113,349,120,342)(114,348,121,341)(115,347,122,340)(116,346,123,339)(117,345,124,338)(118,344,125,337)(119,343,126,350)(127,359,134,352)(128,358,135,351)(129,357,136,364)(130,356,137,363)(131,355,138,362)(132,354,139,361)(133,353,140,360)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,381,162,388)(156,380,163,387)(157,379,164,386)(158,392,165,385)(159,391,166,384)(160,390,167,383)(161,389,168,382)(169,398,176,405)(170,397,177,404)(171,396,178,403)(172,395,179,402)(173,394,180,401)(174,393,181,400)(175,406,182,399)(183,417,190,410)(184,416,191,409)(185,415,192,408)(186,414,193,407)(187,413,194,420)(188,412,195,419)(189,411,196,418)(197,433,204,426)(198,432,205,425)(199,431,206,424)(200,430,207,423)(201,429,208,422)(202,428,209,421)(203,427,210,434)(211,444,218,437)(212,443,219,436)(213,442,220,435)(214,441,221,448)(215,440,222,447)(216,439,223,446)(217,438,224,445), (1,85,29,64)(2,86,30,65)(3,87,31,66)(4,88,32,67)(5,89,33,68)(6,90,34,69)(7,91,35,70)(8,92,36,57)(9,93,37,58)(10,94,38,59)(11,95,39,60)(12,96,40,61)(13,97,41,62)(14,98,42,63)(15,106,50,71)(16,107,51,72)(17,108,52,73)(18,109,53,74)(19,110,54,75)(20,111,55,76)(21,112,56,77)(22,99,43,78)(23,100,44,79)(24,101,45,80)(25,102,46,81)(26,103,47,82)(27,104,48,83)(28,105,49,84)(113,204,141,169)(114,205,142,170)(115,206,143,171)(116,207,144,172)(117,208,145,173)(118,209,146,174)(119,210,147,175)(120,197,148,176)(121,198,149,177)(122,199,150,178)(123,200,151,179)(124,201,152,180)(125,202,153,181)(126,203,154,182)(127,218,162,183)(128,219,163,184)(129,220,164,185)(130,221,165,186)(131,222,166,187)(132,223,167,188)(133,224,168,189)(134,211,155,190)(135,212,156,191)(136,213,157,192)(137,214,158,193)(138,215,159,194)(139,216,160,195)(140,217,161,196)(225,318,261,284)(226,319,262,285)(227,320,263,286)(228,321,264,287)(229,322,265,288)(230,309,266,289)(231,310,253,290)(232,311,254,291)(233,312,255,292)(234,313,256,293)(235,314,257,294)(236,315,258,281)(237,316,259,282)(238,317,260,283)(239,328,275,308)(240,329,276,295)(241,330,277,296)(242,331,278,297)(243,332,279,298)(244,333,280,299)(245,334,267,300)(246,335,268,301)(247,336,269,302)(248,323,270,303)(249,324,271,304)(250,325,272,305)(251,326,273,306)(252,327,274,307)(337,428,367,400)(338,429,368,401)(339,430,369,402)(340,431,370,403)(341,432,371,404)(342,433,372,405)(343,434,373,406)(344,421,374,393)(345,422,375,394)(346,423,376,395)(347,424,377,396)(348,425,378,397)(349,426,365,398)(350,427,366,399)(351,443,380,409)(352,444,381,410)(353,445,382,411)(354,446,383,412)(355,447,384,413)(356,448,385,414)(357,435,386,415)(358,436,387,416)(359,437,388,417)(360,438,389,418)(361,439,390,419)(362,440,391,420)(363,441,392,407)(364,442,379,408)>;
G:=Group( (1,134,15,120)(2,135,16,121)(3,136,17,122)(4,137,18,123)(5,138,19,124)(6,139,20,125)(7,140,21,126)(8,127,22,113)(9,128,23,114)(10,129,24,115)(11,130,25,116)(12,131,26,117)(13,132,27,118)(14,133,28,119)(29,155,50,148)(30,156,51,149)(31,157,52,150)(32,158,53,151)(33,159,54,152)(34,160,55,153)(35,161,56,154)(36,162,43,141)(37,163,44,142)(38,164,45,143)(39,165,46,144)(40,166,47,145)(41,167,48,146)(42,168,49,147)(57,190,78,176)(58,191,79,177)(59,192,80,178)(60,193,81,179)(61,194,82,180)(62,195,83,181)(63,196,84,182)(64,183,71,169)(65,184,72,170)(66,185,73,171)(67,186,74,172)(68,187,75,173)(69,188,76,174)(70,189,77,175)(85,218,106,204)(86,219,107,205)(87,220,108,206)(88,221,109,207)(89,222,110,208)(90,223,111,209)(91,224,112,210)(92,211,99,197)(93,212,100,198)(94,213,101,199)(95,214,102,200)(96,215,103,201)(97,216,104,202)(98,217,105,203)(225,340,239,364)(226,341,240,351)(227,342,241,352)(228,343,242,353)(229,344,243,354)(230,345,244,355)(231,346,245,356)(232,347,246,357)(233,348,247,358)(234,349,248,359)(235,350,249,360)(236,337,250,361)(237,338,251,362)(238,339,252,363)(253,376,267,385)(254,377,268,386)(255,378,269,387)(256,365,270,388)(257,366,271,389)(258,367,272,390)(259,368,273,391)(260,369,274,392)(261,370,275,379)(262,371,276,380)(263,372,277,381)(264,373,278,382)(265,374,279,383)(266,375,280,384)(281,393,305,412)(282,394,306,413)(283,395,307,414)(284,396,308,415)(285,397,295,416)(286,398,296,417)(287,399,297,418)(288,400,298,419)(289,401,299,420)(290,402,300,407)(291,403,301,408)(292,404,302,409)(293,405,303,410)(294,406,304,411)(309,429,333,440)(310,430,334,441)(311,431,335,442)(312,432,336,443)(313,433,323,444)(314,434,324,445)(315,421,325,446)(316,422,326,447)(317,423,327,448)(318,424,328,435)(319,425,329,436)(320,426,330,437)(321,427,331,438)(322,428,332,439), (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), (1,227,8,234)(2,226,9,233)(3,225,10,232)(4,238,11,231)(5,237,12,230)(6,236,13,229)(7,235,14,228)(15,241,22,248)(16,240,23,247)(17,239,24,246)(18,252,25,245)(19,251,26,244)(20,250,27,243)(21,249,28,242)(29,263,36,256)(30,262,37,255)(31,261,38,254)(32,260,39,253)(33,259,40,266)(34,258,41,265)(35,257,42,264)(43,270,50,277)(44,269,51,276)(45,268,52,275)(46,267,53,274)(47,280,54,273)(48,279,55,272)(49,278,56,271)(57,293,64,286)(58,292,65,285)(59,291,66,284)(60,290,67,283)(61,289,68,282)(62,288,69,281)(63,287,70,294)(71,296,78,303)(72,295,79,302)(73,308,80,301)(74,307,81,300)(75,306,82,299)(76,305,83,298)(77,304,84,297)(85,320,92,313)(86,319,93,312)(87,318,94,311)(88,317,95,310)(89,316,96,309)(90,315,97,322)(91,314,98,321)(99,323,106,330)(100,336,107,329)(101,335,108,328)(102,334,109,327)(103,333,110,326)(104,332,111,325)(105,331,112,324)(113,349,120,342)(114,348,121,341)(115,347,122,340)(116,346,123,339)(117,345,124,338)(118,344,125,337)(119,343,126,350)(127,359,134,352)(128,358,135,351)(129,357,136,364)(130,356,137,363)(131,355,138,362)(132,354,139,361)(133,353,140,360)(141,365,148,372)(142,378,149,371)(143,377,150,370)(144,376,151,369)(145,375,152,368)(146,374,153,367)(147,373,154,366)(155,381,162,388)(156,380,163,387)(157,379,164,386)(158,392,165,385)(159,391,166,384)(160,390,167,383)(161,389,168,382)(169,398,176,405)(170,397,177,404)(171,396,178,403)(172,395,179,402)(173,394,180,401)(174,393,181,400)(175,406,182,399)(183,417,190,410)(184,416,191,409)(185,415,192,408)(186,414,193,407)(187,413,194,420)(188,412,195,419)(189,411,196,418)(197,433,204,426)(198,432,205,425)(199,431,206,424)(200,430,207,423)(201,429,208,422)(202,428,209,421)(203,427,210,434)(211,444,218,437)(212,443,219,436)(213,442,220,435)(214,441,221,448)(215,440,222,447)(216,439,223,446)(217,438,224,445), (1,85,29,64)(2,86,30,65)(3,87,31,66)(4,88,32,67)(5,89,33,68)(6,90,34,69)(7,91,35,70)(8,92,36,57)(9,93,37,58)(10,94,38,59)(11,95,39,60)(12,96,40,61)(13,97,41,62)(14,98,42,63)(15,106,50,71)(16,107,51,72)(17,108,52,73)(18,109,53,74)(19,110,54,75)(20,111,55,76)(21,112,56,77)(22,99,43,78)(23,100,44,79)(24,101,45,80)(25,102,46,81)(26,103,47,82)(27,104,48,83)(28,105,49,84)(113,204,141,169)(114,205,142,170)(115,206,143,171)(116,207,144,172)(117,208,145,173)(118,209,146,174)(119,210,147,175)(120,197,148,176)(121,198,149,177)(122,199,150,178)(123,200,151,179)(124,201,152,180)(125,202,153,181)(126,203,154,182)(127,218,162,183)(128,219,163,184)(129,220,164,185)(130,221,165,186)(131,222,166,187)(132,223,167,188)(133,224,168,189)(134,211,155,190)(135,212,156,191)(136,213,157,192)(137,214,158,193)(138,215,159,194)(139,216,160,195)(140,217,161,196)(225,318,261,284)(226,319,262,285)(227,320,263,286)(228,321,264,287)(229,322,265,288)(230,309,266,289)(231,310,253,290)(232,311,254,291)(233,312,255,292)(234,313,256,293)(235,314,257,294)(236,315,258,281)(237,316,259,282)(238,317,260,283)(239,328,275,308)(240,329,276,295)(241,330,277,296)(242,331,278,297)(243,332,279,298)(244,333,280,299)(245,334,267,300)(246,335,268,301)(247,336,269,302)(248,323,270,303)(249,324,271,304)(250,325,272,305)(251,326,273,306)(252,327,274,307)(337,428,367,400)(338,429,368,401)(339,430,369,402)(340,431,370,403)(341,432,371,404)(342,433,372,405)(343,434,373,406)(344,421,374,393)(345,422,375,394)(346,423,376,395)(347,424,377,396)(348,425,378,397)(349,426,365,398)(350,427,366,399)(351,443,380,409)(352,444,381,410)(353,445,382,411)(354,446,383,412)(355,447,384,413)(356,448,385,414)(357,435,386,415)(358,436,387,416)(359,437,388,417)(360,438,389,418)(361,439,390,419)(362,440,391,420)(363,441,392,407)(364,442,379,408) );
G=PermutationGroup([[(1,134,15,120),(2,135,16,121),(3,136,17,122),(4,137,18,123),(5,138,19,124),(6,139,20,125),(7,140,21,126),(8,127,22,113),(9,128,23,114),(10,129,24,115),(11,130,25,116),(12,131,26,117),(13,132,27,118),(14,133,28,119),(29,155,50,148),(30,156,51,149),(31,157,52,150),(32,158,53,151),(33,159,54,152),(34,160,55,153),(35,161,56,154),(36,162,43,141),(37,163,44,142),(38,164,45,143),(39,165,46,144),(40,166,47,145),(41,167,48,146),(42,168,49,147),(57,190,78,176),(58,191,79,177),(59,192,80,178),(60,193,81,179),(61,194,82,180),(62,195,83,181),(63,196,84,182),(64,183,71,169),(65,184,72,170),(66,185,73,171),(67,186,74,172),(68,187,75,173),(69,188,76,174),(70,189,77,175),(85,218,106,204),(86,219,107,205),(87,220,108,206),(88,221,109,207),(89,222,110,208),(90,223,111,209),(91,224,112,210),(92,211,99,197),(93,212,100,198),(94,213,101,199),(95,214,102,200),(96,215,103,201),(97,216,104,202),(98,217,105,203),(225,340,239,364),(226,341,240,351),(227,342,241,352),(228,343,242,353),(229,344,243,354),(230,345,244,355),(231,346,245,356),(232,347,246,357),(233,348,247,358),(234,349,248,359),(235,350,249,360),(236,337,250,361),(237,338,251,362),(238,339,252,363),(253,376,267,385),(254,377,268,386),(255,378,269,387),(256,365,270,388),(257,366,271,389),(258,367,272,390),(259,368,273,391),(260,369,274,392),(261,370,275,379),(262,371,276,380),(263,372,277,381),(264,373,278,382),(265,374,279,383),(266,375,280,384),(281,393,305,412),(282,394,306,413),(283,395,307,414),(284,396,308,415),(285,397,295,416),(286,398,296,417),(287,399,297,418),(288,400,298,419),(289,401,299,420),(290,402,300,407),(291,403,301,408),(292,404,302,409),(293,405,303,410),(294,406,304,411),(309,429,333,440),(310,430,334,441),(311,431,335,442),(312,432,336,443),(313,433,323,444),(314,434,324,445),(315,421,325,446),(316,422,326,447),(317,423,327,448),(318,424,328,435),(319,425,329,436),(320,426,330,437),(321,427,331,438),(322,428,332,439)], [(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)], [(1,227,8,234),(2,226,9,233),(3,225,10,232),(4,238,11,231),(5,237,12,230),(6,236,13,229),(7,235,14,228),(15,241,22,248),(16,240,23,247),(17,239,24,246),(18,252,25,245),(19,251,26,244),(20,250,27,243),(21,249,28,242),(29,263,36,256),(30,262,37,255),(31,261,38,254),(32,260,39,253),(33,259,40,266),(34,258,41,265),(35,257,42,264),(43,270,50,277),(44,269,51,276),(45,268,52,275),(46,267,53,274),(47,280,54,273),(48,279,55,272),(49,278,56,271),(57,293,64,286),(58,292,65,285),(59,291,66,284),(60,290,67,283),(61,289,68,282),(62,288,69,281),(63,287,70,294),(71,296,78,303),(72,295,79,302),(73,308,80,301),(74,307,81,300),(75,306,82,299),(76,305,83,298),(77,304,84,297),(85,320,92,313),(86,319,93,312),(87,318,94,311),(88,317,95,310),(89,316,96,309),(90,315,97,322),(91,314,98,321),(99,323,106,330),(100,336,107,329),(101,335,108,328),(102,334,109,327),(103,333,110,326),(104,332,111,325),(105,331,112,324),(113,349,120,342),(114,348,121,341),(115,347,122,340),(116,346,123,339),(117,345,124,338),(118,344,125,337),(119,343,126,350),(127,359,134,352),(128,358,135,351),(129,357,136,364),(130,356,137,363),(131,355,138,362),(132,354,139,361),(133,353,140,360),(141,365,148,372),(142,378,149,371),(143,377,150,370),(144,376,151,369),(145,375,152,368),(146,374,153,367),(147,373,154,366),(155,381,162,388),(156,380,163,387),(157,379,164,386),(158,392,165,385),(159,391,166,384),(160,390,167,383),(161,389,168,382),(169,398,176,405),(170,397,177,404),(171,396,178,403),(172,395,179,402),(173,394,180,401),(174,393,181,400),(175,406,182,399),(183,417,190,410),(184,416,191,409),(185,415,192,408),(186,414,193,407),(187,413,194,420),(188,412,195,419),(189,411,196,418),(197,433,204,426),(198,432,205,425),(199,431,206,424),(200,430,207,423),(201,429,208,422),(202,428,209,421),(203,427,210,434),(211,444,218,437),(212,443,219,436),(213,442,220,435),(214,441,221,448),(215,440,222,447),(216,439,223,446),(217,438,224,445)], [(1,85,29,64),(2,86,30,65),(3,87,31,66),(4,88,32,67),(5,89,33,68),(6,90,34,69),(7,91,35,70),(8,92,36,57),(9,93,37,58),(10,94,38,59),(11,95,39,60),(12,96,40,61),(13,97,41,62),(14,98,42,63),(15,106,50,71),(16,107,51,72),(17,108,52,73),(18,109,53,74),(19,110,54,75),(20,111,55,76),(21,112,56,77),(22,99,43,78),(23,100,44,79),(24,101,45,80),(25,102,46,81),(26,103,47,82),(27,104,48,83),(28,105,49,84),(113,204,141,169),(114,205,142,170),(115,206,143,171),(116,207,144,172),(117,208,145,173),(118,209,146,174),(119,210,147,175),(120,197,148,176),(121,198,149,177),(122,199,150,178),(123,200,151,179),(124,201,152,180),(125,202,153,181),(126,203,154,182),(127,218,162,183),(128,219,163,184),(129,220,164,185),(130,221,165,186),(131,222,166,187),(132,223,167,188),(133,224,168,189),(134,211,155,190),(135,212,156,191),(136,213,157,192),(137,214,158,193),(138,215,159,194),(139,216,160,195),(140,217,161,196),(225,318,261,284),(226,319,262,285),(227,320,263,286),(228,321,264,287),(229,322,265,288),(230,309,266,289),(231,310,253,290),(232,311,254,291),(233,312,255,292),(234,313,256,293),(235,314,257,294),(236,315,258,281),(237,316,259,282),(238,317,260,283),(239,328,275,308),(240,329,276,295),(241,330,277,296),(242,331,278,297),(243,332,279,298),(244,333,280,299),(245,334,267,300),(246,335,268,301),(247,336,269,302),(248,323,270,303),(249,324,271,304),(250,325,272,305),(251,326,273,306),(252,327,274,307),(337,428,367,400),(338,429,368,401),(339,430,369,402),(340,431,370,403),(341,432,371,404),(342,433,372,405),(343,434,373,406),(344,421,374,393),(345,422,375,394),(346,423,376,395),(347,424,377,396),(348,425,378,397),(349,426,365,398),(350,427,366,399),(351,443,380,409),(352,444,381,410),(353,445,382,411),(354,446,383,412),(355,447,384,413),(356,448,385,414),(357,435,386,415),(358,436,387,416),(359,437,388,417),(360,438,389,418),(361,439,390,419),(362,440,391,420),(363,441,392,407),(364,442,379,408)]])
88 conjugacy classes
class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4
88 irreducible representations
dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + + + - - image C1 C2 C2 C2 C2 C2 C4 Q8 D4 D7 C4○D4 D14 C4×D7 D28 C4○D28 D4⋊2D7 Q8×D7 kernel C4⋊Dic7⋊7C4 C14.C42 C7×C2.C42 C2×C4×Dic7 C2×Dic7⋊C4 C2×C4⋊Dic7 C4⋊Dic7 C2×Dic7 C2×C28 C2.C42 C2×C14 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 3 1 1 1 1 8 2 2 3 8 9 12 12 12 9 3
Matrix representation of C4⋊Dic77C4 in GL5(𝔽29)
28 0 0 0 0 0 13 20 0 0 0 9 16 0 0 0 0 0 12 0 0 0 0 16 17
,
1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 23 0 0 0 0 9 24
,
1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 11 0 0 0 16 17
,
12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,13,9,0,0,0,20,16,0,0,0,0,0,12,16,0,0,0,0,17],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,23,9,0,0,0,0,24],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,16,0,0,0,11,17],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12] >;
C4⋊Dic77C4 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_7\rtimes_7C_4
% in TeX
G:=Group("C4:Dic7:7C4");
// GroupNames label
G:=SmallGroup(448,187);
// by ID
G=gap.SmallGroup(448,187);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^14=d^4=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^7,c*b*c^-1=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations
×
𝔽
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2021-07-28 13:57:02
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https://www.physicsoverflow.org/27661/summary-spinor-bundle-associated-string-worldsheet-correct
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# Is my Summary of a Spinor Bundle Associated with a String Worldsheet Correct?
+ 1 like - 0 dislike
91 views
I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really appreciate it if someone could tell me if the following is true:
Take the worldsheet to be some 2d pseudo-Riemannian orientable manifold $M$.
One can associate with each point $x \in M$ a 2d tangent space $TM_{x}$. The disjoint union of $TM_{x}$ at all $x$ defines the total space $TM$ of a tangent bundle ($TM$, $\pi_{TM}$, $M$) whos projection is given by:
$$\pi_{TM}: TM \rightarrow M$$
The worldsheet $M$ is the base space of the tangent bundle and each $TM_{x}$ is a fibre.
Since the tangent space is 2d, the bases that exist in each $TM_{x}$ are 2d also. Since the base space is pseudo-Riemannian, so is the tangent space and the ordered bases (frames) that exist on each $TM_{x}$ are 'pseudo-orthonormal'. This would mean that the bases transform under an $O(1,1)$ group. However, Since the base space $M$ is orientable, so is each $TM_{x}$ and that means that the frames are oriented pseudo-orthonormal and transform under $SO(1,1)$ instead.
This allows the oriented orthonormal frame bundle (a specific sub-class of principal bundle) to be written as $(F_{SO(1,1)}(M), \pi_{F}, M, SO(1,1))$, where the projection acts as:
$$\pi_{F}: F_{SO(1,1)}(M) \rightarrow M$$
The fibre $F_{x}$ of this frame bundle at a point $x$ on $M$ is the set of all frames of $TM_{x}$ at the same point $x$. $F_{x}$ is homeomorphic to the gauge group $SO(1,1)$ and is said to be an $SO(1,1)$-torsor.
However, now one can define a lift of the group $SO(1,1)$ to $Spin(1,1)$. The corresponding frame bundle is now $(P, \pi_{P}, M, SO(1,1))$ with projection:
$$\pi_{P}: P \rightarrow M$$
The fibre $P_{x}$ of this frame bundle at a point $x$ on $M$ is the set of all frames of $TM_{x}$ at the same point $x$. $P_{x}$ is homeomorphic to the gauge group $Spin(1,1)$ and is said to be an $Spin(1,1)$-torsor.
How can the set of all frames in $TM_{x}$ be homeomorphic to both $SO(1,1)$ and $Spin(1,1)$?
The spinor bundle can then be defined to be given by $(S, \pi_{S}, M, \Delta_{(1,1)} Spin(1,1))$, with projection that acts as:
$$\kappa: S \rightarrow M$$
Here $S$ is given by:
$$S = P \times_{\kappa} \Delta_{(1,1)} = (P \times \Delta_{(1,1)})/Spin(1,1)$$
The fibre is given by $\Delta_{(1,1)}$ which is the Hilbert space of all spinor states. Each section of this bundle then corresponds to a particular Majorana-Weyl spinor field configuration on the worldsheet.
This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Siraj R Khan
+ 1 like - 0 dislike
It is nearly correct. First, $F_x$ is the space of oriented and orthonormal frames of $T_x M$. Furthermore, you gave the correct definition of the $Spin(1,1)$ bundle but confused a little bit its meaning. The $Spin(1,1,)$ bundle is defined as the lift of the $SO(1,1)$ bundle to $Spin(1,1)$. This means we have a principal $Spin(1,1)$-bundle $P$ and a principal bundle map $\phi: P \to F$ which is at the same time a covering (this literally lifts the double covering of $SO(1,1)$ by $Spin(1,1)$ to the bundle picture). For more details see http://en.wikipedia.org/wiki/Spin_structure#Spin_structures_on_Riemannian_manifolds. Now $P$ is not a frame bundle. More precisely, for every oriented, orthonormal frame of $T_xM$ there exists two points $p$ and $p'$ in the fiber $P_x$ which yield this frame by $\phi(p) = \phi(p')$. Thus $P_x$ is not homeomorphic to the frames on $T_xM$.
This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Tobias Diez
answered Mar 4, 2015 by (90 points)
Thanks for the reply Tobias. Wouldn't $F_{x}$ be the space of oriented and pseudo-orthonormal frames of $T_{x}M$, just to be a little more precise? (Since $T_{x}M$ has Lorentzian metric signature). And I understand the $Spin(1,1)$-bundle now, thank you.
This post imported from StackExchange Physics at 2015-03-04 12:46 (UTC), posted by SE-user Siraj R Khan
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2019-04-21 16:18:33
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https://scicomp.stackexchange.com/tags/python/hot
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# Tag Info
52
Python (as of 2.6 and 3.0) now searches in the ~/.local directory for local installs, which do not require administrative privileges to install, so you just need to point your installer to that directory. If you have already downloaded the package foo and would like to install it manually, type: cd path/to/foo python setup.py install --user If you are ...
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I'm going to break up my answer into three parts. Profiling, speeding up the python code via c, and speeding up python via python. It is my view that Python has some of the best tools for looking at what your code's performance is then drilling down to the actual bottle necks. Speeding up code without profiling is about like trying to kill a deer with an ...
32
First, if your undergraduates are like ours and had no prior introduction to computers, expect to spend some time teaching them how to use basic stuff like using a proper editor (i.e., not MS Word), the command line, etc. I think the answer somewhat depends on where you set the focus of your course (or what you are required to teach). For example: How ...
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This is indeed called catastrophic cancellation. In fact, this particular case is very easy: rewrite the function using the equivalent, numerically stable expression $$\frac{t}{1+\sqrt{1-t^2}}.$$ Since you probably need a reference, this is discussed in most numerical methods textbooks in relation to the formula for solving quadratic equations (that ...
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Ease of learning Python and Fortran are both relatively easy-to-learn languages. It's probably easier to find good Python learning materials than good Fortran learning materials because Python is used more widely, and Fortran is currently considered a "specialty" language for numerical computing. I believe the transition from Python to Fortran would be ...
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You have to subclass the rv_continuous class in scipy.stats import scipy.stats as st class my_pdf(st.rv_continuous): def _pdf(self,x): return 3*x**2 # Normalized over its range, in this case [0,1] my_cv = my_pdf(a=0, b=1, name='my_pdf') now my_cv is a continuous random variable with the given PDF and range [0,1] Note that in this example ...
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I am not super familiar with f2py internals, but I am very familiar with wrapping Fortran. F2py just automates some or all of the things below. You first need to export to C using the iso_c_binding module, as described for example here: http://fortran90.org/src/best-practices.html#interfacing-with-c Disclaimer: I am the main author of the fortran90.org ...
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In 2014, I would've said Python. In 2017, I wholeheartedly believe that the language to teach undergraduates is Julia. Teaching is always about a tradeoff. On one hand, you want to choose something that is simple enough that it is easy to grasp. But secondly, you want to teach something that has staying power, i.e. something that can grow with you. The ...
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Joblib does what you want. The basic usage pattern is: from joblib import Parallel, delayed def myfun(arg): do_stuff return result results = Parallel(n_jobs=-1, verbose=verbosity_level, backend="threading")( map(delayed(myfun), arg_instances)) where arg_instances is list of values for which myfun is computed in parallel. The main ...
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Going from MATLAB to Python does introduce quite a bit of syntax overhead. One way to quantify it is the nice QuantEcon cheatsheet which showcases how there's a lot of extra "stuff" going on when trying to write simple linear algebra commands in Python. The verbose NumPy syntax is really just a symptom of how it was not developed as a technical computing ...
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You can use the Python builtin ctypes module as described on fortran90.org. It is pretty straight forward and doesn't require any external dependencies. Also, the ndpointer arg type helper is very handy.
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I will try to answer your question considering that you are asking for Python specifically. I will describe my own method of tackling a simulation problem. Strategies for faster simulations are given in this description. First, I prototype new simulations in Python. Of course, I try to make use of NumPy and SciPy as much as I can. Whereas NumPy provides a ...
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Let me try and break down your requirements: Maintainability Reading/writing text data Strong interfaces/capability for LU factorizations Sparse linear solvers Performance and scalability to large data From this list, I would consider the following languages: C, C++, Fortran, Python, MATLAB, Java Julia is a promising new language, but the community is ...
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There are two issues that you are likely to be encountering. Ill-conditioning First, the problem is ill-conditioned, but if you only provide a residual, Newton-Krylov is throwing away half your significant digits by finite differencing the residual to get the action of the Jacobian: $$J[x] y \approx \frac{F(x+\epsilon y) - F(x)}{\epsilon}$$ If you ...
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Here is the Numba solution. On my machine the Numba version is >1000x faster than the python version without the decorator (for a 200x200 matrix, 'k' and 200-length vector 'a'). You can also use the @autojit decorator which adds about 10 microseconds per call so that the same code will work with multiple types. from numba import jit, autojit @jit('f8[:]...
15
You should check out sympy.stats. It provides an interface to deal with random variables. The following example provides a random variable X defined on the unit interval with density 2x In [1]: from sympy.stats import * In [2]: x = Symbol('x') In [3]: X = ContinuousRV(x, 2*x, Interval(0, 1)) In [4]: P(X>.5) Out[4]: 0.750000000000000 In [5]: Var(X) # ...
15
There are a number of issues in your question. Do not use Gaussian Elimination (LU factorization) to calculate the numerical rank of a matrix. LU factorization is unreliable for this purpose in floating-point arithmetic. Instead, use a rank-revealing QR decomposition (such as xGEQPX or xGEPQY in LAPACK, where x is C, D, S, or Z, though those routines are ...
15
A difficulty with any of these types of questions is that the answer is highly community-dependent. To answer some of your questions in haphazard order: MATLAB is used a lot both in academia and in industry. One of the reasons it's used quite a bit in industry is because it is taught in academia. I know for a fact that MATLAB is used at Lincoln Laboratory ...
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The question has two very different subquestions. I will address the first one only. Matlab's version runs on average 24 times faster than my python equivalent! The second one is subjective. I would say that letting know the user that there is some problem with the integral is a good thing and this SciPy behavior outperforms the Matlab`s one to keep it ...
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For the first part of my question, I found this very useful comparison for performance of different linear interpolation methods using python libraries: http://nbviewer.ipython.org/github/pierre-haessig/stodynprog/blob/master/stodynprog/linear_interp_benchmark.ipynb Below is list of methods collected so far. Standart interpolation, structured grid: http:/...
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Here is R1, as computed in MATLAB: 1.0e+07 * -7.382605957465515 -9.599867106092937 -2.830412177259742 -0.000000000002830 -0.000000000002830 -1.230434326244253 -1.599977851015490 -0.471735362876624 -0.000000000000472 -0.000000000000472 3.691302978732758 4.799933553046468 1.415206088629871 0.000000000001415 0.000000000001415 -5....
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First, see Mark L. Stone's answers, which is completely correct. Second, realize that this is the reason why people told you to use relative errors in your numerical analysis class. :) Third, the real question here is why the results do not coincide exactly, since both languages call some BLAS library functions for their computations. There are several very ...
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Algorithms for rank-1 updates of the SVD (also called incremental SVD) do exist, but I haven't been able to find a LAPACK-like implementation anywhere. The one I've seen mentioned repeatedly is that of Brand (2003). Judging from this website, it seems as though Brand's algorithm is relatively simple to implement using existing LAPACK and BLAS routines as ...
13
I will address only the comparison of C to C++. While it is true that anything written in C can be ported to C++ with a few syntactic touch-ups, the communities have different values. The C library community, more than almost any other, values binary stability. Binary stability is critical for low-level libraries to avoid inflicting constant pain on those ...
13
What you're looking for is Numba, which can auto parallelize a for loop. From their documentation from numba import jit, prange @jit def parallel_sum(A): sum = 0.0 for i in prange(A.shape[0]): sum += A[i] return sum
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To the best of my knowledge, Numpy does not support independent streams. Indeed, getting independent streams from the Mersenne Twister (Pythons RNG) is notoriously difficult although it can be done. Consider using the RandomGen package. It is fully compatible with Numpy, and provides you with the PCG64 generator, supporting up to $2^{63}$ independent ...
13
The problem you are encountering is likely not a consequence of your choice of algorithm, but in fact a consequence of the resulting dynamical system after applying time reversal. Per the definition of an attractor, all points in some neighborhood of the attractor will converge to the attractor under the flow of the dynamical system as $t\to\infty$. However, ...
13
There are libraries that you can use in Python that will give you all (or at least nearly all) of the functionality of MATLAB. For example, scipy.integrate.solve_ivp() supports a number of methods for ODE integration that are comparable to what you can get with the various odexxx() functions in MATLAB. So no, you wouldn't have to write your own ODE ...
12
For logging that allows full reproducibility, I highly recommend the Sumatra python package. It nicely links the version control commit number, machine state, and output files to each program run and has a django web interface to interact with the database of run info. The python API makes it very easy to include logging in my scripts.
12
The degeneracy of some eigenvalues looks to me like the hallmark of the breakdown of the Lanczos algorithm. The Lanczos algorithm is one of the more commonly used methods to approximate the eigenvalues and eigenvectors of Hermitian matrices; it's what scipy.eigsh() uses, through a call to the ARPACK library. In exact arithmetic, the Lanczos algorithm ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2020-03-31 03:00:35
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https://www.oreilly.com/library/view/acoustics-sound-fields/9780128152287/XHTML/B9780128152270150022/B9780128152270150022.xhtml
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## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.
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Appendix II
# Mathematical formulas [1,2]
Abstract
This chapter presents the functions and formulas used in this text.
In the following formulas, m and n are integers, and μ and ν can have any value.
### Binomial theorem
$MathML$
(A2.1)
$MathML$
(A2.2)
$MathML$
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2019-11-14 21:01:35
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https://electronics.stackexchange.com/questions/279089/current-mirror-design-gate-to-drain-feedback
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# Current mirror design - gate to drain feedback
In my texbook [Analog VLSI: Circuits and Principles] there are two circuits that function as a current mirror. The first uses two nfets with a shared gate, and the drain of the first nfet (M1) is tied to the common gate. The current through M1 is then mirrored through M2.
The book also shows a Winner Take All circuit that uses a current mirror where the gate of M2 is tied to the drain of M1. And the gate of M1 is tied to the source of M2.
The book states "We should remember that Vg is determined by the gate charge, which cannot be directly influenced by the input current due to the infinite impedance between channel and gate." For the second current mirror design, how is the current getting mirrored to M2 if the gate can not be directly set by the input current in M1?
simulate this circuit – Schematic created using CircuitLab
This is most succinctly explained in Carver Mead's 1989 book on neuromorphic engineering, Analog VLSI and neural systems. In the two circuits above, the top one is a current mirror and the bottom one a current conveyor. The both assume saturation of the devices. In the current conveyor, the assumption is that the voltage across M1 is larger than $4U_t$ for saturation. Firstly, you also need to "untie" the bulk from the source of M2 and M1 and not assume that the source is tied to GND. You then can setup the two current equations for the devices.
• ps: I'd give you more but this is very much like homework that I've assigned in the past, so I only wanted to get you started. – b degnan Jan 8 '17 at 18:09
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2020-06-06 11:07:33
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https://zbmath.org/?q=an:1297.11028
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# zbMATH — the first resource for mathematics
Congruences between Hilbert modular forms: constructing ordinary lifts. (English) Zbl 1297.11028
Summary: Under mild hypotheses, we prove that if $$F$$ is a totally real field, and $$\overline{\rho}:G_{F}\to\mathrm{GL}_2(\overline{\mathbb F}_l)$$ is irreducible and modular, then there is a finite solvable totally real extension $$F'/F$$ such that $$\overline{\rho}|_{G_{F'}}$$ has a modular lift which is ordinary at each place dividing $$l$$. We deduce a similar result for $$\overline{\rho}$$ itself, under the assumption that at places $$v|l$$ the representation $$\overline {\rho}|_{G_{F_v}}$$ is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti-Tate representations and the Buzzard-Diamond-Jarvis conjecture. The proof makes use of a novel lifting technique, going via rank 4 unitary groups.
##### MSC:
11F33 Congruences for modular and $$p$$-adic modular forms 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
Full Text:
##### References:
[1] J. Arthur and L. Clozel, Simple algebras, base change, and the advanced theory of the trace formula , Ann. of Math. Stud. 120 , Princeton Univ. Press, Princeton, 1989. · Zbl 0682.10022 [2] T. Barnet-Lamb, T. Gee, and D. Geraghty, The Sato-Tate conjecture for Hilbert modular forms , J. Amer. Math. Soc. 24 (2011), 411-469. · Zbl 1269.11045 [3] T. Barnet-Lamb, Serre weights for rank two unitary groups , preprint, [math.NT] 1106.5586v1 [4] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight , preprint, [math.NT] 1010.2561v1 · Zbl 1310.11060 [5] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy, II , Publ. Res. Inst. Math. Sci. 47 (2011), 29-98. · Zbl 1264.11044 [6] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, ten years on , London Math. Soc. Lecture Note Ser. 249 , Cambridge Univ. Press, Cambridge, 1998. · Zbl 0568.20001 [7] L. Clozel, Changement de base pour les représentations tempérées des groupes réductifs réels , Ann. Sci. École Norm. Sup. (4) 15 (1982), 45-115. · Zbl 0516.22010 [8] L. Clozel, M. Harris, and R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations , Inst. Haute Études Sci. Pub. Math. 108 (2008), 1-181. · Zbl 1169.11020 [9] H. Darmon, F. Diamond, and R. Taylor, “Fermat’s last theorem” in Elliptic Curves, Modular Forms and Fermat’s Last Theorem (Hong Kong, 1993) , Int. Press, Cambridge, Mass., 1997, 2-140. [10] T. Gee, A modularity lifting theorem for weight two Hilbert modular forms , Math. Res. Lett. 13 (2006), 805-811; erratum , Math. Res. Lett. 16 (2009), 57-58. · Zbl 1185.11030 [11] T. Gee, Automorphic lifts of prescribed types , Math. Ann. 350 (2011), 107-144. · Zbl 1276.11085 [12] T. Gee, On the weights of mod p Hilbert modular forms , Invent. Math. 184 (2011), 1-46. · Zbl 1280.11029 [13] T. Gee and D. Geraghty, Companion forms for unitary and symplectic groups , Duke Math. J. 161 (2012), 247-303. · Zbl 1295.11043 [14] D. Geraghty, Modularity lifting theorems for ordinary Galois representations , preprint, 2009. [15] R. Guralnick, F. Herzig, R. Taylor, and J. Thorne, Adequate subgroups , preprint, [math.NT] 1107.5993v1 [16] M. Harris, “Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications” in Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin, Vol. II , Progr. Math. 270 , Birkhäuser, Boston, 2009, 1-21. · Zbl 1234.11068 [17] M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy , Ann. of Math. (2) 171 (2010), 779-813. · Zbl 1263.11061 [18] M. l. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties , with an appendix by Vladimir G. Berkovich, Ann. Math. Stud. 151 , Princeton Univ. Press, Princeton, 2001. · Zbl 1036.11027 [19] P. N. Hoffman and J. F. Humphreys, Projective Representations of the Symmetric Groups: Q-Functions and Shifted Tableaux , Oxford Univ. Press, New York, 1992. · Zbl 0777.20005 [20] C. Khare and J.-P. Wintenberger, On Serre’s conjecture for 2- dimensional mod p representations of the absolute Galois group of the rationals , Ann. of Math. (2) 169 (2009), 229-253. · Zbl 1196.11076 [21] M. Kisin, “Modularity for some geometric Galois representations” in L - Functions and Galois Representations , with an appendix by Ofer Gabber, London Math. Soc. Lecture Note Ser. 320 , Cambridge Univ. Press, Cambridge, 2007, 438-470. · Zbl 1171.11035 [22] M. Kisin, “Modularity of 2-dimensional Galois representations” in Current Developments in Mathematics, 2005 , Int. Press, Somerville, Mass., 2007, 191-230. · Zbl 1218.11056 [23] M. Kisin, Potentially semi-stable deformation rings , J. Amer. Math. Soc. 21 (2008), 513-546. · Zbl 1205.11060 [24] M. Kisin, The Fontaine-Mazur conjecture for GL 2 , J. Amer. Math. Soc. 22 (2009), 641-690. · Zbl 1251.11045 [25] M. Kisin, Moduli of finite flat group schemes, and modularity , Ann. of Math. (2) 170 (2009), 1085-1180. · Zbl 1201.14034 [26] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur , Ann. of Math. (2) 156 (2002), 115-154. · Zbl 1076.11035 [27] J.-P. Serre, Local Fields , translated from the French by M. J. Greenberg, Grad. Texts in Math. 67 , Springer, New York, 1979. · Zbl 0423.12016 [28] A. Snowden and A. Wiles, Bigness in compatible systems , preprint, [math.NT] 0908.1991v3 · Zbl 1407.11073 [29] R. Taylor, On the meromorphic continuation of degree two L-functions , Doc. Math. 2006 , Extra Vol., 729-779. · Zbl 1138.11051 [30] J. Thorne, On the automorphy of l-adic Galois representations with small residual image , to appear J. Inst. Math. Jussieu, preprint, [math.NT] 1107.5989v1 [31] P.-J. White, m-bigness in compatible families , C. R. Math. Acad. Sci. Paris 348 (2010), 1049-1054. · Zbl 1229.11086 [32] R. A. Wilson, The finite simple groups: An introduction , Grad. Texts Math. 251 , Springer, London, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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https://hkxb.buaa.edu.cn/CN/html/20200817.html
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文章快速检索 高级检索
1. 厦门大学 航空航天学院, 厦门 361005;
2. 山东科技大学 电气与自动化工程学院, 青岛 266510
Fusing method for dynamic derivatives and added mass of airships
LIN Xianwu1, WANG Shichao1, LI Zhibin2, LAN Weiyao1
1. School of Aerospace Engineering, Xiamen University, Xiamen 361005, China;
2. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266510, China
Abstract: Dynamic stability derivatives and added mass represent an airship's aerodynamics in viscous flow and acyclic potential flow. To develop a method for fusing these two aerodynamic coefficients in airship modeling, the classification method for dynamic stability derivatives and added mass and the fusing method for same aerodynamic coefficient ingredient are studied. By introducing a unified theory for analyzing aerodynamics in incompressible flow, a conclusion is drawn that the viscous aerodynamics should be reserved and the corresponding results in acyclic potential flow should be abandoned in fusing the same ingredient of aerodynamics. By studying the relationship between aerodynamics and the present motion parameter of an airship, the classification method for aerodynamics and the aerodynamic coefficient is built. To keep the aerodynamic classification method same in the two flow fields, a new method for reconstructing the present motion parameters of airships is proposed so that the aerodynamic coefficients in the two flow fields can be both classified according to the new parameters, and each ingredient can be evaluated. Based on these studies, a new fusing method for dynamic stability derivatives and added mass is proposed and its difference with the present fusing method is discussed. A numerical example is presented to show the effect of different fusing methods on the dynamic characteristics of an airship's longitudinal perturbation motion and illustrate the necessity for adopting this new fusing method.
Keywords: airship dynamic derivatives added mass fusing vorticity dynamics linearization dynamic characteristics
1 不可压缩流中通用的气动力和力矩表达式
Lamb[1]所介绍无旋无环流场中对应的气动力和力矩方法是基于能量守恒的原理推导得到的；因为有黏流场中存在黏性损耗，因此这种方法难以被推广到有黏流中去。运动体在有黏流场中运动时所对应的气动力和力矩可以用Wu提出的涡量矩定理[24-26]来计算和分析，这种方法是基于动量守恒方法得到的。然而在流场黏性减少为零的情况下，涡量矩定理并不能收敛于Lamb的研究结果[27]。这种不兼容性的原因之一在于涡量矩定理的推导过程中将运动体和流场视为一体。在无旋无环流的情况下，滑移边界条件将导致速度等变量分布在流体和固体所占领的空间区域上不连续，从而导致涡量矩定理的推导过程和结果不成立。涡量矩定理不能兼容无旋无环流结果的另一个原因是目前许多涡动力学理论[28-31]中的气动力和力矩表达式并没有考虑力矩参考点可动的情况；然而在飞行力学中，运动体所受气动力矩的参考点一般是固联在运动体上的，因而是运动的。考虑到这些问题和不足，重新推导运动体的气动力和力矩表达式可得到能同时兼容涡量矩定理和无旋无环流结果的气动力FA和力矩MA的公式为
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{\rm{A}}} = - \frac{{{\rm{d}}{\mathit{\boldsymbol{Q}}_{f,\infty }}}}{{{\rm{d}}t}}}\\ {{\mathit{\boldsymbol{M}}_{\rm{A}}} = - \frac{{{\rm{d}}{\mathit{\boldsymbol{K}}_{f,\infty }}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{Q}}_{f,\infty }}} \end{array}} \right.$ （1）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{{N - 1}}\int_{{R_{f,\infty }}} \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{\omega }}{\rm{d}}R + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\rho }{{N - 1}}\int_{{S_{\rm{i}}}} \mathit{\boldsymbol{r}} \times (\mathit{\boldsymbol{v}} \times \mathit{\boldsymbol{n}}){\rm{d}}S}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{v}}{\rm{d}}R = - \frac{\rho }{2}\int_{{R_{f,\infty }}} {{\mathit{\boldsymbol{r}}^2}} \mathit{\boldsymbol{\omega }}{\rm{d}}R + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\rho }{2}\int_{{S_{\rm{i}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （2）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{{N - 1}}\int_{{S_{\rm{i}}}} \mathit{\boldsymbol{r}} \times (\mathit{\boldsymbol{v}} \times \mathit{\boldsymbol{n}}){\rm{d}}S}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{2}\int_{{S_{\rm{i}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （3）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \frac{\rho }{{N - 1}}\int_{{R^\prime }_{f,\infty }} \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{\omega }}{\rm{d}}R + \frac{\rho }{{N - 1}}\int_{{S_{\rm{b}}}} \mathit{\boldsymbol{r}} \times }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (v \times n)dS}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = - \frac{\rho }{2}\int_{{R^\prime }_{f,\infty }} {{\mathit{\boldsymbol{r}}^2}} \mathit{\boldsymbol{\omega }}{\rm{d}}R + \frac{\rho }{2}\int_{{S_{\rm{b}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （4）
2 气动系数计算与融合
2.1 无旋无环流中的气动系数计算
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{{M_\omega }}} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array}} \right.$ （5）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - \frac{{{\rm{d}}({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})}}{{{\rm{d}}t}}}\\ {{\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - \frac{{{\rm{d}}({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})}}{{{\rm{d}}t}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array}} \right.$ （6）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\lambda }}_{Fv}} = \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}}&0&0\\ 0&{{\lambda _{22}}}&0\\ 0&0&{{\lambda _{33}}} \end{array}} \right],{\mathit{\boldsymbol{\lambda }}_{F\omega }} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{{\lambda _{26}}}\\ 0&{{\lambda _{35}}}&0 \end{array}} \right]}\\ {{\mathit{\boldsymbol{\lambda }}_{Mv}} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{{\lambda _{26}}}\\ 0&{{\lambda _{35}}}&0 \end{array}} \right],{\mathit{\boldsymbol{\lambda }}_{{M_\omega }}} = \left[ {\begin{array}{*{20}{c}} {{\lambda _{44}}}&0&0\\ 0&{{\lambda _{55}}}&0\\ 0&0&{{\lambda _{66}}} \end{array}} \right]} \end{array}} \right.$ （7）
${\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - \left[ \begin{array}{l} {\lambda _{11}}\dot u + {\lambda _{35}}{q^2} - {\lambda _{26}}{r^2} - {\lambda _{22}}vr + {\lambda _{33}}wq\\ {\lambda _{26}}\dot r + {\lambda _{22}}\dot v + {\lambda _{11}}ur - {\lambda _{33}}w{\rm{p }} - {\lambda _{35}}pq\\ {\lambda _{35}}\dot q + {\lambda _{33}}\dot w - {\lambda _{11}}uq + {\lambda _{22}}vp + {\lambda _{26}}pr \end{array} \right]$ （8）
${\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - \left[ \begin{array}{l} {\lambda _{44}}\dot p - ({\lambda _{22}} + {\lambda _{33}})vw + ({\lambda _{35}} + {\lambda _{26}})vq - ({\lambda _{26}} + {\lambda _{35}})wr{\rm{ }} - ({\lambda _{55}} - {\lambda _{66}})qr\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _{55}}\dot q + {\lambda _{35}}\dot w + ({\lambda _{11}} - {\lambda _{33}})uw - {\lambda _{26}}vp - {\lambda _{35}}uq + ({\lambda _{44}} - {\lambda _{66}})pr\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _{66}}\dot r + {\lambda _{26}}\dot v + ({\lambda _{22}} - {\lambda _{11}})vu + {\lambda _{26}}ur + {\lambda _{35}}wp + ({\lambda _{55}} - {\lambda _{44}})qp \end{array} \right]$ （9）
2.2 有黏流中的气动系数计算
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_v} = \bar QS({C_x}{\mathit{\boldsymbol{e}}_1} + {C_y}{\mathit{\boldsymbol{e}}_2} + {C_z}{\mathit{\boldsymbol{e}}_3})}\\ {{\mathit{\boldsymbol{M}}_v} = \bar QSL({m_x}{\mathit{\boldsymbol{e}}_1} + {m_y}{\mathit{\boldsymbol{e}}_2} + {m_z}{\mathit{\boldsymbol{e}}_3})} \end{array}} \right.$ （10）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{qr,v}} = \bar QS({C_{yr}}\mathit{\boldsymbol{j}} + {C_{zq}}\mathit{\boldsymbol{k}})}\\ {{\mathit{\boldsymbol{M}}_{qr,v}} = \bar QU({m_{yq}}\mathit{\boldsymbol{j}} + {m_{zr}}\mathit{\boldsymbol{k}})} \end{array}} \right.$ （11）
$\left\{ {\begin{array}{*{20}{l}} {{C_{yr}} = C_y^rr,{C_{zq}} = C_z^qq}\\ {{m_{yq}} = m_y^qq,{m_{zr}} = m_z^rr} \end{array}} \right.$ （12）
$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{{a_y}{a_z},v}} = \bar QS({C_{y{a_y}}}\mathit{\boldsymbol{j}} + {C_{z{a_z}}}\mathit{\boldsymbol{k}})}\\ {{\mathit{\boldsymbol{M}}_{{a_y}{a_z},v}} = \bar QU({m_{y{a_z}}}\mathit{\boldsymbol{j}} + {m_{z{a_y}}}\mathit{\boldsymbol{k}})} \end{array}} \right.$ （13）
$\left\{ \begin{array}{l} {C_{y{a_y}}} = C_y^{{a_y}}{a_y} = C_y^{{a_y}}(\dot v + ru - pw)\\ {C_{z{a_z}}} = C_z^{{a_z}}{a_z} = C_z^{{a_z}}(\dot w + pv - qu)\\ {m_{y{a_z}}} = m_y^{{a_z}}{a_z} = m_y^{{a_z}}(\dot w + pv - qu)\\ {m_{z{a_y}}} = m_z^{{a_y}}{a_y} = m_z^{{a_y}}(\dot v + ru - pw) \end{array} \right.$ （14）
2.3 动导数与附加质量的融合
$\left\{ {\begin{array}{*{20}{l}} {{v_{\rm{D}}} = \sqrt {{u^2} + {v^2} + {w^2}} }\\ {\alpha = - {\rm{arctan}}\frac{v}{u}}\\ {\beta = {\rm{arcsin}}\frac{w}{{{v_{\rm{D}}}}}} \end{array}} \right.$ （15）
$\left\{ {\begin{array}{*{20}{l}} {\dot \alpha = \frac{{v\dot u - \dot vu}}{{{u^2} + {v^2}}} \approx - \dot v/{v_{\rm{D}}}}\\ {\dot \beta = \frac{{\dot w{v_{\rm{D}}} - w{{\dot v}_{\rm{D}}}}}{{{v_{\rm{D}}}\sqrt {{w^2} + v_{\rm{D}}^2} }} \approx \dot w/{v_{\rm{D}}}} \end{array}} \right.$ （16）
$\left\{ {\begin{array}{*{20}{l}} {\dot \alpha = {{\dot \alpha }_1} + {{\dot \alpha }_2}}\\ {\dot \beta = {{\dot \beta }_1} + {{\dot \beta }_2}} \end{array}} \right.$ （17）
$\begin{array}{l} {\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{Q}}_{f,\infty }} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} + {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})} \end{array} \end{array}$ （18）
$\begin{array}{l} {\mathit{\boldsymbol{F}}_{qr,p}} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot [(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times {\mathit{\boldsymbol{v}}_{\rm{D}}}] - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times ({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times [{\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})]} \end{array} \end{array}$ （19）
$- {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}})$ （20）
${\mathit{\boldsymbol{F}}_{qr1,p}} = {\mathit{\boldsymbol{F}}_{qr,p}} - {\mathit{\boldsymbol{F}}_{qr,p}} \cdot \mathit{\boldsymbol{i}}$ （21）
$\begin{array}{l} {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} = - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{\rm{d}}(v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})}}{{{\rm{d}}t}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot (\dot v\mathit{\boldsymbol{j}} + \dot v\mathit{\boldsymbol{k}}) - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot [{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times (\dot v\mathit{\boldsymbol{j}} + \dot w\mathit{\boldsymbol{k}})] \end{array}$ （22）
${\mathit{\boldsymbol{F}}_{{a_y}{a_z}1,p}} = {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} - {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} \cdot \mathit{\boldsymbol{i}}$ （23）
$\begin{array}{l} {\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}}) - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}) - }\\ {{\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array} \end{array}$ （24）
$\begin{array}{l} {\mathit{\boldsymbol{M}}_{qr,p}} = {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot [(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times {\mathit{\boldsymbol{v}}_{\rm{D}}}] - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times ({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times [{\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})] - }\\ {{\mathit{\boldsymbol{v}}_0} \times [{\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})]} \end{array} \end{array}$ （25）
${\mathit{\boldsymbol{M}}_{qr1,p}} = {\mathit{\boldsymbol{M}}_{qr,p}} - {\mathit{\boldsymbol{M}}_{qr,p}} \cdot \mathit{\boldsymbol{i}}$ （26）
$\begin{array}{l} {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} = - {\mathit{\boldsymbol{\lambda }}_{Mv}}\frac{{{\rm{d}}(v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})}}{{{\rm{d}}t}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot (\dot v\mathit{\boldsymbol{j}} + \dot w\mathit{\boldsymbol{k}}) - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot [{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times (v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})] \end{array}$ （27）
${\mathit{\boldsymbol{M}}_{{y^a}{z^1},p}} = {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} - {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} \cdot \mathit{\boldsymbol{i}}$ （28）
3 算例分析
图 1 两种气动系数融合方法对比 Fig. 1 Comparison of two fusing methods of aerodynamic coefficients
${\mathit{\boldsymbol{M}}_{{\rm{ln}}}}{\mathit{\boldsymbol{\dot x}}_{{\rm{ln}}}} = {\mathit{\boldsymbol{a}}_{{\rm{ln}}}}{\mathit{\boldsymbol{\dot x}}_{{\rm{ln}}}} + {\mathit{\boldsymbol{b}}_{{\rm{ln}}}}{\mathit{\boldsymbol{u}}_{{\rm{ln}}}}$ （29）
$\begin{array}{l} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_{{\rm{ln}}}} = [\Delta {v_{\rm{D}}},\Delta \alpha ,r,\Delta \vartheta ]}\\ {{\mathit{\boldsymbol{M}}_{{\rm{ln}}}} = } \end{array}\\ \left[ {\begin{array}{*{20}{c}} {m + {\lambda _{11}}}&0&{m{c_y}}&0\\ 0&{ - m{v_{\rm{D}}} - \bar QSC_y^{\dot \alpha }}&{ - m{c_x} - \bar QSC_y^{\dot r}}&0\\ {m{c_y}}&{m{c_x}{v_{\rm{D}}} - \bar QUm_z^{\dot \alpha }}&{ - \bar QUm_z^{\dot r} + {J_z}}&0\\ 0&0&0&{m{v_D}} \end{array}} \right] \end{array}$ （30）
$\begin{array}{l} {\mathit{\boldsymbol{a}}_{{\rm{ln}}}} = \\ \left[ {\begin{array}{*{20}{c}} { - (\rho {v_{\rm{D}}}{C_{x0}} + 3\bar QC_x^{{v_{\rm{D}}}})S + 2{T^v}{\rm{cos}}{\mu _{\rm{e}}}}&0&{m{v_{\rm{D}}}{\alpha _{\rm{e}}}}&{ - (mg - \rho U){\rm{cos}}{\vartheta _{\rm{e}}}}\\ {2{T^v}{\rm{sin}}{\mu _{\rm{e}}}}&{\bar QSC_y^\alpha }&{\bar QSC_y^r + m{v_{\rm{D}}}}&{(mg - \rho U){\rm{sin}}{\vartheta _{\rm{e}}}}\\ {2{T^v}{d_x}{\rm{sin}}{\mu _{\rm{e}}} - 2{T^v}{d_y}{\rm{cos}}{\mu _{\rm{e}}}}&{\bar QUm_z^\alpha }&{\bar QUm_z^r + m{v_{\rm{D}}}{c_x} - m{v_{\rm{D}}}{c_y}{\alpha _{\rm{e}}}}&{\begin{array}{*{20}{l}} {(mg{c_y} - \rho U{b_y}){\rm{cos}}{\vartheta _{\rm{e}}} - }\\ {( - mg{c_x} + \rho U{b_x}){\rm{sin}}{\vartheta _{\rm{e}}}} \end{array}}\\ 0&0&{m{v_{\rm{D}}}}&0 \end{array}} \right] \end{array}$ （31）
$\left\{ \begin{array}{l} {\mathit{\boldsymbol{u}}_{{\rm{ln}}}} = {[\Delta {\delta _z},\Delta p]^{\rm{T}}}\\ {\mathit{\boldsymbol{b}}_{{\rm{ln}}}} = \left[ {\begin{array}{*{20}{c}} 0&{2{T^p}{\rm{cos}}{\mu _{\rm{e}}}}\\ {\bar QSC_y^{{\delta _z}}}&{2{T^p}{\rm{sin}}{\mu _{\rm{e}}}}\\ {\bar QUm_z^{{\delta _z}}}&{2{T^p}{d_x}{\rm{sin}}{\mu _{\rm{e}}} - 2{T^p}{d_y}{\rm{cos}}{\mu _{\rm{e}}}}\\ 0&0 \end{array}} \right] \end{array} \right.$ （32）
$\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}l/{v_{\rm{D}}} + C_y^{{a_y}}{v_{\rm{D}}}}\\ {C_y^{\dot \alpha } = - C_y^{{a_y}}{v_{\rm{D}}},C_y^{\dot r} = C_y^{\dot {\bar r}}l/{v_{\rm{D}}}}\\ {m_z^r = m_z^{{\bar r}}l/{v_{\rm{D}}} + m_z^a{v_{\rm{D}}}}\\ {m_z^{\dot \alpha } = - m_z^{{a_y}}{v_{\rm{D}}},m_z^{\dot r} = m_z^{\dot {\bar r}}l/{v_{\rm{D}}}} \end{array}} \right.$ （33）
$\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}l/{v_{\rm{D}}},C_y^{\dot \alpha } = - C_y^{{a_y}}{v_{\rm{D}}},C_y^{\dot r} = C_y^{\dot {\bar r}}l/{v_{\rm{D}}}}\\ {m_z^r = m_z^{\bar r}l/{v_{\rm{D}}},m_z^{\dot \alpha } = - m_z^{{a_y}}{v_{\rm{D}}},m_z^{\dot r} = m_z^{\dot {\bar r}}l/{v_{\rm{D}}}} \end{array}} \right.$ （34）
参数 数值 m/kg 453 U/m3 370 l/m 7.18 cx/m 0 cy/m -2.35 bx/m 0 vD/(m·s-1) 10 μe 0 by/m 0 Jz/(N·m2) 145 000 Tv/(N·s·m-1) 15 dx/m 0 dy/m -3.70 ρ/(kg·m-3) 1.225 ϑe/(°) 0 αe/(°) 0
$\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{\lambda _{11}} = {K_{11}}\rho U = 35.34{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{kg/}}{{\rm{m}}^{\rm{3}}}}\\ {C_y^{\dot \alpha } = - {v_{\rm{D}}}C_y^{{a_y}} = - {v_{\rm{D}}}\frac{{ - {K_{22}}\rho U}}{{\rho {U^{2/3}}v_{\rm{D}}^2/2}} = 1.338{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}\\ \begin{array}{*{20}{l}} {C_y^{\dot r} = \frac{{ - {K_{26}}\rho {U^{4/3}}}}{{\rho {U^{2/3}}v_{\rm{D}}^2/2}} = 0.098{\kern 1pt} {\kern 1pt} {\kern 1pt} 95{\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{s}}^{\rm{2}}}}\\ {m_z^{\dot \alpha } = - {v_{\rm{D}}}m_z^{{a_y}} = - {v_{\rm{D}}}\frac{{ - {K_{62}}\rho {U^{4/3}}}}{{\rho Uv_{\rm{D}}^2/2}} = - 0.137{\kern 1pt} {\kern 1pt} {\kern 1pt} 8{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}\\ \begin{array}{*{20}{l}} {m_z^{\dot r} = \frac{{ - {K_{66}}\rho {U^{5/3}}}}{{\rho Uv_{\rm{D}}^2/2}} = - 0.485{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{s}}^2}}\\ {{C_{x0}} = 0.055{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{m/s}},C_x^{{v_{\rm{D}}}} = - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 4{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{m/s}}} \end{array}\\ \begin{array}{*{20}{l}} {C_y^\alpha = 0.82,m_z^a = 0.52}\\ {C_y^r = C_y^{\bar r}\frac{l}{{{v_{\rm{D}}}}} = 0.581{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}}\\ {m_z^r = m_z^{\bar r}\frac{l}{{{v_{\rm{D}}}}} = - 0.7538{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array} \end{array} \right.$ （35）
$\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}\frac{l}{{{v_{\rm{D}}}}} - C_y^{\dot \alpha } = - 0.756{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}}\\ {m_z^r = m_z^{\bar r}\frac{l}{{{v_{\rm{D}}}}} - m_z^{\dot \alpha } = - 0.616{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}} \right.$ （36）
图 2 纵向扰动系统的根轨迹 Fig. 2 Root locus of longitudinal perturbation system
图 3 复特征根随飞行速度的变化 Fig. 3 Variation of complex characteristic roots with flight velocity
图 4 实特征根随飞行速度的变化 Fig. 4 Variation of real characteristic roots with flight velocity
4 结论
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Pressure-distribution measurements on the hull and fins of a 1/40-scale model of the US airship: NACA 443[R]. Washington, D.C.: NACA, 1934. [9] ZAHM A F, SMITH R H, LOUDEN F A. Air forces, moments and damping on model of fleet airship Shenandoah: NACA 215[R]. Washington, D.C.: NACA, 1926. [10] GOMES S B V. An investigation of the flight dynamics of airships with application to the YEZ-2A[D]. Cranfield: Cranfield Institute of Technology, 1990: 10-21. [11] KING A, DELAURIER J. An experimental investigation of the aerodynamic effects on a body of revolution in turning flight[C]//6th Lighter-Than-Air Systems Conference, 1985: 866. [12] COWLEY W L, GLAUERT H. The effect of the lag of the downwash on the longitudinal stability of an aeroplane and on the rotary derivative Mq[R]. Washington, D.C.: NACA, 1921. [13] JONES S P, DELAURIER J D. Aerodynamic estimation techniques for aerostats and airships[J]. Journal of Aircraft, 1983, 20(2): 120-126. Click to display the text [14] WANG X. 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Click to display the text [18] 基里林·阿列克桑德拉·尼卡伊维奇.现代飞艇设计导论[M].吴飞, 王培美, 译.北京: 国防工业出版社, 2009: 5-26. [19] MUELLER J B, PALUSZEK M A, ZHAO Y Y. Development of an aerodynamic model and control law design for a high altitude airship[C]//AIAA 3rd "Unmanned Unlimited" Technical Conference, Workshop and Exhibit. Reston: AIAA, 2004. [20] LI Y W, NAHON M, SHARF I. Airship dynamics modeling:A literature review[J]. Progress in Aerospace Sciences, 2011, 47(3): 217-239. Click to display the text [21] SEBBANE Y B. Lighter than air robots[M]. Netherlands: Springer, 2012: 16-34. [22] 徐忠新. 平流层预警探测飞艇[M]. 北京: 国防工业出版社, 2017: 140-148. XU Z X. Stratospheric airship for early warning and detection[M]. Beijing: National Defense Industry Press, 2017: 140-148. (in Chinese) [23] ZHENG W, YANG Y N. Flight dynamics and control of airship[M]. Beijing: Science Press, 2016: 43-56. [24] WU J C. A theory for aerodynamic forces and moments[R]. Atlanta: Georgia Institute of Technology, 1978. [25] WU J C. Theory for aerodynamic force and moment in viscous flows[J]. AIAA Journal, 1981, 19(4): 432-441. Click to display the text [26] WU J C. Elements of vorticity aerodynamics[M]. Shanghai: Shanghai Jiaotong University Press, 2014: 54-55, 130. [27] 吴子牛, 王兵, 周睿, 等.空气动力学.下册[M].北京: 清华大学出版社, 2008: 85. WU Z N, WANG B, ZHOU R, et al. Aerodynamics. Volume ii.[M]. Beijing: Tsinghua University Press, 2008: 85(in Chinese). [28] WU J Z, MA H Y, ZHOU M D. Vorticity and vortex dynamics[M]. Berlin: Springer Science and Business Media, 2006: 26, 52-53. [29] WU J Z, MA H Y, ZHOU M D. Vertical flows[M]. Berlin Heidelberg: Springer Science and Business Media, 2015. [30] QUARTAPELLE M L. Force and moment in incompressible flows[J]. AIAA Journal, 1983, 21(6): 911-913. Click to display the text [31] HOWE M S. On the force and moment on a body in an incompressible fluid with application to rigid bodies and bubbles at high and low Reynolds numbers[J]. The Quarterly Journal of Mechanics and Applied Mathematics, 1995, 48(3): 402-426. Click to display the text [32] HESS J L, SMITH A M O. Calculation of potential flow about arbitrary bodies[J]. Progress in Aerospace Sciences, 1967, 8: 1-138. Click to display the text [33] LIN X W, LIN X W, LAN W Y. On the fluid dynamic force of a solid body moving in the incompressible flow[J]. Journal of Xiamen University (Natural Science), 2018, 57(1): 124-129. Click to display the text [34] 林新武.不可压缩流中飞艇艇体气动力的计算方法研究[D].厦门: 厦门大学, 2018: 39-41. LIN X W. On the calculation methods of aerodynamic force for the airship hull in incompressible flow[D]. Xiamen: Xiamen University, 2018: 39-41(in Chinese). [35] 刘智丽.飞艇建模中的气动力和力矩计算方法研究[D].厦门: 厦门大学, 2019: 37-64. LIU Z L. On the calculation methods of aerodynamic force and moment in airship modeling[D].Xiamen: Xiamen University, 2019: 37-64(in Chinese). [36] BRYAN G H. Stability in aviation, macmillan[M]. London: Macmillan Co., 1911: 19-37. [37] 埃特肯. 大气飞行动力学[M]. 北京: 科学出版社, 1979: 157. ETKIN B. Dynamics of atmospheric flight[M]. Beijing: Science Press, 1979: 157. (in Chinese) [38] 方振平. 航空飞行器飞行动力学[M]. 北京: 北京航空航天大学出版社, 2005: 223-224. FANG Z P. Aircraft flight dynamics[M]. Beijing: Beihang University Press, 2005: 223-224. (in Chinese) [39] 徐明友, 丁松滨. 飞行动力学[M]. 北京: 科学出版社, 2003: 7. XU M Y, DING S B. Flight dynamics[M]. Beijing: Science Press, 2003: 7. (in Chinese) [40] KHOURY G A, GILLETT J D. Airship technology[M]. 2nd ed. New York: Cambridge University Press, 2012: 60-86.
http://dx.doi.org/10.7527/S1000-6893.2020.23648
0
#### 文章信息
LIN Xianwu, WANG Shichao, LI Zhibin, LAN Weiyao
Fusing method for dynamic derivatives and added mass of airships
Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 123648.
http://dx.doi.org/10.7527/S1000-6893.2020.23648
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2023-02-03 09:25:30
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https://sbseminar.wordpress.com/2008/06/16/request-long-distance-collaboration/
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Request: long-distance collaboration
Nathan Dunfield (a new commenter!) supplies our first request:
How about a discussion of long-distance collaboration tools and methods, beyond just using email and talking on the phone? It seems like there are a lot things that might work, e.g. pointing a cheap webcam at piece of paper, using collaborative text editors (e.g. SubEthaEdit), IM’ing (some clients have LaTeX support, I tnink), virtual whiteboards (e.g. Scriblink.com), but which might also turn out to be useless in practice for all sorts of annoying technical reasons. So it would be interesting to hear from people who have had success or failure with various methods.
Unfortunately, I have nothing insightful to say on this topic (I would be really excited to hear if any one else has exciting ideas, for reasons which will be clear below). This is a little sad, since I’m a perfect candidate for having done something interesting in this area. I’m pretty technophilic, even for a mathematician, and am currently writing two different papers with two different people in Germany, and working on another paper in a group of 4 where I don’t think more than 2 of have been in the same state simultaneously in over a year. Almost all this work has been done over email, or face-to-face, with occasional phone conversations and one video chat on Skype (but with no attempt to write anything on a board or paper, just gesticulation). In particular, the last paper I mentioned has been written entirely while we were all permanently in different locations (me in Princeton and Boston, one in Oregon, one in California, and one in Amherst), and generated an enormous number of emails, I think around 500 (thank Gbus for Gmail).
So, why haven’t I done anything more exciting? Well, as Nathan said, the main reason is I just haven’t found the killer app that seemed worth investing in. There are online collaborative word processing programs, but none which do LaTeX well, to the best of my knowledge, so it’s easier to just pass files around via email.
The other problem is that if I did find a program I liked, I would then have to convince my coauthors it was worth using, and they’re, on the whole, more skeptical about these things than I am. I mean, with some them, getting BibTeX was something of a fight, and I think BibTeX is about the best LaTeX add-on in history. More generally, coauthors disagree about what technology is useful or convenient. I mean, Nick Proudfoot and I still have half-joking arguments about whether emacs or vi is superior (for the record: I’m on the side of emacs).
I think the other issue is my work-style. Even when collaborating on research with someone, I’m a very solitary worker. I prefer to go think on my own for a while, and then meet again if I get stuck or find something interesting (or vice versa), which means that I often don’t find email too limiting. I mean, it would obviously be better to be able to meet face-to-face whenever it felt necessary, but I think it feels necessary for me less often than it does for many people.
37 thoughts on “Request: long-distance collaboration”
1. You got a beef with vi?
2. onymous says:
LaTeX-equipped wikis can be pretty nice, as a supplement to (but not replacement of) email/phone/Skype/IM/whatever….
3. What about subversion? I’m thinking about setting up a subversion server to track changes on my own papers; it seems like a pretty good call for a paper being jointly done. More for writing the paper than working out the argument, though.
4. I’d like to see more knowledge of version control software, such as Subversion, Mercurial, and so on — especially distributed version control software (Mercurial, git, Bazaar, darcs, and so on). Mathematicians seem very ignorant of such software, even though it can make collaboration much easier.
If you can learn a complex software system like LaTeX, you can learn to push, pull, and merge changes with your coauthor(s) using version control software!
5. toomuchcoffeeman says:
When I was a PhD student, some of my applied math contemporaries raved about the benefits of using a version control system such as CVS. The idea is that rather than emailing copies around and having to keep track of whose version is latest, the CVS system locks copies, tracks previous edits and so on. Plus people could log in and make changes as and when they wanted, provided someone else wasn’t simultaneously doing so.
Having said that I never used it, and it certainly sounds like a non-starter for colleagues who are reluctant even to go for BibTeX.
6. I recently set up SVN for a budding collaborator of mine; in order to make revisions of papers-in-progress easier. With TortoiseSVN (he’s on Windows), seeing changes and dealing with them ends up being quite easy. Generally, I make sure I keep all my own LaTeX code in a backed up source code repository at all times; it gives me editing history and storage security: things that aren’t all bad for collaborations either.
Furthermore, I very recently discovered that both Miranda and Pidgin support a LaTeX plugin: Instant Messaging with pretty wide-reaching LaTeX support!
7. Mikael: which Pidgin plugin do you use? I tried the pidgin-latex project at sourceforge but had trouble getting it to work properly. Is that the one you use, or is there a better one?
8. huh says:
mathim.com FTW
9. I’ve also been using SVN for a number of years now (and CVS before). The motivation was mostly to have easy backup and history, but it’s quite useful at least for the write-up part of a collaboration, even if only one person uses it to keep the “master” copy of a paper (or a book; in fact, I think for collaborating on a book, where many writing issues are magnified, such a setup is really extremely useful).
I’ve discussed about my setup briefly in this post:
http://blogs.ethz.ch/kowalski/2008/02/23/version-control/
10. Jay: I use the pidgin-latex plugin. Haven’t done much with it yet; but trying it out made it work neatly.
You have to take care only to use
$$[some formula]$$
as it won’t recognize $..$.
What problems have you had?
11. Anthony Bak says:
i use svn and am continuously surprised that other people don’t. I’ve tried to use it for colaboration but for most people it’s seems too difficult. I feel the same thing about emailng documents back and forth….
kopete also has a latex plugin for IM.
12. Like the previous posters, I’ve had good experiences using version control systems (e.g. CVS, subversion) to manage the LaTeX files of papers I’m writing collaboratively. I’ve found the benefits are especially large when there are more than one other person involved, or when there are more than one or two files, e.g. if you have lots of figures.
While the person who manages the central repository where the paper is stored needs a certain level of technical acumen, the other users don’t; the Tortoise front-ends that Mikael mentions make it quite easy to use, even on Windows. Also, the distributed systems that Dan mentions give you a way around the problem of needing for all collaborators to have accounts on a common machine, or one of the collaborators running a server, as is the case for traditional tools like CVS.
The PracTeX journal had an issue last year devoted to collaborative issues, which includes an introduction to a common version control system, Subversion.
13. Also, regardless of how one moves the LaTeX files around, I’ve found the “latexdiff” program very useful for seeing what one’s collaborators have been up to. It produces output like this where you can clearly see what changes have been made. I also find it helpful to double-check my own edits, to make sure I haven’t introduced any new typos in my attempts to remove the old ones.
14. “Having said that I never used it, and it certainly sounds like a non-starter for colleagues who are reluctant even to go for BibTeX.”
As the colleague who was reluctant to go for BibTeX, I will say that this subversion thing sounds like a good idea, provided that Ben walks me through the procedure. Emailing documents back and forth ten times a day along with itemized descriptions of changes sure was a pain in the ass.
15. One of my long-term collaborations has a yahoo discussion group dedicated to our projects; the ability to upload files and search through archived posts is very useful.
Another collaboration has a wordpress group blog. The ability to scribble on each other’s posts is particularly useful (we even agreed on a colour scheme to distinguish each of our comments). And of course, the LaTeX support is a big plus.
Of course, both of these are members-only, for obvious reasons. These are also collaborations involving three or more people; with just two people, it seems that one can get by on good old email with attachments, plus the occasional phone call, fax, and of course face-to-face meeting (preferably near a blackboard).
16. I’m sure Scott will have a lot more to say about this, but all the projects I’m collaborating with him on (up to 3 at the moment) we’ve been using SVN both for the LaTeX files and for the Mathematica files. It’s really nice for LaTex, and kinda nice for Mathematica. On the other hand, I wouldn’t have been able to start using it without an expert in the room, and even now would be a little apprehensive about using it without Scott being reachable (though I mostly have the hang of it).
SVN is also nice when you are in the same room. Scott, Emily, and I have had various paper writing days where we’re all working on a different section and every time you update and compile it’s added all the stuff that the other people have been working on. This is very satisfying and makes writing somewhat less intimidating.
As an added bonus if I’m not at a computer I can look up any of the images in our papers on my iPhone via Scott’s SVN server.
I also do a *lot* of math via IM, so I’m excited to hear about this IM clients which have LaTeX support.
17. Todd Trimble says:
I’ve been using skype together with Windows Netmeeting for a virtual whiteboard, which is okay for just shooting ideas around (if both/all parties are running Windows). We’ve also used whiteboards (like jarnal) which permit use of an electronic pen and easy creation of pdf files, but is not too generous with space (meaning screen area). I myself would be interested in flexible and spacious whiteboards; haven’t tried scriblink yet.
18. If you want to try Subversion (SVN) for free, one simple website to do it at is http://beanstalkapp.com/.
On Mac OS X, check out Versions (http://www.versionsapp.com/) which is some new program for working with Subversion repositories. People seem to like it, although I have no personal experience with it.
19. Mikael: took me a while to get it to compile, but that’s my fault (I’m running Linux). Since I’m running Pidgin 2.4.1 I had to update to the latest version in the CVS. Last night every time I tried to trigger it I got an error message saying the .png file didn’t exist where it should. Today I’m not getting an error message, but every math item I send displays as just the image-not-found red x. I’ll probably play around with it a little more this evening–although mathim sounds more likely to convince my less techie friends–but any suggestions you have would be appreciated.
I’m also glad to see so much support for the version-control stuff. I’m planning on setting up a server for myself anyway, so svn should be pretty effective whenever I need it. (And gives me a free off-site backup).
20. A.J. Tolland says:
The pidgin latex plugin works pretty nicely for me. Compiles cleanly, and installs easily.
However, you should note that pidgin-latex doesn’t display any graphics unless you have the right image conversion programs installed. You can find out which programs by clicking the ‘plugin details’ button in pidgin’s Plugins configuration window.
21. rmb says:
Which of these solutions (aside from the pidgin-latex plugin) are open source? CVS has the advantage of being available (and standard!) for people running Linux/Unix…
22. Jay: You might want to check that you have ImageMagick installed and working on your system. IIRC, the kind of magic that the pidgin latex plugin relies on is basically the image conversion tools in ImageMagick.
rmb: All the version control systems mentioned above are open source, and easily available on Linux/Unix/MacOSX.
23. Tim Wescott says:
Johanna asked me to look through this thread and provide some input. I’m not sure if this is what you’d want to do, but there are some pretty decent whiteboarding applications out there, several of which are free. Basically, the software does what you’d expect: it’s a virtual white board where all participants can see what’s being written. Some have VOIP capability, too (or, if not, you can use Skype concurrently).
http://socialsourcecommons.org/search/query?q=whiteboard&submit=Search
I’ve actually used Vyew, albeit for a non-mathematical group, and I thought it worked fairly well.
Hope this helps.
24. G’day y’all.
I recently switched to a version control system for my papers (in fact, for just about everything – dotfiles, tex style files, programs, even my website is version controlled). As well as the easy backup and collaboration features, it also makes it easy to transfer files from one machine to another. I use bazaar. I wrote up my initial experiences with the shift on my website and, if anyone’s interested, you can find it here. I did try subversion initially but found that bazaar was much more flexible – it took me a few goes to find a system that I liked and bazaar was easier to rub out and start again with. I’d never used a version system before and consider bazaar extremely easy to use (mind you, I am a bit of a Linux geek so take “easy” in that light; however, as canonical are developing it then I think that they are trying to make it as easy to use as possible whilst retaining all the features. The online documentation – see the ‘links’ section of my website – is extremely useful and I recommend reading it before setting up a system). One extremely useful feature of bazaar (may be in others as well) is that it can use sftp (ie ssh) as a protocol.
On the page linked above you’ll also find a link to a wikibook about using versioning systems for LaTeX papers. That gave me a few ideas before setting out.
I would also add that it is extremely useful even if you don’t have collaborators. Having a ‘ChangeLog’ helps a lot too.
Oh, and Emacs has an extremely useful macro: ‘add-entry-to-changelog’ so you can easily keep that up to date as well.
(For the record, I use both emacs and vim).
For those using, or considering using, such a system I’d recommend reading the section on line breaks on the page on my website.
Another program that I have recently installed and instantly found useful is xournal. Amongst other things, it provides for easy annotation of PDFs. This might be useful if, say, in the collaboration you have a “designated writer”. Not sure if it has a LaTeX mode, though.
Regarding long-distance interactive tools, I’d love to hear more about what works. For ‘real time’ interaction, I suspect that something LaTeX-enabled isn’t going to work. Where you need LaTeX capabilities is when the mathematics gets too hard for ascii (I’d be willing to bet that most readers of this can read a line of simple LaTeX code without needing it to be compiled) and then it can take a few goes to get it to look right, spoiling the immediacy of real time interaction. What I’d be most interested in is something like a mini-interactive whiteboard, so a graphics tablet connected to some program that everyone involved could see the output.
For non-‘real time’ collaboration, something more wiki-ish is probably better. Though I’d be interested to know why a wiki is better than a forum and why everyone is so keen to be able to put LaTeX on the actual webpage. Although mathml is great, I don’t think that the current system is ideal. The general method seems to be to code something that is a little but not entirely unlike LaTeX which is then processed into mathml (eg via the amazing iTeX2MML). This looks nice, but if one wants to quote something then one has to essentially reverse engineer the code. Moreover, the various filters are not extendible so typing a long section of mathematics quickly becomes tedious (and cut-and-pasting from a genuine LaTeX document is fraught with difficulty). Wouldn’t it be so much simpler to upload a PDF?
Here’s a thought to close with: imagine doing a spectral sequence calculation with version control. Each commit would be a new ‘page’ so you could step backwards and forwards through the sequence at will to see where all the differentials vanish. He he.
PS Sorry if this post got a bit long …
25. Andrew-
Certainly most mathematicians can read bare LaTeX but it’s a bit like following a formula spoken out-loud: it gets hard fast. It happens to me semi-regularly that I don’t have the patience to read a MathSciNet review in plain-text; I’m sure I could, but it just doesn’t seem worth it. Math is often hard to follow under optimal conditions, so taking even one step out of the processing can make it better.
26. Ben,
You misunderstood me, so I apologise for not stating my point clearly enough. I was not saying “Why do we want LaTeX->(math|ht|x)ml when we have ASCII?”, I was saying “Why do we want LaTeX->(math|ht|x)ml when we have LaTeX->PDF?”. The obvious first objection is that it’s too much hassle to compile a PDF for a simple expression such as -exp^{\pi i}=1. I was attempting to anticipate this by saying that for simple expressions, ASCII is good enough. But for anything for which ASCII is not good enough, probably neither is LaTeX->(math|ht|x)ml. The reason being that if ASCII is not good enough, the expression is probably complicated enough that it should be previewed before posting and at that point the advantages of LaTeX->PDF vastly outweigh the disadvantages (IMHO).
The primary advantages are the following:
WYSIWIP: What you see is what I posted. One of the major problems, for mathematics, for (math|ht|x)ml is that the display is not sufficiently under the author’s control. A simple example can be found on the n-cafe TeXnical issues thread on placement of primes.
Speed. Any post should be previewed before submitting but that goes double if it has maths in it. Having to send the data and wait for it to be processed then sent back often takes quite a while. My most complicated LaTeX document took 2.56s to compile. I don’t think I’ve ever had a preview request from the cafe that quickly. Of course, I could install the correct filter locally, run my post through that until I’ve gotten it right, and then post but how many people have the know-how to do that? But that brings me on to my next point.
Not having to learn yet another markup. I’m very impressed by filters such as iTeX2MML. But they are very definitely not LaTeX->mathml. They are a new markup language and you have to learn new syntax to use them (I’m always having to look up how to do aligned equations in iTeX). I don’t know whether or not it is possible to write a converter that does honest LaTeX->.*ml (tex4ht seems pretty close, but that may be overkill for a blog filter) – I’ve tried writing one myself (babytex) so I appreciate the difficulties. In addition, since there are several filters, different sites will use different filters and one has to remember how to write certain equations each time.
The Power of Emacs (or vi). Writing any markup without using Emacs feels like trying to play squash with both hands tied behind my back. It’s related to my last point: I know Emacs, I know how to use it. Writing in a tiny box on a webpage just isn’t the same.
This, of course, all relates to non-real time interaction. For real time interaction then I don’t think any system involving LaTeX will do and one would be better off with a graphics tablet and an artistic package (proving that maths is an art, after all).
27. I don’t think I’ve ever had a preview request from the cafe that quickly.
My impression is that the delays involved with posting comments on the Cafe are not related to the processing of th math typesetting, but result from a wealth of other tasks that run in the background, such as email notification and notably anti-spam measures.
28. I did not mean to disparage the system at the cafe! I used the cafe as an example because that is the one I have most experience of using but my remarks were intended as being a comparison of local vs remote. Namely, whatever task I wish to do – whether it be preview a post on a blog, compile a LaTeX document, or anything – it is highly likely to be faster if I can do it locally rather than remotely.
It’s mildly interesting that there are two seemingly opposite trends in current computing: distributed computing and remote applications. Actually, they are quite complementary. After all, while I’m waiting for the remote system to notice that I’ve typed another character into my amazing article, the computer in front of me may as well be using its resources to scan for intelligent life on this planet (“Here I am, brain the size of a planet, and they ask me to pick up a piece of paper.”).
If I were posting to the cafe at a much higher rate than I currently do (something like your rate, Urs!) then I would set up a system like I describe above with a local version of the filters to convert my post to mathml and then post the raw mathml.
Actually, that wouldn’t be hard to do …
Mind you, the delays you mention and the delays that I mentioned have null intersection. I was talking about the delay between first hitting “Preview” and (after several reviews) finally hitting “Post” whilst you are talking about the delay between hitting “Post” and the comment actually appearing. The delays in the former set are, I imagine, purely due to data processing and transfer.
29. As the maintainer of itex2MML, let me make a few comments.
1) It is really fast — much faster than any TeX implementation you will ever see. There are lots of things that are slow at the n-Category Café but itex2MML isn’t one of them. Because it only concerns itself with equations, rather than general page-layout, even a relatively un-optimized parser can beat the pants off TeX.
2) It’s certainly not LaTeX, but it is is designed to be as close as possible to AMSLaTeX. For aligned equations, it supports the standard AMSLaTeX environments:
aligned gathered split
It doesn’t support eqnarray because eqnarray is crap. I’m very happy to add other idioms of AMSLaTeX, as users demand them.
3) Currently, it operates as a Unix stream filter, and has native bindings for Ruby. I’m happy to add native bindings for Perl or Python if anyone wants them. I want to make it as easy as possible to incorporate in other blogging/wiki/… systems.
4) I am sympathetic with your idea of doing more of the requisite processing locally, rather than remotely. But the likelihood, say, of getting something like itex2MML to work in Javascript is rather slim.
30. Jacques, Urs, and anyone else still reading, …
I have no criticism to make of the cafe or any of the machinery that underlies it! I’m extremely impressed by all of it, especially itex2MML (by the way, I have a modified version of your MT plugin for blosxom if you’d like it).
What is slow about the cafe is that if I want to check that what I’ve written looks right, I have to send the entire post to the remote machine, let it process it, and wait for the result to be sent back. That machine may be fast, the link may be over a high-speed line, but still it’s over there and that’s what makes it slow.
I’m only using the cafe as an example because I’ve used it.
I’ve seen people, such as Urs, saying that they tend to prepare their comments in a separate editor and then paste them in. All they need is a previewer to check that it looks right and ta-da! That’s the model I’m suggesting, not a javascript implementation (which still involves network communication, if only to send the initial script).
My point, which is getting a little lost, is that – great as they are – maybe remote filters are the wrong way to implement this. And maybe part of the problem is that “this” has not been clearly defined.
So, just to try to ensure that I’m not misunderstood (and to make sure that neither Jacques nor Urs add an extra line to the cafe filters directing all my comments to /dev/null):
The cafe, and the supporting infrastructure, are fantastic.
But I’m not sure that they make a good prototype for wide-spread adoption because, as far as I can see, it works by taking existing systems and adapting them for maths rather than building something specifically designed for maths from the ground up. If one is going to adapt something, I’d rather adapt LaTeX to the web than the web to LaTeX.
PS I agree – eqnarray is not worth implementing. What I have to keep reminding myself is that it uses ‘aligned’ and not ‘align’. Your offer is great, but exposes one of the problems I mentioned earlier: to extend itex, I have to email you with a recommendation and hope that you implement it.
31. …by the way, I have a modified version of your MT plugin for blosxom if you’d like it
By all means, send me a link, and I will publicize it. (If you are interested, I could even include it in my distribution — see below.)
I’ve seen people, such as Urs, saying that they tend to prepare their comments in a separate editor and then paste them in. All they need is a previewer to check that it looks right and ta-da! That’s the model I’m suggesting, not a javascript implementation (which still involves network communication, if only to send the initial script).
AbiWord uses itex2MML and GtkMathView for its Math editing. So you could edit things in that work processor.
A Javascript-based solution (like jsMath) would not involve round-tripping the content to the server. But it would still be slow. And I’m unimpressed by the Javascript-based previewers, like the plugin for WordPress that allows some sort of rudimentary preview of one’s comments before posting.
…it works by taking existing systems and adapting them for maths rather than building something specifically designed for maths from the ground up.
That’s indeed a problem. But, it’s really a second-order problem. The first-order problem is that I have my own research to do, and haven’t the time to write a comparable system from scratch.
What I have to keep reminding myself is that it uses ‘aligned’ and not ‘align’.
The simple mnemonic is that I’ve implemented those environments which work inside an equation context. Thus: gathered instead of gather, aligned instead of align, and split (which, despite not having an “ed” suffix, only works inside an equation context).
Your offer is great, but exposes one of the problems I mentioned earlier: to extend itex, I have to email you with a recommendation and hope that you implement it.
Actually, you can check out a copy of the source code BZR repository, make whatever changes you have in mind, and then send me an email with the URL of your BZR repository, and I’ll pull your changes from there.
32. Great post and interesting discussion!
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2022-12-04 21:21:07
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http://gunnarpeipman.com/2007/10/testing-sorting-algorithms/
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# Testing sorting algorithms
Some time ago I had to deal with sorting algorithms. Besides my main task I found a good way how to test custom sorting algorithms. This blog entry is one of early birds, more about sorting algorihtms is coming soon. Hopefully some time after TechEd 2007 for Developers. The procedure I wrote to test sorting algorithms is simple and works. Of course, I am always opened for better ideas if somebody wants to suggest some. Here’s the little overview about what I’ve done.
### Is array sorted?
At first, let’s look at method that checks if array is sorted or not. To make things simpler I expect that all members in array implement IComparable interface. This expectation has very strong point: it is easy to compare objects of classes that have comparison operators defined but it is impossible to compare objects of classes that doesn’t have those operators defined. To make my method as universal as possible I move comparison logic out of it.
About implementation. Don’t look for any highly respected architectural decisions from this example. This is example about testing, not about nice and respected architecture. But here is my IsSorted() method.
public static bool IsSorted<T>(T[] array) where T : IComparable<T>{ if (array.Length <= 1) return true; for (int i = 1; i < array.Length; i++) if (array[i - 1].CompareTo(array[i]) >= 0) return false; return true;}
The other point I found mentally interesting is question about empty array – is it sorted or not? 🙂
### Testing IsSorted() method
As a next step let’s write test for IsSorted() method. In this example I’m using Int32 as a type of arrays to test. Of course you can use any other types you need. To make our tests better we test IsSorted() method with array that contains some elements with same values. Then we can be sure that we can find also CompareTo() methods that make comparisons incorrectly somewhy. And one note – I’m using NUnit for tests.
I wrote to tests – one test for sorted array and the other for unsorted array. In this case we see behavior of IsSorted() for both cases that may occure. The tests are as follows.
[Test(Description = "Integer array is sorted")]public void TestInt32ArrayIsSorted(){ Assert.IsTrue(Program.IsSorted<int>(new int[] { 10, 20, 20, 120 }));} [Test(Description = "Integer array is not sorted")]public void TestInt32ArrayIsNotSorted(){ Assert.IsFalse(Program.IsSorted<int>(new int[] { 10, 1, 44, 44 }));}
Now we can be sure that isSorted() method will be tsted against sorted and unsorted arrays. Before we can test sorting algorithms we need to write some tests more.
### Testing comparisons
Specially in the case of custom objects with comparers we need to test if comparers are working well. If they don’t we may get wrong results from other tests. So let’s test them now.
[Test(Description = "Integer is less than other")]public void TestInt32LessThan(){ int i = 2; Assert.Less(i.CompareTo(3), 0);} [Test(Description = "Integer is greater than other")]public void TestInt32GreaterThan(){ int i = 2; Assert.Greater(i.CompareTo(1), 0);} [Test(Description = "Integers are equal")]public void TestInt32Equal(){ int i = 1; Assert.AreEqual(i.CompareTo(1), 0);}
Now we can make sure comparisons are tested and if something went wrong our tests will tell us so.
### Sorting algorithm
Now let’s write sorting algorithm we want to test. I have many algorithms and I selected SelectionSort() for this example. Here’s the code.
public static void SelectionSort<T>(T[] array) where T : IComparable<T>{ int i, j, min; T temp; for (i = 0; i < array.Length - 1; i++) { min = i; for (j = i + 1; j < array.Length; j++) if (array[j].CompareTo(array[min]) < 0) min = j; temp = array[i]; array[i] = array[min]; array[min] = temp; }}
I wrote all my sorting algorithms using generics so I can use the same algorithms with differents types of objects.
### Testing Sorting Algorihtm
Now let’s write test for SelectionSort() algorithm using integers.
[Test(Description = "Test selection sort on Int32 array")]public void TestInt32SelectionSort(){ int[] array = new int[] { 10, 2, 34, 76, 23, 34 }; Program.SelectionSort<int>(array); Assert.IsTrue(Program.IsSorted<int>(array));}
Now we have test for SelectionSort() that uses array of integers. Same way we can test also other sorting algorithms we want because IsSorted() method is not dependent of sorting algorithm we are testing. Also it has no strong dependence of types of objects we are testing because we require that all objects used here implement IComparable interface.
### 3 thoughts on “Testing sorting algorithms”
• Petar Petrov says:
What will happen if the array contains duplicates ? Your IsSorted() method will return false and for me the array {1,2,2,2,3} is sorted.
I think you need to change the comparision from
(array[i – 1].CompareTo(array[i]) > 0) to
(array[i – 1].CompareTo(array[i]) >= 0).
• Gunnar says:
Thanks Petar, it was my bug. Now it is fixed 🙂
• Sergey says:
That’s great, but your procedure can’t guarantee that the sorting function works correctly. What if you algorithm makes all elements that same, or just duplicates some elements? (what happens quite often)
For example:
Initial array: 3 1 4 0 9
Output "sorted" array: 0 1 1 4 9
IsSorted will return true, but sorting method is wrong! (1 replaced 3 in the output)
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2017-02-23 11:40:53
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https://iesc.io/608/S20/lab08a
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# Lab 08a: Secret Sharing
## Spring 2020
The questions below are due on Sunday April 05, 2020; 11:59:00 PM.
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Tearing up Secrets into Shards
Goals:Today you'll get some exposure to git in advance of the final project and then implement a secret sharing scheme between you and some fake people. This lab will also serve as a review/more practice with writing some server side scripts that are a bit more time-sensitive than ones we've previously been messing with.
This lab can be done on your own, just like in Lab06A. You can either get a live-checkoff with a staff member over video chat at the end (one large checkoff), or submit a writeup report like you did for Lab06A, which will be graded witha bit of delay.
## 1) Git
When working on files either alone or with collaborators, there's often the desire to prevent overwriting/forgetting/messing up certain versions of files. Let's say you got some functionality working with your code, and then you decided to go further and add more functionality, but all of the sudden nothing works. You try to undo and still nothing works. We've all been there (if Piazza posts are any accurate indication). Similarly, if you're working on a shared file in Word with a friend and you both edit the same paragraph at the same time and don't keep in constant communication, when you try to put your work together, it can be a mess.
Version control software is one solution to this and we'll be using git in this lab. Version control allows you to keep track of previous versions of files as well as collaborate with others in a way that ensures changes made by any user aren't necessarily lost right away, but instead will exist in various revisions and branches. The basic idea of how we'll be using it (highly simplified) is that a central repository (repo) of files exists on a centralized server (we'll be setting up a repo on 608dev-2.net) and then individual users clone this repo onto their local computer using the git clone command so they have their own copy of everything on their computer which they are free to work with and change and modify as needed via the git add and git commit commands. This could be very beneficial for team-based work.
Users can, when they want, contribute their changes from their local copy to the master copy using the git push command. If other users are contributing, then a user can also harvest/collect the changes pushed to the master repo using the git pull command. While all of this is happening, git is making sure that no conflicts between different versions of files happen and nothing gets lost. Git makes an attempt at resolving any conflicts that have a generally accepted solution (you edited the top of a document, and your friend edited the bottom of a document...in this case it combines these). You can try this with a git merge. More difficult-to-judge conflicts (edits in the same block of code), must be merged manually.
git != github. git is a free software created initially created by Linus Torvalds (also creator of Linux). Github is a company that provides a service based on git and its own web-based stuff. They are now owned by Microsoft and probably steal your data. You can use git without Github (and many do!). You cannot use Github without git. For lab we'll create a git repo on the server, then clone it locally on our computer. You'll then add some files locally, commit them, then push them up to the server. Real easy.
First you'll need to get git (lol) on your computer. Links to do that for the three operating systems are below (Many of you will already have it in place though if you installed the ESP32 core so proceed into these links with caution to make sure you don't redo stuff you've already done).
Once installed, we will be using the git shell/terminal. In Mac and Unix/Linux systems this will just be the terminal/shell. On Windows this will be the git bash shell. This is a Linux shell so you need to be familiar with how to work in it a bit. It is basically the same as what we've been doing on the server/interfacing with it so far, so it shouldn't be too foreign to you!
If you aren't comfortable moving around in a terminal, here are a few basic commands as a refresher. You should be at least somewhat used to these from working on/with the 608dev.net server. Ask a staff member for help!
• cd: Move your current view to "home directory"
• cd DIRECTORY: move into a directory
• cd ..: Move up one directory level
• cd -: Move to the previous location you were at
• cd ~: Move to home directory
• ls: List contents (files, etc.) of specified directory
• pwd: Print the path of the current working directory
Check Yourself:
Can you move up and down in the file structure on your computer using these commands?
We'll be working quite a bit on the server/shell today. Make sure to go back to Exercise 03 if you need a refresher on that stuff.
## 2) Create a Repo on the Server
A Python Error Occurred:
Error on line 1 of Python tag (line 59 of source):
print('%s' % (cs_user_logos[kerberos],))
NameError: name 'kerberos' is not defined
in case you forgot.
OK let's first ssh into our server and make a git repo. In your home directory make a folder (mkdir) called lab08a. Move into that folder and then run the command:
touch README.md
What this will do is create a file inside lab08a that is called README.md with nothing in it. Cool. If you run ls you should now see your empty directory has README.md in it.
Now, next, run the following command while in lab08a:
git init
Some stuff will fly by but it should look like the following (don't worry about the templates not being found):
<div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 77 of source):
print('%s' % (kerberos,))
NameError: name 'kerberos' is not defined
<p></font></div>@608dev2:~/lab08a$git init warning: templates not found /usr/share/git-core/templates Initialized empty Git repository in /home/<div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 79 of source): print('%s' % (kerberos,)) NameError: name 'kerberos' is not defined <p></font></div>/lab08a/.git/ Cool, you've just created a git repository. The repository is empty on creation by default, however. Let's add that file we made. Run the following command: git add README.md This adds the file to the repo. Now let's commit this file into the repo. To do this we'll use the git commit command, but we'll also provide a message while we're doing it which notes what's happening during this commit so that if we need to look at our history in the future, we'll be able to know at a high level what we were doing here. An inline message is done with a -m 'MESSAGE' tacked on to the command. Finally, let's specify that we're commiting the changes that we've implemented with README.md, namely its creation. All together this means type: git commit -m 'adding in README.md' README.md When you run this you'll get something like the following: <div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 98 of source): print('%s' % (kerberos,)) NameError: name 'kerberos' is not defined <p></font></div>@608dev2:~/lab08a$ git commit -m 'bringing in README.md' README.md
[master (root-commit) 44e729b] bringing in README.md
Committer: 6.08 Student <<div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 100 of source):
print('%s' % (kerberos,))
NameError: name 'kerberos' is not defined
<p></font></div>@608dev2.608dev-2.net>
You can suppress this message by setting them explicitly. Run the
git config --global --edit
After doing this, you may fix the identity used for this commit with:
git commit --amend --reset-author
1 file changed, 0 insertions(+), 0 deletions(-)
You may also get a message about your email not being auto-detected looking like this:
*** Please tell me who you are.
Run
git config --global user.email "you@example.com"
git config --global user.name "Your Name"
to set your account's default identity.
Omit --global to set the identity only in this repository.
fatal: unable to auto-detect email address
If you do, go ahead and follow the instructions that it prompts you with:
git config --global user.email "you@example.com"
and then:
git config --global user.name "Your Name"
where you should replace the content in quotes with stuff that is relevant to you.
If you'd like to set your email you can, but the message above isn't going to prevent other stuff from working.
Cool you've now successfully set up a git repository on the server! Before we leave the server, let's check on the size of that README.md file. It should be 0 since there's nothing in it. Run the following command:
ls -s README.md
You should get back:
0 README.md
which confirms the size is 0. Good.
Finally we need to change one more thing on our repo. Inside of your repo if you do ls, you'll see just README.md. This isn't the whole story. If you type:
ls -a
you'll see several other things show up:
. .. .git README.md
This is everything that's in the folder. . means the current directory. .. means the parent directory (these two are always there). We can see README.md, but the other one is .git. Files in *Nix systems that start with a period are "hidden" files. When you turned this folder into a git repository, what git did was set up a hidden folder with all the appropriate settings stored in it. If you move into that folder with cd .git and then run ls you'll get:
COMMIT_EDITMSG config HEAD index logs objects refs
These are all the settings and history and important information about your repo. When you run git commands in it, they will be using/modifying information stored in there.
We need to change one of the files in this folder, the config file. Onto your local machine, download this config file HERE (don't rename it. Save it just as "config").
Do not save the file as "config.txt" or anything with an extension. It should just be "config". If it automatically sets it something with an extension (I'm looking at you, Windows kids), you need to change it
scp it up to your repo, and replace the current config file. For example, if your saved config in you home directory on your local computer you could do:
scp ~/config <div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 186 of source):
print('%s' % (kerberos,))
NameError: name 'kerberos' is not defined
<p></font></div>@608dev-2.net:~/lab08a/.git/config
You could also just scp it into your home directory on the server, and then move it (mv) it manually to where it needs to go.
Make sure you do the last step above where you replace config. Failure to do this will lead to confusing results later.
## 3) Clone the Repo
Now let's open up a new terminal/shell local on your computer. In terminal or git bash shell, move to your home directory (or another directory if you prefer, but try to avoid something in Dropbox or OneDrive, GDrive, or iCloud to avoid syncing issues) and type the following:
git clone PATH_FROM_SERVER name_youd_like
This says, "make a local clone of the repo located at PATH_FROM_SERVER and call it name_youd_like.
Specifically, if you put the files where you were instructed to above, you'd do:
git clone <div><font color='red'><b>A Python Error Occurred:</b><p>Error on line 1 of Python tag (line 206 of source):
print('%s' % (kerberos,))
NameError: name 'kerberos' is not defined
<p></font></div>@608dev-2.net:~/lab08a ~/lab08a_local
It will prompt you for your password, and once that's entered, it will clone! If you don't specify a name for the local copy, git will create a directory with the same name as the remote repository. As typed above, this will create a local copy of the repo called lab08a_local in the home directory.
Move into the repo using a command like the following (replacing lab08a with whatever you called your local copy)
cd lab08a_local
Once inside show the files by doing:
ls
You should see the following show up:
README.md
Now open that README.md (either through a GUI file manager/editor or a command line editor if you are comfortable with that) and add in a line of text about something random. Use whatever text editor you like (Sublime, notepad, !Word). When you are done, save it.
Then in the terminal/shell enter:
git status
You will get back something like:
On branch master
Your branch is up-to-date with 'origin/master'.
Changes not staged for commit:
(use "git add <file>..." to update what will be committed)
(use "git checkout -- <file>..." to discard changes in working directory)
no changes added to commit (use "git add" and/or "git commit -a")
There's some helpful instruction messages, but of particular note is that git is telling us we have modified README.md. Let's solidify these changes by commiting them. To do that type:
git commit -m 'modifying README' README.md
It'll say something like:
[master 375e910] editing README
1 file changed, 1 insertion(+)
If you now check the status again you'll see:
On branch master
(use "git push" to publish your local commits)
nothing to commit, working tree clean
This is essentially saying that we've made some changes and commited them locally which are not yet reflected on our remote server (the origin/master). Let's actually move them up there. To do that we'll run push. Specifically do:
git push origin master
In response you'll see some stuff fly by like:
Counting objects: 3, done.
Writing objects: 100% (3/3), 260 bytes | 0 bytes/s, done.
Total 3 (delta 0), reused 0 (delta 0)
To 608dev-2.net:~/lab08a
44e729b..375e910 master -> master
which is a success message.
If you now jump back into you ssh session on the server and type ls -s README.md you should see something like (note you may not have an 8...the number there will be based on the size of text you put into the file!)
8 README.md
This is showing us that README.md has now increased in size from before. Awesome.
### 3.1) Add in a File
We've edited a file that was already there and saw evidence that our remote change was successfully brought into the live repo on the server.
Let's create a new file from scratch remotely, add it, commit it, then push it up to the server. Move back onto your local computer. Inside of lab08a_local create a new python file called lab08a.py. Give it a simple request_handler function like so:
def request_handler(request):
return "Hello, there!"
Returning to your local shell, we need to add that file to your local repo. To do this run:
git add lab08a.py
Then commit that new file:
git commit -m 'adding in a starter file for lab08a.py' lab08a.py
Then push to your master repo using the appropriate command. Returning to the server, and you should now see that a file called lab08a.py lives there! Not only that, visiting http://608dev-2.net/sandbox/sc/
A Python Error Occurred:
Error on line 1 of Python tag (line 321 of source):
print('%s' % (kerberos,))
NameError: name 'kerberos' is not defined
/lab08a/lab08a.py shows that the file is working and live!
Every time you make a change, or a series of changes, you can do it locally. And then commit and push them! Where it gets really cool is that all of your changes are saved (if you commit often), so that if you ruin your code later on, you can always move back to a prior commit to get working code again. (we'll add a brief note about this in the future for reference.)
### 3.2) Quick Set of Commands
A short list of commands we just used (and one or two new ones is below):
• git clone: Used to make local copy of remote repo
• git status: Returns the current status of your directory (are you up to date with the master repo?, do you have uncommitted changes?)
• git commit -m 'message' files: Adds changes to a repository's history
• git add file1: Adds file1 into current branch, to be included in your next commit
• git rm file1: Removes file1 from being tracked by git (does not actually remove file) (need to commit after that)
For more juicy details on git the 6.031 git overview is a good one to read through.
When done, onto the next part!
## 4) Secret Sharing
In exercises this week we look at some really simple means of encryption and decryption of messages. In other words, as shown in the figure below, we were developing a way (albeit an easily cracked one) to enable two parties to securely communicate while preventing outside parties from understanding what was being transmitted
Today we're going to investigate another form of data security. We want to develop a secret-sharing infrastructure. This solves a fundamentally different problem than the encryption up above. Instead of ensuring two parties can talk securely together, we want to ensure that two or more parties can securely "consent" on a topic while making sure no party can act on its own. In effect we are going to have a secret be required for some action to take place, but are going to subdivide that secret among multiple parties so that no single user has enough information to get/use the secret. A prototypical situation is where you want to make sure a system acts only if there is agreement among parties. How can we design something electronic to work like that? The problem is illustrated below:
Further can we design the system where we only require a subset of approved parties to consent?
We can find an answer to this predicament in a simple line on a 2-dimensional plane:
As we've probably learned in multiple math classes from growing up, in order to fully define a line, you need to only provide two points. This is sufficient to uniquely determine a line. Along with a line comes a y-intercept, of which there will only be one per line, provided we don't have a perfectly vertical line located at the origin. Another way of thinking about this is, that given two points (x,y), they will uniquely specify a y intercept, but if you have only one point or the other point, you can't figure that out. More importantly, if you give each point to a separate entity, and tell them to keep it secret, only when they provide their two points to some central entity will their actions be able to result in the secret y-intercept being determined. We'll call these points "shards", as in shards of glass, or shards of the secret.
If we treat this y-intercept as the secret we want to keep safe, then what we've done is spread a part of that secret across several distinct entities. Neither entity has enough information to know the secret, thus ensuring that only a critical mass of provided shards will allow determination of the secret.
What's interesting about this setup is that you don't need to provide only those two specific secret shards. Since a line has infinitely many points, one could choose another pair of points to generate shards and they'd be equally as effective:
Furthermore, if you simultaneously deploy more than two shards like in the figure below, you can have a system where any two shards can be combined to generate the key:
If we'd like to be more restrictive in the number of required parties, we only need to increase the order of our polynomial. For example if we want a consensus of at least three entities, we can use a second-order polynomial (a quadratic).
A quadratic can be uniquely specified by any of its three points, so the same sort of approach can be undertaken, just with more shards:
This entire approach is known as Shamir's Secret Sharing Scheme (SSSS), developed by Adi Shamir. What's the first letter in Adi's last name? S, you say? Is it coincidence that there's also an S in RSA? Nope. He is that S. He was one of the guys (along with Ron Rivest for the R, Leonard Adleman for the A) who developed RSA and has done some other fantastic work cryptography and security!
So this SSSS has a lots of uses. Let's say you run a big company with four chief officers (CEO, CTO, etc...) and you only want to be able to pay out money to vendors if all four chief officers agree. You choose a secret code needed to release funds that you give to a trusted third party, perhaps some automated secure Python script. You make that secret code the constant term of a third order polynomial and then randomly choose the other three coefficients to generate a hidden 3rd-order polynomial. You then produce four coordinate points (x,y values) and provide each of your four officers with one of them (the shard...a point coordinate pair). You then make sure your secure script will only release the money if:
1. four shards are provided
2. the four shards provided describe a 3rd order polynomial with an identical constant (the y-intercept) to what the secret code is.
The result is that all four points will need to be provided by your chief officers!
Another use for SSSS, which is perhaps a bit more relevant of the times, is for backing up the seed value needed to recover your wallet for the highly-stable and functional crypto-currency Bitcoin in multiple geographic locations. Let's say you have five servers spread throughout the world and you want to use them to securely store backups of some critical number for your bitcoin wallet (if you lose that number you lose your bitcoin). Keeping the entire secret value on one computer (or keeping it completely on five separate computers) is potentially insecure (someone could steal it). Breaking the secret value up across the five servers is also problematic since if one goes down and/or gets stolen, you won't have the full secret anymore. One solution is to use SSSS to break it up so that any three of the five servers is sufficient to recover the seed by giving each server a part of the key. If someone steals one of your servers, whatever...the thief doesn't have enough information to steal your cryptocoinz and you still have enough information in your remaining servers to get at your cryptocoinz.
Now in a real-life implementation, they usually use much higher-dimension geometries/polynomials over finite fields, but the general idea holds true! It is beyond the scope of this short lab to go into a more modern and viable implementation of SSSS, but maybe that could be a final project element?
## 5) Implementation
So now it is time to implement a version of SSSS. We're not going to implement a super-secure version...just one that basically uses the thought-experiments above with polynomials in a plane. We'd like you to create a system where a critical number of users (three) must (via a POST from the ESP32 lab kit and two spoofed users via POSTman) submit their shards and usernames within a 30 second sliding window in order to enable a GIF of your choosing to be available upon visiting your site in a web-browser via a GET request. The system must not only prevent viewing of the GIF if less than three shards have been provided in the allotted time, but should also prevent viewing if the three shards specify a polynomial with a y-intercept (the constant term) that is different from the secret value.
You're going to need to fit a polynomial for this assignment. May we suggest numpy? Use the internet to look up how to fit a polynomial using numpy.
Again, to reiterate and restate what you must do:
• If any three valid users submit three valid shards within a 30 second window...
• Visiting the python script via a GET in your web browser should reveal a super-secret valuable GIF of your choice that should be available until the 30 second window has passed.
• At all other times, visiting your python script via a GET in the web browser will provide a sad face text emoticon.
Additionally, if a user submits the same valid shard several times within a 30 second window, and two other users submit their unique shards as well within that same window, the GIF should also be released
Additionally, if a user submits a bad shard and a good shard within a thirty second window, the system should default to not accepting anything until only a good shard exists.
Additionally, you must be sending your shards up in the body of your POST. Do not put it in the query arguments. Body. Failure to do this will cause you to fail the class.
In terms of what you put on your labkit, you don't need to do much. The ESP32 code (provided here) should only need minor modification to be working, including ensuring your wiring is matched with the button it assumes, and that your POST body content is correct with how you are setting up your server-side script (remember you must be POSTing).
Details of your implementation are up to you. However, you will collectively develop TWO Python scripts:
• The first script will generate a second-order polynomial and three user-provided keys as well as the secret value. This can just be a simple local script you run which prints this stuff out. In real-life we'd take care to not publically share its output (since both you and your partner are seeing all the points, and therefore have enough information to separately each specify the secret), but for our toy example today this is fine.
• The second script will be for use on the server and handle the incoming POST requests and GET requests. Remember that your server-side implementation will need to be able to remember who has POSTed their secret shard within the last thirty seconds. Any sort of "remembering" like this will need a database, so make sure you generate a structure for that. Your system must use the POST request body to transfer information, not the POST request query arguments.
• You will need to solve for a polynomial given several points. There are several ways to do that, but we've already actually done one of these previously. Not doing this (and just waiting for three shards to be submitted without checking) voids the whole point of doing this exercise!
<checkoff_writeup>Briefly discuss Shamir's Secret Sharing Scheme and a general idea of how to implement the assignment </checkoff_writeup>
Get your system working. Choose a GIF to display in the browser (via a simple GET) when all required users have submitted their shards (via a POST using the body of the POST) within the last thirty seconds. In order to return a GIF, you need to just return some HTML in your Python!
return """ <iframe src="https://giphy.com/embed/sbG9nMttKvPYA" width="480" height="266" frameBorder="0" class="giphy-embed" allowFullScreen></iframe><p><a href="https://giphy.com/gifs/hacf-gordon-clark-good-boy-gordo-sbG9nMttKvPYA">via GIPHY</a></p>"""
will return the GIF below when visited in a browser (from the best television show of the last forty years, Halt and Catch Fire). You should pick a different GIF (that is appropriate).
via GIPHY
<checkoff_writeup>Demonstrate your working SSSS where three separate users must have logged in within the last thirty seconds in order for a GIF to be available in the browser. </checkoff_writeup>
<checkoff_writeup>Demonstrate in a short video (~30 seconds) your system performing the proper secret sharing scheme. Describe how your code (both the ESP32 and the server code) works and include it in your writeup. Make sure you describe what you did. Don't just dump it into the document. Make sure to specifically address the following things:
• How your system ensures enough shards have been submitted
• How your system gets its key from the points
• How the system handles a bad shar
</checkoff_writeup>
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2021-01-23 10:51:13
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https://www.r-bloggers.com/2020/11/rvspython-5-using-monte-carlo-to-simulate-%CF%80/
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The number $\pi$, while being an irrational and transcendental number is a central number to which much of mathematics and science at large relies on for many calculations.
A whole book can be written on this matter alone, but today we are going to focus on approximating the true value of $\pi$ using Monte Carlo simulations in R and Python!
# Disclaimer
This problem has been covered extensively across the internet and serves as a benchmark example of what Monte Carlo can do. What we are going to do is highlight how this method preforms in both R and Python.
# The General Algorithm
The formula for the unit circle is:
$x^2+y^2=1$
$\iff y = \sqrt{1-x^2}$
For $(x, y)\in[0,1]$, the length of a quarter of the unit circle $\pi\over{4}$. Thus, to approximate $\pi$, the function that we will be using is:
$Y_i = 4\sqrt{1-X_i^2}$
To approximate $\pi$, we use the following Algorithm:
1. Generate $X_1,X_2$ $\dots,$ $X_n \sim{}U(0,1)$
2. Calculate $Y_i = 4\sqrt{1-X_i^2}$
3. Take the mean of $Y_1,$ $\dots$, $Y_{n}$: ${\bar{{Y}} =} {{\sum_{i=1}^{n}Y_i}\over{n}}$ – this is our approximation
The code for this is relatively straight forward. But the question is, which code will run faster?
Let’s go!
# The Test
For our challenge we are going to be writing code which is as intuitive as possible in each language. We are going to seek to approximate the value of $\pi i$ using the above algorithm. For R we will be using lapply to implement the Monte Carlo algorithm and for Python we will using for loops.
# The Solution with R
#' Define Number of points we want to estimate
n<-c(10,100,1000,10000,100000,1000000)
#' Generate our random uniform variables
x<-sapply(n,runif)
#' Our Transformation function
y<- function(u) {
4*sqrt(1-u^2)
}
startTime<-Sys.time()
yvals<-lapply(x,y)
endTime<-Sys.time()-startTime
avgs<-lapply(yvals,mean)
endTime
## Time difference of 0.01399588585 secs
data.frame(n, "MC Estimate"=unlist(avgs), "Difference from True Pi"= abs(unlist(avgs)-pi))
## n MC.Estimate Difference.from.True.Pi
## 1 10 3.281637132 0.1400444782036
## 2 100 3.391190973 0.2495983193740
## 3 1000 3.090265904 0.0513267494211
## 4 10000 3.143465663 0.0018730098616
## 5 100000 3.141027069 0.0005655842822
## 6 1000000 3.141768899 0.0001762457079
# The Solution with Python
import numpy as np
import pandas as pd
import time
# Define Number of points we want to estimate
n = [10, 100, 1000, 10000, 100000, 1000000]
# Generate our random uniform variables
x = [np.random.uniform(size=n) for n in n]
# Our Transformation function
def y(x):
return 4 * np.sqrt(1 - x ** 2)
startTime= time.time()
yvals = []
for array in x:
yval=[]
for i in array:
yval.append(y(i))
yvals.append(yval)
avgs=[]
for array in yvals:
avgs.append(np.mean(array))
endTime= time.time()-startTime
# How long it took to run our code
print("Time difference of "+ str(endTime) + " secs\n")
# Output
## Time difference of 3.146182060241699 secs
## Estimated Values of Pi
pd.DataFrame({"n":n,
"MC Estimate":avgs,
"Difference from True Pi": [np.abs(avg-np.pi) for avg in avgs]})
## n MC Estimate Difference from True Pi
## 0 10 3.320525 0.178933
## 1 100 3.172290 0.030698
## 2 1000 3.156044 0.014451
## 3 10000 3.141675 0.000083
## 4 100000 3.147255 0.005662
## 5 1000000 3.141400 0.000193
# Comparing R with Python
From the following ratio we can see how much faster R is than Python:
library(reticulate)
reticulate::py\$endTime/as.numeric(endTime)
## [1] 224.7933496
Woah! Using my machine- R is over 220 times faster than Python!
I think it’s pretty clear to see who the winner is as far as speed is concerned.
# Concluding Remarks
While R most of the time sits on the sidelines in the Python-dominant world of Data Science- we need to keep in mind where Python’s weaknesses lie and when to pivot from and use R.
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2021-11-29 03:48:11
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https://www.zybuluo.com/zhongdao/note/1533290
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@zhongdao 2019-08-16T08:21:02.000000Z 字数 32992 阅读 1905
# 每个永续农场的水资源管理中英对照
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# 每个永续农场的水资源管理
April 17, 2017 by papprentice 4 Comments
When I was a kid, my brother and I used to enjoy the winter weather at my grandparents’ farm in the mountains. In a beautiful interplay of natural forces, throughout the winter and early spring we would play in half a meter of snow, walk on frozen lakes and then run from the floodwaters once the snow had thawed.
We always looked forward to wintertime and knew what to expect. Now that we’ve grown, however, these natural cycles of snow, ice, and floodwater have changed significantly. It’s not like there are not they no longer exist, but everything has become so much less predictable, and so much more extreme.
This weird weather is the new normal for many of us. It is beyond doubt that we all are experiencing more extreme and erratic weather cycles, with huge downpours and flash floods on the one hand and extreme droughts on the other.
Now, maybe meteorologists will one day call this period of weird weather some fancy term such as the ‘mid-millennium little warming period’ and, looking long term, it will just be blamed on a statistical error, but for us living through this period, it is the new normal. If this new normal continues for just 50 years, while it may a tiny speck of time in the Earth’s history from a climate science perspective, for many of us it’s going to be the rest of our lives, and the lives of our children.
It is this new reality to which we have to adapt and create resilient extreme weatherproof systems that are able to handle either too much or not enough water, all in one growing season. In these circumstances, where water from rainfall is generally in short supply while at other times there is run-off from rainfall, the plan for the control of water is paramount and involves a combination of earthworks, soil-building techniques and irrigation pipes.
Let’s dig deeper...
Permaculture Water Management 永久性水资源管理
Assessing Your Site's Water Needs and Resources 评估场地的用水需求和资源
Storing Water in the Soil 在土壤中储存水
Storing Water on the Surface 在地表储存水
Conclusion 总结
### Permaculture Water Management 永久性水资源管理
Nothing defines the nature of a place more than water! Without water we have deserts almost devoid of life, but with an abundance of water we have rainforests that are the hubs of the Earth’s biodiversity.
Water is always the number one priority for any permaculture system, as Mark Shepard would say: No matter where you go and what mineral deficiencies you have, there are plants who can adapt to these conditions, but no plant can live without water! That’s why Permaculture design tries to harvest, retain and rescue as much water as possible before it is lost from the system.
There are two basic strategies of water conservation on a permaculture farm: storing water in the soil and the diversion of surface water to dams/ponds and tanks for later use; storing it on the surface.
First we want to slow, spread, and sink water as it falls from the sky into the soil. Following this, our secondary goals, as Ben Falk writes in Resilient Farm and Homestead, are to: (1) capture as much water as is reasonably possible, (2) store that water for dry periods, and (3) distribute that water when necessary across the site. Whether you’re going to use one or both of these strategies depends on your site conditions: climate, terrain, soil, your context…
However, as I said, the first objective of landscape design is to control and better use the water that falls on the surface of the land. You want to disperse the flow of water so it can slow down and infiltrate into soil, turning this runoff into soak-in. Essentially, you want to make the water stroll, not run, through the landscape and for this we must shape the land in such a way (more on that later) that it facilitates getting water into the ground and storing it there.
Once you’ve made the best use of the fallen rainfall and stored that water in the soil, you’ll get runoff as the field capacity of soil is reached. Truth be told, you might get this runoff straight away if your site’s watershed is in a bad shape; however, whatever the case, you can begin diverting and storing that water on the surface in ponds and tanks. How much water you’ll be able to store on the surface depends on many factors: your climate, terrain, soil, budget…
Here, I’ll outline the entire process of water management for a permaculture farm, and this includes, but is not limited, to:
• Assessing your site’s water needs and resources 评估场地的用水需求和资源
• Developing water storages (both in the ground and on the surface) 开发储水设施(地面和地面)
• Harvesting water 收获水
• Reticulating water 网状的水
### Assessing Your Site's Water Needs and Resources 评估场地的用水需求和资源
The first issues to address are what water resources are available to your property, and what exactly are your needs?
1) Your goals and context - what are your water needs, and how do you plan to use your harvested water?
1)你的目标和环境——你的水需求是什么，你计划如何使用你收获的水？
Can we do anything without being clear on what our goals and context are? NO… of course not!
You’ll have to be clear on what you want to achieve with your water system from the outset, because you want to know what size of storage you’ll have to build and, most importantly, whether they’ll be possible to build due to your terrain and your budget.
First, think about how you are planning to use the water: do you need water for household use, livestock, irrigation, fish production, fire protection, recreation…?
Following this, try to get a ballpark estimate of how much water you’ll need for each of these activities - calculate how much you need.
Finally, think about what, realistically, you can build. Here, your budget, available space and aesthetics are all factors you need to consider.
This thinking process can eliminate a great deal of unnecessary planning and will help you prioritize based on the reality of your situation. That’s why we always start with being clear on your context and your goals: The best way to save money on a project is not to start it in the first place!
2) Identify the sources of water
2)确定水的来源
Once you have an idea on your water needs and how you plan to use your harvested water, let’s see what water sources are available to your farm. As I explained in the last post, you can find this out by searching for information online and by reading the landscape.
So, the first question you should be asking yourself is: how much precipitation am I getting within the year - what is my average annual rainfall measured on mm or inches? Secondly, how is that precipitation distributed throughout the year? Is it being delivered in heavy downpours, only during the winter, or equally distributed throughout the year?
Your water systems will be completely different based on these numbers. If you get 600mm as an average rainfall and most of it falls in a few huge summer storms, this will require a completely different approach than 1200mm equally distributed throughout the year.
The precipitation and its distribution will be the foundation for your planning, and you can find this crucial information easily on the Internet just a few clicks away. Now, for other sources of water on your property and beyond, you’ll have to do some permaculture detective work.
You’ll want to distinguish any streams that are running across your property. This flowing water is essentially runoff from outside the boundaries of your property and within your watershed. You can’t control how this water gets onto your site, but you can use it for your water needs if necessary. That’s why you need to know the precise reliability of your water source. Is it perennial or just seasonal? Can you count on it when there is a drought?
Lastly, consider if there is undergoing water that’s available to you. As I outlined in the landscape post, you can’t reliably tell how much water you’ll have under your feet unless you drill a well, but there is a tell-tale sign you can spot in the landscape: If you have any groundwater, it’s another water source which you should take into account.
3) Watershed - determining your place in the hydrologic cycle and your site’s watershed
3)分水岭——决定你在水文循环中的位置和你的场地的分水岭
Okay, so once you’ve got an idea about the precipitation you’re getting and other water sources available to you, you can start by analyzing your watershed and determining your place of the hydrologic cycle.
Every piece of land belongs to a watershed, and it’s defined as an area of land that drains runoff from rain or snow downhill from the highest geographical barriers, such as hills, ridges and mountains, to a specific low point, generally a tributary outlet to a larger river or a lake.
On a larger scale, your land is almost sure to be a part of a regional watershed that drains thousands of square miles or kilometers of land, creating streams and rivers. Although knowing your regional watershed might not have an immediate use to you, I would recommend that you first look at the broader watershed.
Water movement on your site or within your area is a function of where you are in the overall watershed. For example, if you’re high in the hills, you’ll have a small flow of water, probably some small creeks, but on the other hand, if you’re low in the landscape, there be lots of water, probably rivers rather than creeks.
However, to access your site’s actual water resources, you’ll have to look at your site’s watershed or the sub-watershed. You might belong to an extensive watershed, but the precise quantities will depend on the local site’s terrain.
Nothing can be more critical to this process of identifying your site’s watershed than understanding the land patterns represented by topographical maps. For this, you’ll have to be able to recognize the contours for their definition of ridges, saddles and valleys/gullies. This is essential for the effective calculation of catchments.
To start assessing your site’s watershed, you’ll have to define boundaries of your property and the watershed directly affecting your site. You can do this by looking at a topographic map and identifying the divide lines (or center lines) on the ridges. The lines located at the tip of the ridges determine if water is flowing toward or away from your location. Find those lines…
Once you know where they are you’ll have an idea of the boundaries of that catchment and, by using simple math or online tools, you’ll get an estimate of the size of this surface area.
4)计算你的场地的降雨量-你的用水预算
After you have an idea of the size of your watershed and the average rainfall you’ll receive, it’s easy to calculate your water budget. The first step in doing so is to multiply these two numbers. Calculating rainfall volumes or total rainwater you can get = watershed area x average yearly rainfall.
This number gives you the total rainwater volume assuming there is a 100% runoff, and since we’re not calculating runoff from a concrete patio or a metal roof, we need to adjust that number. You can get a ballpark estimate of runoff volume from any sloped surface by multiplying the volume of rain that falls on that surface by its runoff coefficient.
The runoff coefficient is the average percentage of rainwater that runs off a certain type of surface, and it all depends on what the surface is composed of and the rain intensity - the higher the intensity, the higher the runoff coefficient. I won’t go into details here about calculating rainfall volumes using different coefficients. Instead, if you’re going to run some equations yourself, you can use the following tables from this Darren Doherty’s article…
Now that you’ve calculated you site’s rainfall volume, and taken into consideration any surface streams and underground water, it’s time to circle back to your goals and needs to see is this water budget is sufficient to sustain your needs, along with needs of your crops and livestock. Here’s where reality kicks in and you’ll see what’s possible and what isn’t…
### Storing Water in the Soil 在土壤中储存水
Okay, so let’s now start with storing the water in the soil. The cheapest place to store water is in the soil - it’s the largest storage resource available on most sites. Maybe you have big plans for interconnected network of cascading ponds but let’s first cover the essentials that won’t cost that much money. Our initial efforts should always be to get water into the ground and store it there.
To store water in the soil you have to focus on two objectives. The first is to slow, spread and sink the rainfall so that the water takes the longest possible path across your land, rubbing to as many things as possible, spreading where it’s needed, giving it time to infiltrate before it eventually leaves your site and drains away.
Your second objective is to build the soil’s organic matter, because the key to the soil’s capacity to hold water is the organic matter. The organic matter acts as a sponge and absorbs the water that’s slowly moving across the landscape. So it’s imperative that, if you want to store more water in the soil, you must promote organic-matter-rich topsoil.
Research shows that soil with as little as 2% organic matter can reduce the irrigation needed by 75% when compared to poor soils having less than 1% organic matter. Therefore, you’ll want to focus on developing the soil sponge.
You’ll also need to shape the land in such a way as to slow-spread and sink water for that sponge to absorb. To do this you can use two very famous techniques: 1. keyline plowing/subsoiling and, 2. Swales on the contour. Let’s start with keyline…
Keyline plowing /subsoiling - Keyline Pattern Cultivation
Keyline耕作 / 深松. Keyline模式栽培
The concept of keyline agriculture emerged from the drylands of Australia thanks to P.A. Yeoman. This now legendary Australian bloke has shaped how we permaculturists think about managing water on the farm.
While keyline agriculture contains many concepts, its most fundamental is to spread the abundance of water from where it is concentrated in we****t areas to areas that are consistently too dry. You see, normally water flows from ridges into valleys. The ridges stay dry, and the valleys accumulate moisture.
However, by using a keyline cultivation pattern, you can channel the water away from the valleys and towards the ridges, and, by so doing, distribute it evenly over the land and increase the infiltration. This is achieved by using the tractor and ripping lines (opening up furrows in the soil) with a keyline plow parallel to keyline (thus giving the name keyline cultivation pattern).
These small water channels in the soil, these hundreds of small drains, will then intercept water that flows down toward the valleys and move it in the other direction, toward the ridges. The net effect is that rip lines hold water for infiltration, instead of the water running down the slope. With more water in the soil, plant growth and soil microbes increase.
Keyline cultivation is also a soil improvement system, as it promotes rapid topsoil formation. As you create furrows in the soil and rip the subsoil you allow water and air to infiltrate deeper into the soil where they can be used by plants. This can break up the hard pan and build rich fertile soils, and, as you already know, as soil becomes fertile, more water can be absorbed and stored.
So, now you can see why keyline pattern cultivation is such a great tool in managing water on a permaculture farm. It can harvest rainwater, distribute it equally and build rich, fertile soils by turning subsoil into topsoil. I don’t want to make this post longer than it has to be, so I won’t go into how to find a keyline at this point, instead you can read P.A. Yeoman’s book Water For Every Farm.
Swales on contour
Your second strategy for storing water in the soil is by using swales. Swales also help us to slow, spread, and sink water, allowing us to hold off the runoff water and allowing it to seep into the soil, thus storing it there.
In his book Gaia's Garden, Toby Hemenway describes a swale as: a shallow trench laid out dead level along the land’s contours. It can be anything from one to several feet across, a foot or so deep, and whatever length necessary. The earth dug from the swale is piled on the downhill side to make a raised mound or berm.
During the rain event, once the soil can’t absorb the falling rain any longer, overland flow occurs. Whatever water the soil can’t absorb flows downhill as runoff. As that surface water and rainwater runs downhill it is intercepted by the swale, spreads out along its length, and slowly percolates into the soil.
This underground water then seeps downslope, forming a lens of moisture. The stored water creates an underground reservoir that aids plant growth for tens of feet below the swale. Most importantly, swales are tree-growing systems; by planting trees or other crops on the mound (berm) on the downhill side of the swale (or just below it) they’ll be able to take advantage of this soil moisture during dry periods.
We primarily use swales for this purpose, but swales also prevent gullies from forming by intercepting rainwater, slowing it, spreading it, essentially decreasing its erosive potential. Swales also trap organic matter and the ditch becomes a rich, thick layer of humus which holds a considerable volume of water. Moreover, once you have it dug out, you can bring that organic matter in, fill it out with wood chips, dead branches….
Now I know that, after hearing about swales, you’ll be eager to implement them on your land, but would they work on your property? Swales are the most widely used and abused permaculture water-management technique. There are many factors that influence whether or not you swale your property, depending on your slope, soils, hydrology, type of management, ecosystem’s condition and resource base.
Generally, swales are most appropriate for slopes of 5% or less. The size of watershed, the climate, the soil type, and the land use determine how much water flows off the land and into swales. Small watersheds, sandy soil, and forested areas won’t produce much runoff. Conversely, large watersheds, and soils with clay and loam, shed more water. The location’s climate also plays a part, because some areas are more likely to experience intense storms with more runoff.
Here is a nice infographic from Ben Falk I found online, which explains whether or not you should swale.
### Storing Water on the Surface 在地表储存水
Okay, once you’re done with storing water in the soil, and developing that cheap water storage in the soil, let’s move to storing water on the surface. Here we’ll be developing water systems that will store, harvest and reticulate surface water.
Water Storage (options)
On the surface, you can store water in the ponds/dams and in the water tanks…
If you need to store anything less than around 100 000 liters of water or just need drinking water, then a water-storage tank is potentially a cheaper and better option. Moreover, your site terrain might indicate that a pond construction would be too expensive so, yet again, a tank is a better option.
You can construct water tanks from various materials and, if you position them somewhere on the top of your property, at highest practical point, you’ll have a source of effective gravity storage in conjunction with, for example, a lower level pond, stream or groundwater.
The cheapest way of storing large volumes of water (more than 100 000 liters) is in a water-storage dam or pond. In a changing climate, water in a pond is an enormous asset to have, you can use it for many different purposes at once - for aquaculture, irrigation, stock and domestic storage, wildlife habitat, recreation and more…
Generally speaking, there are two type of ponds/dams - an embankment pond and an excavated pond. An embankment, as the name suggests, is made by building an embankment or dam across a stream or watercourse where the stream valley is depressed enough to permit storing reasonable amounts of water.
An excavated pond is made by digging a pit or dugout in a nearly level area. Because the water capacity is obtained almost entirely by digging, excavated ponds are used where only a small supply of water is needed. Some ponds are built in gently to moderately sloping areas and the capacity is obtained both by excavating and by building a dam.
Now, what type of a pond you’ll be able to construct and, most importantly, where, depends on your site’s terrain. Different pond types and locations have different storage ratios (the volume of excavation versus the volume of storage) and this is the most important factor in determining how viable a potential site will be. When constructing a pond, what you want to ensure is to make a minimal investment in both time and earthworks for a maximal amount of storage.
The type and dimensions of the pond will also depend upon the climate and the amount of average evaporation losses. In semi-arid and arid zones the amount of evaporation will be quite significant in comparison with cooler climates. Ponds in the hotter zones need to be deep in order to overcome annual evaporation losses…
With this in mind, let’s go through different pond types from the most economical and easiest to dig to the more expensive ones that require more extensive earthworks. In so doing, the first rule of working with water is to keep it in its place of highest potential on the landscape, up high if it can be economically placed there. So, we’ll start from the locations up the hill and go downhill.
• Gully/ 沟渠 /Keypoint Ponds 池塘
These are probably the most common of all dams and one of the easiest storage options. Since they are constructed by building an embankment in a gully or in a drainage depression, they are also the most economic option. The earthworks required come down to building a dam wall that needs to be capable of keeping the water in a gully/valley behind it.
The right way to pinpoint the best pond location in the gully/valley would again be by using keyline design principles. In this context, this means first identifying the major keypoint of the slope (where the gully/valley slope section changes from a concave to convex profile), once you know where the keypoint is located, the contour line on the landscape that goes through the keypoint is the keyline.
This keyline is the highest contour in the gully/valley that can efficiently hold water, and usually the highest overall practical point in the landscape to hold water. The main use for keypoint dams/ponds is to store irrigation water. This irrigation water is then generally released though the large pipe going underneath the dam’s wall.
A saddle is a topographic feature - this is simply a dip or break along a level ridge crest. Since it’s on a ridge, this is the high ground and the highest available water storage in the landscape**.** This pond has a much smaller watershed than a gully/valley pond, but still can collect water runoff from both sides of the ridge crests. The primary use of a saddle dam is for wildlife and domestic stock, not so much for irrigation.
• *Hillside/Contour Ponds 山坡 / 等高池塘*
Contour or hillside ponds are built on the side of hills and usually have a three-sided or curved bank or long, curved bank straight across the across the hillside slope (on the contour). The best way to locate these types of dam is to look at your topographic map and check for any widening of the contours along the hillside. Widening means that the terrain is flattening and this might be a good location for the pond.
These ponds are relatively expensive to build since you have to do more digging for less water storage, but they’ll still provide you with a gravity storage. Gravity-fed water still gets priority over ponds in the flat. They are usually filled by diversion drains or graded catch drains and have the same use as a saddle dam: for wildlife and domestic stock.
• Ponds for the flat sites: Excavated tanks*, Ring tanks, ‘Turkey’s nest’ ponds*
•平地的池塘: 挖出的水箱、环形水箱、“土耳其巢”池塘
Excavated tanks
Ring tank / dam
‘Turkey’s nest’ dam
All these are suitable for flat sites, and since they cannot capture runoff, they need to be filled from external sources. In excavated tanks, the excavation becomes the water storage, below the surface level. Earth removed is stockpiled nearby, unless additional dam walls are constructed for additional storage above ground level.
Ring tanks are constructed by using earth from inside the ring (circular or shaped to suit topography) to build the surrounding embankment. Water is generally stored above the natural surface. ‘Turkey’s nest’ dams are a variation of the ring tank where the borrow pit is located outside the embankment. Water is stored above ground level.
Water Harvesting
Once your water storage is ready, you need to develop and expand upon the methods of harvesting the water. Sure, you can fill your ponds with water from a well, but before you go deep and tap into the underground aquifer, you’ll want to use the surface flows and rainfall runoff to fill your water storage.
You can capture water with water-harvesting drains that will divert the runoff, stream flow or pumped water into your ponds, and subsequently tanks. Bill Mollison in his Designers Manual explains: These drains are actually trenches in the soil that aren’t dead level: they are placed in the landscape off-contour and have a slight gradient, with the goal of moving water to a specific location such as your pond.
You can think of diversion drains/ditches as being giant earthen gutters placed across the landscape to harvest and move water in a manner similar to rain gutters on a house. They differ from swales in that they are built to flow after rain and, unlike swales, which are normally built on permeable soils, diversion drains work better when the base and sides are clay-lined.
However, swales or ditches on contour can also harvest water for you, and if they are connected to your pond as they fill up, they will overflow to your ponds. Also, if you have a series of ponds connected with swales, then the overflow of one pond enters the feeder channel/swale of the next. Having a spillway for a pond is a must and this way you’ll once more be slowing, spreading and sinking water across your landscape.
Once installed, your roads themselves become a very important and efficient water-harvesting system. Since the roads are compacted, graded and often made of impervious materials, they have very high runoff coefficient. In certain landscapes like karts, the roads will be the only available runoff surface. The roads and the adjacent water collection drains can be then also integrated with other harvesting drains and/or swales, contributing to the overall hydration of the farm.
Water Distribution
When developing the water resources of a farm, there are two primary water channels in addition to already mentioned harvesting drains. The other type of water channel is there for irrigation purposes. These diversion drains, which are essentially the same thing as harvesting drains (placed in the landscape off-contour, have a slight gradient, moving the water) here function as irrigation channels for flow irrigation.
Water from a pond is directed into the drain, it fills up and overflows the top side, along its length, and cascades downhill over crops or paddocks… Basically, you can use these types of drain for your field irrigation of crops such as potatoes, corn and beans, or water your pastures.
In his book The Bio-Integrated Farm, Shawn Jadrnicek recommends that in order to build this drain, start at the outlet - a pond, retention basin, swale, or some other area with the capacity to hold and safely release the harvested water, and move down the slope towards the desired irrigated area.
Another way of reticulating water is the release of the water stored in ponds and water tanks by a gravity-fed pipe network. In adopting this approach, you will use your header water tank located at the highest point in your landscape and release the stored water to irrigate your orchards and gardens through the network of irrigation pipes. You could do the same on a bigger scale by using a pond as a source of stored water and a series of the irrigation reticulation pipes connected to it.
### Conclusion 总结
Managing water is crucial in designing and setting up a permaculture farm. No permaculture site has been properly planned unless it first considers how to use the available water resources.
The water systems you establish become a permanent feature of the new landscape and the base of permaculture land development planning. All the water lines: diversions, swales, terraces, dams/ponds, channels, form the foundation that other infrastructure components (structures, farm roads, fencing) will follow.
In summary:
• You want to boost the overall resilience of the farm by first storing water in the soil, this is done by building soil’s organic matter, using swales and keyline cultivation pattern. 你想通过首先在土壤中储存水来提高农场的整体弹性，这是通过建立土壤的有机物质，使用洼地和 集水线keyline 耕作模式来实现的。
• Once you have evenly hydrated the whole landscape you can capture the runoff rainfall with harvesting drains and fill up your water tanks and ponds. Keep this water for times of drought and release it through irrigation channels and pipes to rehydrate the landscape while you’re waiting for the water droplets to return.
一旦你均匀地给整个场地补水，你就可以用排水沟收集径流降雨，捕获的并填满你的水箱和池塘。在干旱无雨时期保存这些水，并通过灌溉渠道和管道释放这些水，给土地和植物补充水分。
I hope this post has helped you to better understand water management on a permaculture farm. If you found it useful, please share it with anyone interested in permaculture water management.
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2023-03-21 17:34:52
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https://chat.stackexchange.com/transcript/3740/2019/6/17
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6:46 AM
A new tag was created by M.Mass.
2
I have the system (took very basic example on purpose, to understand the idea): $$\begin{cases} \dot{x} = x \\ \dot{y} = 2x -y \end{cases}$$ so I have plot phase plane. what have been done so far: $$A = \begin{pmatrix} 1 && 0 \\ 2 && -1 \end{pmatrix}$$ \det A = \begin{vmatrix}1 - \lambda &&...
A new tag was created by Yanior Weg. The same user created a tag-excerpt.
6 hours later…
12:33 PM
Two new tags and created by supremum in the same question. (In fact, was created there too, but removed in a subsequent edit.)
0
How do we prove the similarity dimension equals the Hausdorff dimension if the self-similar set satisfies the open set condition? Which article contains this proof?
3 hours later…
3:30 PM
@MartinSleziak Ugh... I really don't think that we need any of those tags... :\
But subsequent edits have gotten rid of all of them.
3:42 PM
FYR: self-similar solution is very commonly used in PDE, so it's probably not a good idea to have the "self-similar" tags.
@ArcticChar Indeed. I could, perhaps, see the use for a "self-similar-fractals" tag, but my guess is that the set of questions with that tag and the "fractals" tag would be nearly identical.
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2019-08-24 01:14:16
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https://socratic.org/questions/how-much-heat-is-produced-when-29-6-g-of-methanol-react-with-45-9-g-of-oxygen
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# How much heat is produced when 29.6 g of methanol react with 45.9 g of oxygen?
Nov 19, 2016
You need to supply the enthalpy of combustion of methanol.
#### Explanation:
Methanol combusts according to the reaction:
${H}_{3} C O H + \frac{3}{2} {O}_{2} \left(g\right) \rightarrow C {O}_{2} \left(g\right) + 2 {H}_{2} O + \Delta$
This reaction will be exothermic. Just how exothermic it will be is subject to measurement, which you have not quoted.
$\text{Moles of methanol} = \frac{29.6 \cdot g}{32.04 \cdot g \cdot m o {l}^{-} 1} = 0.924 \cdot m o l$.
$\text{Moles of dioxygen} = \frac{45.9 \cdot g}{32.0 \cdot g \cdot m o {l}^{-} 1} = 1.41 \cdot m o l$.
Given the molar quantities, there is sufficient dioxygen for complete combustion. However, we still need the molar enthalpy of combustion.
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2019-08-21 00:47:17
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http://latexref.xyz/Command-line-input.html
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Next: , Previous: , Up: Command line [Contents][Index]
28.2 Command line input
As part of the command line invocation pdflatex options argument you can specify arbitrary LaTeX input by starting argument with a backslash. This allows you to do some special effects.
For example, this file (which uses the hyperref package for hyperlinks) can produce two kinds of output, one for paper and one for a PDF.
\ifdefined\paperversion % in preamble
\newcommand{\urlcolor}{black}
\else
\newcommand{\urlcolor}{blue}
\fi
...
\href{https://www.ctan.org}{CTAN} % in body
...
Compiling this document book.tex with the command line pdflatex book will give the ‘CTAN’ link in blue. But compiling it with pdflatex "\def\paperversion{}\input book.tex" has the link in black. (Note the use of double quotes to prevent interpretation of the symbols by the command line shell; your system may do this differently.)
In a similar way, from the single file main.tex you can compile two different versions.
pdflatex -jobname=students "\def\student{}\input{main}"
pdflatex -jobname=teachers "\def\teachers{}\input{main}"
The jobname option is there because otherwise both files would be called main.pdf and the second would overwrite the first.
A final example. This loads the package graphicx with the option draft
pdflatex -jobname=aa "\RequirePackage[draft]{graphicx}\input{aa.tex}"
so the graphic files are read for their size information but not incorporated into the PDF. (The jobname option is there because otherwise the output file would be graphicx.pdf, as \RequirePackage does an \input of its own.)
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2019-02-20 19:14:03
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https://proofwiki.org/wiki/Definition:Inverse_of_Subset_of_Group
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# Definition:Inverse of Subset/Group
Jump to navigation Jump to search
## Definition
Let $\struct {G, \circ}$ be a group.
Let $X \subseteq G$.
Then the inverse of the subset $X$ is defined as:
$X^{-1} = \set {x \in G: x^{-1} \in X}$
or equivalently:
$X^{-1} = \set {x^{-1}: x \in X}$
## Examples
### Subset of $\R$ under Multiplication
Let $\struct {\R, \times}$ be the multiplicative group of (non-zero) real numbers.
Let $S = \set {-1, 2}$.
Then the inverse $S^{-1}$ of $S$ is:
$S^{-1} = \set {-1, \dfrac 1 2}$
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2020-07-16 00:02:31
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https://www.electro-tech-online.com/threads/beginner-circuit-design-how-do-i-finish-this-thing-off.2908/
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# Beginner circuit design - how do I finish this thing off!
Status
Not open for further replies.
#### CNO81
##### New Member
The diagram attatched is for a 3 bit counter ive designed but I want to be able to see it working by simulating it in a program called 'Electronics Workbench' - how do I finish it off i.e wire it up (yes, its probably a laughable question) with an appropriate visual output so I can see it working on screen, bearing in mind I can't see any obvious visual components to add?
#### Attachments
• 38.1 KB Views: 235
#### e
##### New Member
well connect the A,A,B,B.... to the respective D flip-flop, A is the Q output of the A FF, A is the Q output... then put an indicator of some sort, LED.. on the outputs of the D-FFs
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Not open for further replies.
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2020-09-24 23:47:43
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https://ilovephilosophy.com/viewtopic.php?f=2&t=186424&p=2798478
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## What is your most recent purchase? - pics required
This is the place to shave off that long white beard and stop being philosophical; a forum for members to just talk like normal human beings.
### Re: What is your most recent purchase? - pics required
You see my advertising brilliance at work there. With the humorous approach the prospective buyer is disarmed to a degree, and really without knowing it. He's already trusting me because I'm comfortable enough to make a funny. I'm not forcing anything, I remain casual, there are no tricks, etc.
Next - and this for the younger buyers - I trigger subconscious response to an array of masculine symbols and subtexts. Mad max, the punisher, machine guns, military laboratories, etc.
Every buyer under the age of 25 is going to respond to this in a more favorable way; he's going to see himself in that blue jacket on that bike... and when he rides he becomes mad max or the punisher. I put this reverie into his mind.
promethean75
Philosopher
Posts: 4049
Joined: Thu Jan 31, 2019 7:10 pm
### Re: What is your most recent purchase? - pics required
Dude I'm gonna be honest with you, it was better the way it was before.
Now it sounds like you're pushing it, and with a bike that beautiful the question arises why possibly.
Maybe it's a negotiation stance? You will obviously not drop the price on the Punisher bike?
Pedro I Rengel
ᛈᛖᛉᛖᛉ
Posts: 8792
Joined: Mon Feb 05, 2018 2:55 pm
### Re: What is your most recent purchase? - pics required
phoneutria wrote:nope i don't need some hack
fair
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
Mr Reasonable
resident contrarian
Posts: 28461
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
### Re: What is your most recent purchase? - pics required
promethean75 wrote:Reas and/or Peez, I got a deal for ya....
https://raleigh.craigslist.org/mcy/d/ga ... 07889.html
2 things. 1, the ad is too wordy. just put SV1000 X miles, well maintaned. $price firm 2 im not sure of the market there, but it'll be spring before you get 3700 for it. its possible, but youll have to find just the right buyer. my friend bought this one for 2900 and sold it for 3300 after riding it for a season. good tires new chain and sprockets etc. at a certain point, the$ value of them gets down to a level where you really have to ask yourself, "why am i selling this?" you got a bike you could pull a little under 4k out of. what could you buy with that money that would be better than that bike?
here's an sv1000 next to the legendary RC51 honda. that RC is super mean.
Attachments
1007191219_resized.jpg (186.88 KiB) Viewed 85 times
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
Mr Reasonable
resident contrarian
Posts: 28461
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
### Re: What is your most recent purchase? - pics required
looks like yours is a litter newer with the front end upgrade so maybe you're closer on the price than i thought. still man, in the spring they are always worth more.
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
Mr Reasonable
resident contrarian
Posts: 28461
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
### Re: What is your most recent purchase? - pics required
i have an answer to my previous question about what you would spend the money on that youd get from selling that bike. you could spend it on one of these.
no one really rides these without asking themselves at least once if they are actually mental. it's like flipping a coin with your life. we all know this. get this gsxr1000 and just go all in. fuck leaving a beautiful corpse. it aint happening. ive been on a lot of fast bikes and theyre all a little different with the torque range and the power to weight and the balance and handling and this thing doth not suffer fools. it aint forgiving, and its twitchy af. but a rocket if there ever was one. im too old for this shit. i just want a remote starter on my car now and heated seats. this one was 5500 and sold for 6200 give or take. there may have been a gun or something included as part of the deal who even remembers.
Attachments
1028191033_resized.jpg (230.6 KiB) Viewed 84 times
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
Mr Reasonable
resident contrarian
Posts: 28461
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
### Re: What is your most recent purchase? - pics required
I want one with the fucking tricycle handles.
Pedro I Rengel
ᛈᛖᛉᛖᛉ
Posts: 8792
Joined: Mon Feb 05, 2018 2:55 pm
### Re: What is your most recent purchase? - pics required
whose motorcycle is this?
it's not a motorcycle, it's a chopper
whose chopper is this?
zed's
who's zed?
phoneutria
purveyor of enchantment, advocate of pulchritude AND venomously disarming
Posts: 4141
Joined: Fri May 23, 2014 5:37 am
### Re: What is your most recent purchase? - pics required
The low rider is a little slower.
Pedro I Rengel
ᛈᛖᛉᛖᛉ
Posts: 8792
Joined: Mon Feb 05, 2018 2:55 pm
### Re: What is your most recent purchase? - pics required
"looks like yours is a litter newer with the front end upgrade so maybe you're closer on the price than i thought"
Lol... no bike with 26,000 miles on it is worth $3,800. Remember I don't need to sell it... so I'm putting it out there at a ridiculous price to see if some knucklehead takes it. Real value is around$2,700 therebout.
promethean75
Philosopher
Posts: 4049
Joined: Thu Jan 31, 2019 7:10 pm
### Re: What is your most recent purchase? - pics required
_
I once sold a second-hand BMW for double what I paid for it, and all because some guy wanted it/was looking for that exact make colour and model.. Prom has faith in the other too, in being out there, in wanting something specific.
It had a shot diff, but that was minor for this guy, as he owned his own car repair shop and so replaced it for free, then gave the car to his younger sister as a birthday present.
The possibility of anything we can imagine existing is endless and infinite.. - MagsJ
I haven't got the time to spend the time reading something that is telling me nothing, as I will never be able to get back that time, and I may need it for something at some point in time.. Huh! - MagsJ
You’re suggestions and I, just simply don’t mix.. like oil on water, or a really bad DJ - MagsJ
MagsJ
The Londonist: a chic geek
Posts: 21523
Joined: Wed Nov 01, 2006 2:59 pm
Location: Suryaloka / LDN Town
### Re: What is your most recent purchase? - pics required
You see...a pimp's love is very different from that of a square.
Dating a stripper is like eating a noisy bag of chips in church. Everyone looks at you in disgust, but deep down they want some too.
Mr Reasonable
resident contrarian
Posts: 28461
Joined: Sat Mar 17, 2007 8:54 am
Location: pimping a hole straight through the stratosphere itself
Previous
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2021-01-24 19:32:52
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https://security.stackexchange.com/questions/179229/is-it-ok-to-use-6-as-a-cost-of-bcrypt-algortihm-for-64-chars-string
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# Is it ok to use 6 as a cost of bcrypt algortihm for 64 chars string? [duplicate]
This question already has an answer here:
What should the ideal $cost be in this case to prevent brute force attack even by ASIC or an FPGA $cost = '06';
$secret_key = bin2hex(random_bytes(64));$store_data = crypt($secret_key, '$2a$.$cost.$'.bin2hex(random_bytes(16)).'$')
Scenerio
Company gives user $secret_key which is generated by above code to give access to withdraw some money and this secret key should never be guessed by hacker. $store_data is the string that we stored in our database.
So if an attacker gets access to the database and learns the $store_data variable he should not be able to obtain $secret_key.
I have used the bcrypt algorithm to hash but I am not sure if $cost = 6 is secure enough in this case. Note: I didn't use password_hash because I will convert all code into Java in the future and in Java there is no equivalent of password_hash ## marked as duplicate by Sjoerd, Steffen Ullrich, forest, Tobi Nary, DKNUCKLESFeb 9 '18 at 2:49 This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question. • If you are worried about hardware attacks you may consider using scrypt, which is harder to optimize in hardware than bcrypt. – Sjoerd Feb 6 '18 at 9:44 • I read that scrypt is less secure than bcrypt – Alex Robertson Feb 6 '18 at 11:07 • For 64 bytes of random data bcrypt isn't even necessary, just use a fast cryptographic hash like SHA256. – AndrolGenhald Feb 6 '18 at 15:00 • what about in the future? I mean I read that it is unlikely someone will find the way to broke bcrypt but they can broke SHA256 @AndrolGenhald – Alex Robertson Feb 6 '18 at 15:51 ## 1 Answer If you have 64 truly random (i.e., cryptographically random) bytes, you can use any hashing algorithm that provides preimage resistance to securely store it and prevent an attacker from learning the input. Algorithms like scrypt and bcrypt provide work factors to make it harder to perform dictionary attacks against the hashes. This is because user-selected passwords tend to have much lower entropy then is ideal for their use case. With 64 bytes of random data, you can use any cryptographic hashing algorithm with 256 bits of output without any concern about brute force attacks, because it is physically impossible to brute force a 256-bit value. Just iterating over all possible 256-bit values takes more energy than the sun can produce in its lifetime. Enumerating all possible 128 bit keys requires the total energy usage of Spain for 1 year. Calculating the hashes is even more. Going from 128 to 129 bits doubles the energy requirement, 256 bits requires more energy than the sun can produce. This is a minimal limit of energy storage called the Landauer Limit. (See How Confident Can We Be That Nobody Will Crack a 128-bit key and Wikipedia: Brute Force Theoretical Limits) My advice, then, is to use SHA-256. • I wouldn't say it's physically impossible, even if it is well out of reach for humanity today. – forest Feb 7 '18 at 2:27 • Added more notes on the physics limits – David Feb 7 '18 at 2:44 • if I use bcrypt, what you can suggest to use as an $cost param? – Alex Robertson Feb 7 '18 at 8:34
• @erickson I read your answer in stackoverflow.com/questions/1561174/… What could you suggest in this case? – Alex Robertson Feb 7 '18 at 8:37
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2019-10-16 07:33:36
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https://www.physicsforums.com/threads/is-in-the-one-way-current-have-selfinduction-and-if-yes-why-if-no-why.169050/
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# Is in the one-way current have selfinduction? and if yes why, if no why?
## is in the one-way current have selfinduction?
0 vote(s)
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0 vote(s)
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1 vote(s)
100.0%
1. May 6, 2007
2. May 6, 2007
### ZapperZ
Staff Emeritus
If you wish someone to put some EFFORT in answer your question, you should at least put in the same amount of effort in present a clear and unambiguous question. This will not cut it. The geometry of the current make a heck of a difference here. The self-inductance of a straight line current will be different than a coil. Your question gives no such clarification.
Zz.
3. May 6, 2007
### scientist91
But isn't it that the electrons which are going accross conductor have magnetic field, so they are moving and the magnetic field is moving so there is some electromagnetic induction? Logically.
4. May 6, 2007
### lalbatros
Please, before discussing anything, start by translating this sentense:
in an intelligible language.
5. May 6, 2007
### scientist91
If u are from Mars u will not understand for sure.
6. May 6, 2007
### lpfr
I must be martian.
7. May 6, 2007
### jambaugh
I think the best way to express this problem unambiguously is to imagine two widely separated oppositely charged spheres connected through a switch by a straight wire. Then the question is:
When the switch is thrown will the spheres once discharged, recharge somewhat with opposite polarity due to the self induction of the current through the wire. In short is this an LC circuit.
The answer I am certain is: Yes. My well trained intuition tells me this must be the case else electromagnetic waves could not propagate. Even the virtual Maxwell current due to a changing E field must have a component of "self induction" in order for the E-M wave to propagate. However there is a slight chance I am wrong. So do the math or better yet do the physical experiment.
Further I think the calculation of the self inductance per unit length will be the same as if you take the limit on a square loop of wire as the size goes to infinity.
Also try working out the self-inductance per unit length of a coaxial cable as the radius goes to infinity.
Regards,
James Baugh
8. May 6, 2007
### scientist91
Can u explain simpler please. Thank u.
9. May 6, 2007
### Xezlec
If you are asking whether a wire, by itself, has any self-inductance, then yes.
If you are asking whether a constant current flowing in that wire is at all affected by that self-inductance, then no, because inductance doesn't "do" anything unless you have time-varying voltages/currents.
If you are asking anything else, I'm afraid you'll need to spend a little more time working out the English grammar of your sentences. We can't really understand you. Sorry!
10. May 6, 2007
### lpfr
Hard to make simpler than this. If u are from Mars u will not understand for sure.
I totally agree with jambaugh.
The self-inductance per unit length in a coaxial cable is:
$${L\over\ell}={\mu\over2\pi}\ln{R_2\over R_1}$$
R1 and R2 are the radius of the internal and external conductor. When the external radius goes to infinity, self-inductance per unit length tends also to infinity.
11. May 6, 2007
### scientist91
is in the direct current have selfinduction (electromagnet induction)? and if yes why, if no why? Anybody from here have studied physics?
12. May 6, 2007
### Xezlec
OK, I'm pretty sure I already answered your question. Is there some part of my response you are having trouble understanding?
Also, it's irritating that you keep asking the question exactly the same way, after you've been told that your English is completely incomprehensible. Please rephrase if you are not satisfied with any of these answers.
Last edited: May 6, 2007
13. May 6, 2007
### scientist91
What you can't understand in the question? btw- I didn't understand nothing.
14. May 6, 2007
### Office_Shredder
Staff Emeritus
The problem is that your sentence makes no grammatical sense.
For example, if you said "Is there self-inductance by a direct current in a cable?", that would be much clearer. It appears to be approximately what you're asking, but since that question was already answered, you must not mean that.
15. May 6, 2007
### lpfr
Yes. Longtime and better than you.
We have also learned from our parents to be polite.
16. May 6, 2007
### lalbatros
English is not my mother tongue.
But I am quigte sure that this sentense:
- is gramatically incorrect
- cannot be understood, even approximately, by a majority of people
- could not describe a real question in physics, even if corrected for its poor grammar
If physicsits are martians, then I am one for sure, I have a complete pedigree in physics and engineering as well.
I like precision.
17. May 6, 2007
### scientist91
what is incorrect?
self-induction?
ANSWER ON MY QUESTION IF U KNOW THE ANSWER, DON't TRY TO IMPROVE MY ENGLISH THERE IS ANOTHER TOPIC FOR THAT. SO BE POLITE PLEASE, IF U WANT THE SAME FROM ME>
18. May 6, 2007
### lalbatros
The problem with this sentense is that there are two verbs.
If you were polite, you would at least try to understand why nobody answers your question and you would not assume people here are not physicists, which is really laughable.
All you gave in return is insults.
You are not a gentlemen, nor a physicist.
Or show us if you are.
Last edited: May 6, 2007
19. May 6, 2007
### lpfr
I support totally lalbatros.
I think that a moderator should close this thread.
20. May 6, 2007
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2017-02-26 10:32:45
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https://math.stackexchange.com/questions/3427013/is-o-in-bodmas-really-for-of
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# Is 'O' in BODMAS really for 'of'?
We were trying to solve a BODMAS question (with many solutions like always) when I came across this link.
https://www.mathsisfun.com/operation-order-bodmas.html
For the entire life, I have studied (and think that you might have also studied) that O in BODMAS is for 'of' meaning multiplication when * is not specified but bracket form is mentioned, like: 2(2+2) where 'of' makes it 2*(2+2). However, this website is claiming that 'O' means order, i.e., equations containing powers and roots. Is this really true? Seems like the world is shaking.
Also, can you please tell me what should be the answer of the following equation according to you?
3*{8/2(2+2)}+2
Note: Please bear with me if have been studying wrong my entire life.
• See Order of operations : BODMAS : it is only mnemonic; nothing really "deep" here. See Pedmas, where "E" stands for "Exponents". – Mauro ALLEGRANZA Nov 8 '19 at 9:57
• If $*$ is exponentiation, 2*(2+2) will be $2^{(2+2)}=2^4$. This is not about order of operations (we have enough parentheses to disambiguate the formula) but about the way to express exponentiation. Mult is $\times$, e.g. $2 \times 2$, while Exp is $a^b$. – Mauro ALLEGRANZA Nov 8 '19 at 10:02
• For your last question, see math.stackexchange.com/questions/33215/what-is-48%c3%b7293. It's ambiguous, and that's it. – Arthur Nov 8 '19 at 10:03
• Well, I learnt it as "BEDMAS" with "E" standing for "Exponents" in my computer textbook, which was the order if operations carried out by the computer. – AryanSonwatikar Nov 8 '19 at 10:03
• The answer regarding 3*{8/2(2+2)}+2 is very simple: Never write an expression this way that bears the risk of being ambiguous – Hagen von Eitzen Nov 8 '19 at 10:52
Through out my life I have used o for 'of' , meaning the bracket operation Solution :-
3×{8÷2(2+2)} +2 .......( If l am interpreting same you want to ask)
3×{8÷2(4)}+2
3×{8÷8}+2
3×1+2
3+2
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2020-02-20 11:49:50
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https://www.knowpia.com/knowpedia/Genus_(mathematics)
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BREAKING NEWS
Genus (mathematics)
## Summary
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface.[1] A sphere has genus 0, while a torus has genus 1.
A genus-2 surface
## Topology
### Orientable surfaces
The coffee cup and donut shown in this animation both have genus one.
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.[2] It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ = 2 − 2g − b. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort.
For instance:
• The sphere S2 and a disc both have genus zero.
• A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke "topologists are people who can't tell their donut from their coffee mug."
Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.
In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has.[3]
### Non-orientable surfaces
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.
For instance:
### Knot
The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K.[4] A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
### Handlebody
The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it.
For instance:
• A ball has genus 0.
• A solid torus D2 × S1 has genus 1.
### Graph theory
The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of the genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles.[5]
In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept. The genus of a group G is the minimum genus of a (connected, undirected) Cayley graph for G.
## Algebraic geometry
There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus.[7] When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points). For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.
By the Riemann–Roch theorem, an irreducible plane curve of degree ${\displaystyle d}$ given by the vanishing locus of a section ${\displaystyle s\in \Gamma (\mathbb {P} ^{2},{\mathcal {O}}_{\mathbb {P} ^{2}}(d))}$ has geometric genus
${\displaystyle g={\frac {(d-1)(d-2)}{2}}-s,}$
where s is the number of singularities when properly counted.
## Differential geometry
In differential geometry, a genus of an oriented manifold ${\displaystyle M}$ may be defined as a complex number ${\displaystyle \Phi (M)}$ subject to the conditions
• ${\displaystyle \Phi (M_{1}\amalg M_{2})=\Phi (M_{1})+\Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1}\times M_{2})=\Phi (M_{1})\cdot \Phi (M_{2})}$
• ${\displaystyle \Phi (M_{1})=\Phi (M_{2})}$ if ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ are cobordant.
In other words, ${\displaystyle \Phi }$ is a ring homomorphism ${\displaystyle R\to \mathbb {C} }$ , where ${\displaystyle R}$ is Thom's oriented cobordism ring.[8]
The genus ${\displaystyle \Phi }$ is multiplicative for all bundles on spinor manifolds with a connected compact structure if ${\displaystyle \log _{\Phi }}$ is an elliptic integral such as ${\displaystyle \log _{\Phi }(x)=\int _{0}^{x}(1-2\delta t^{2}+\varepsilon t^{4})^{-1/2}dt}$ for some ${\displaystyle \delta ,\varepsilon \in \mathbb {C} .}$ This genus is called an elliptic genus.
The Euler characteristic ${\displaystyle \chi (M)}$ is not a genus in this sense since it is not invariant concerning cobordisms.
## Biology
Genus can be also calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.[9]
## Citations
1. ^ Popescu-Pampu 2016, p. xiii, Introduction.
2. ^ Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.
3. ^ Weisstein, E.W. "Genus". MathWorld. Retrieved 4 June 2021.{{cite web}}: CS1 maint: url-status (link)
4. ^ Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
5. ^ Graphs on surfaces.
6. ^ Thomassen, Carsten (1989). "The graph genus problem is NP-complete". Journal of Algorithms. 10 (4): 568–576. doi:10.1016/0196-6774(89)90006-0. ISSN 0196-6774. Zbl 0689.68071.
7. ^ Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 978-3-540-58663-0. Zbl 0843.14009.
8. ^ Charles Rezk - Elliptic cohomology and elliptic curves (Felix Klein lectures, Bonn 2015. Department of Mathematics, University of Illinois, Urbana, IL)
9. ^ Sułkowski, Piotr; Sulkowska, Joanna I.; Dabrowski-Tumanski, Pawel; Andersen, Ebbe Sloth; Geary, Cody; Zając, Sebastian (2018-12-03). "Genus trace reveals the topological complexity and domain structure of biomolecules". Scientific Reports. 8 (1): 17537. Bibcode:2018NatSR...817537Z. doi:10.1038/s41598-018-35557-3. ISSN 2045-2322. PMC 6277428. PMID 30510290.
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2023-03-28 05:33:59
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https://socratic.org/questions/59ad7647b72cff37c78ca53b
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# Question #ca53b
Sep 4, 2017
Approximately $8.82 \cdot {10}^{-} 3 m o l$ or $1.72 g$ of platinum.
#### Explanation:
This is a very vague question, but I will assume you're looking for how much they weigh.
$\left(5.31 \cdot {10}^{21} m o l .\right) \cdot \frac{m o l}{6.023 \cdot {10}^{23} m o l .} \approx 8.82 \cdot {10}^{-} 3 m o l$
Note that mol. is referring to molecules and mol (without the period) is referring to moles.
$8.82 \cdot {10}^{-} 3 m o l \cdot \frac{195.1 g}{m o l} = 1.72 g$
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2019-08-22 02:32:57
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http://mathhelpforum.com/calculus/1895-try-iota.html
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# Math Help - Try iota
1. ## Try iota
Just try fittin in iota in your calculus equations.
eg. x^2 = - (ix)^2
you can then use e^i@ = cos(@) + i sin(@)
try it this idea is a serious whirlwind.
2. ## Serious?
Can you further explain what you are trying to say here?
3. Originally Posted by MathGuru
Can you further explain what you are trying to say here?
Perhaps he's trying to say that the use of complex variables makes many
problems simpler.
RonL
4. have you ever heard of i being referred to as iota?
5. Originally Posted by MathGuru
have you ever heard of i being referred to as iota?
Perhaps in Greece?
RonL
6. ## iota
I think abu is from India (I looked up the IP address), but ok.
7. ## Yes I Am An Indian
Yes i am saying it makes problemz simpler, you can also use it if u dont remember a formula etc.
for example (let | be the integration sign)
| root of(a^2 - x^2) dx = | root of(x^2 + a^2) dx
this worx as i is not dependent on x
but is i a const?
8. Originally Posted by abu
Yes i am saying it makes problemz simpler, you can also use it if u dont remember a formula etc.
for example (let | be the integration sign)
| root of(a^2 - x^2) dx = | root of(x^2 + a^2) dx
this worx as i is not dependent on x
but is i a const?
If this were an identity it would be true when $a=0$,
but:
$
\int \sqrt(-x^2)\ dx=\int \sqrt(x^2)\ dx
$
so:
$
i\ \int |x| dx = \int |x| dx
$
which is false even if we throw in a random $\pm$ sign or two
RonL
9. Cut and paste from wikipedia:
"The lowercase Iota symbol is sometimes used to write the imaginary unit but more often latin i or latin j are used."
Sounds good to me!
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2014-03-13 23:26:16
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https://www.gamedev.net/forums/topic/384976-pass-parameters-to-php-include/
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# [web] pass parameters to php include
This topic is 4464 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
how can i pass parameters to an include file in php? lets say i have an id i want to pass to it, usually i could pass parameters on the command line via: myphp.php?id=4 which doesnt work when im in another php script and i want to include myphp.php.
##### Share on other sites
You don't need to pass parameters to include files, think of an include and literally including one file into another there for you can use the variable which are in scope where the include() was called...
for example...
start.php:
<?php// Set myVar to 27$myVar = 27;// Include myinclude.php into start.phpinclude('myinclude.php');?> myinclude.php: <?php// echo the value of a variable called myVarecho 'myVar = '.$myVar;?>
However, I would like to point out that I feel this technique is bad coding practice, it's not wrong and it does work and is completely legal, but I feel you should really have functions/classes in the files you and include and pass the variables in by parameters to the functions/members when you call them rather than rely on variables which are(or maybe) declared elsewhere. Hope that makes sense.
##### Share on other sites
hrmm yeah its really ugly, i'll post my problem:
i have this page, which takes as parameters the body page, and adds on a header and footer automatically. So the header getts written, then my php script is included, then the footer is added on.
Problem here is if i want to redirect users. I cant use the header redirect, unless i use this:
http://au3.php.net/manual/en/function.ob-start.php
which also looks ugly. Any ideas/suggestions appreciated
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Quote:
Original post by supaguhrmm yeah its really ugly, i'll post my problem:i have this page, which takes as parameters the body page, and adds on a header and footer automatically. So the header getts written, then my php script is included, then the footer is added on.Problem here is if i want to redirect users. I cant use the header redirect, unless i use this: http://au3.php.net/manual/en/function.ob-start.phpwhich also looks ugly. Any ideas/suggestions appreciated
My approach to this problem is not to output anything to the browser until you have completed all the logic and therefore know if you're going to do a redirect or display a page. Since I'm also someone who belives that the code and the html should not be mixed I always use some sort of templating system (either write my own, use Smarty or use XSL), Smarty is a very good complete templating system and is available from http://smarty.php.net/ check it out if you're working on reasonable sized projects!
I don't know if I've really answered you're question but hopefully provided some food for thought ;)
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A common solution to this making the whole include a function.
Example:
MyPage.php:
<?phpinclude("header.php");writeheader("My PHP Page"); //Write page with Title "My PHP Page"?>
<?function writeheader($title) {?><html><head><title><?=$title?></title></head><?}?>
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Even easier is an output buffer:
ob_start();// do stuff// output stuff// include stuff// do more stuff// output more stuff// Hey, I can still do a header() redirect here!$contents = ob_get_contents();echo$contents;
The "output stuff" doesn't actually output anything. It's captured by the output buffer (ob). This means that you can still do header() redirects at the end of the script. The last two statements actually output all the captured contents.
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Yuck. A far better approach is to follow xstreme2000's advice and implement your own simple management system, or to use an existing one.
$backend = new Backend(new MyDatabaseImpl);$backend->run(); // Does pre-processing for later use:
// session management, grabs the requested
// page, etc.
if (is_empty($backend->config['page'])) header('HTTP/1.0 404 Not Found'); // Supposed redirect. include 'template/mytemplate.phtml';$backend->shutdown(); // Shouldn't be required.
Then...
// mytemplate.phtml
include \$backend->config['page']; // Should return an absolute path.
// Include footer
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• ### Forum Statistics
• Total Topics
631419
• Total Posts
2999981
×
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2018-06-23 22:17:34
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http://en.wikipedia.org/wiki/Space_diagonal
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# Space diagonal
AC' (shown in blue) is a space diagonal while AC (shown in red) is a face diagonal
In a rectangular box or a magic cube, the four space diagonals are the lines that go from a corner of the box or cube, through the center of the box or cube, to the opposite corner. These lines are also called triagonals or volume diagonals.
the pic demonstrates how to graphically build a spacediagonal and mathematically calculate it with Pythagoras Theorem
For the cube to be considered magic, these four lines must sum correctly.
The word triagonal is derived from the fact that as a variable point travels down the line, three coordinates change. The equivalent in a square is diagonal, because two coordinates change. In a tesseract it is quadragonal because 4 coordinates change, etc.
The space diagonal of a cube with side length $a$ is $a\sqrt {3}$.
## r-agonals
This section applies particularly to Magic hypercubes.
The magic hypercube community has started to recognize an abbreviated expression for these space diagonals. By using r as a variable to describe the various agonals, a concise notation is possible.
If r =
• 2 then we have a diagonal. 2 coordinates change.
• 3 = a triagonal. 3 coordinates change
• 4 = a quadragonal. 4 coordinates change
• n = the dimension of the hypercube, the 2n-1 agonals are required to sum correctly for the hypercube to be considered magic.
... By extension, if r =
• 1, the line is parallel to a face. Only 1 coordinate changes. A 1-agonal may be called a monagonal, in keeping with a diagonal, a triagonal, etc. Lines parallel to the faces of the hypercube have, in the past, also been referred to as i-rows.
Because the prefix pan indicates all, we can concisely state the characteristics or a magic hypercube.
For example;
• If pan-r-agonals sum correctly for r = 1 and 2, we know the square is pandiagonal magic.
• If pan-r-agonals sum correctly for r = 1 and 3, we have a pantriagonal magic cube (the equivalent of a pandiagonal magic square).
• If the r-agonals sum correctly for r = 1 and n, then the magic hypercube is simple magic regardless of what dimension it is.
The length of an r-agonal of a hypercube with side length a is $a\sqrt {r}$.
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2014-08-23 14:38:41
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https://brilliant.org/discussions/thread/question-why-the-static-charge-dont-give-magnetic-/
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×
Question : Why don't static charge give a magnetic field, but moving charges do?
Note by Nandan Gowda
3 years, 5 months ago
Sort by:
Hmm...very interesting question, I should have thought of this before!
It turns out that I Googled this and got this link. · 3 years, 5 months ago
I know why Moving charges have MF but I wanted to know why static charges don't ?? · 3 years, 5 months ago
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2017-05-23 22:37:43
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https://math.stackexchange.com/questions/linked/142381
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10k views
### The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. ...
10k views
### Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
12k views
### Lebesgue measurable but not Borel measurable
I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
9k views
### Achilles and the tortoise paradox?
Let's say we decide to race on a track $1000$ km long. You are a $100$ times faster than me, meaning if we both start at the beginning, you obviously win. To make things more fair you give me a head ...
3k views
### between Borel $\sigma$ algebra and Lebesgue $\sigma$ algebra, are there any other $\sigma$ algebra?
Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra? How about not in between the two, but in general, are there any other $\sigma$ ...
### Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel $\sigma$-...
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2019-05-25 21:33:50
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http://motls.blogspot.com/2012/04/exceptional-lie-groups.html
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## Wednesday, April 25, 2012 ... /////
### Exceptional Lie groups
Twenty years ago, when I started to read various articles on particle physics, I often encountered exceptional Lie groups in various articles on grand unified theories and related topics. And I couldn't understand why they exist and what they really are. Even when I got to the college, it took half a year to find the right sources that made it clear.
There might be some people who are facing the same puzzles so let me offer an exposition that would have clarified my confusions when I was a kid. This blog entry will try to clarify the composition of all the simple compact Lie groups.
Groups, compact Lie groups
Pretty much everyone may understand the group $SO(N)$. It is the group of symmetries of the $N$-dimensional unit ball$x_1^2 + x_2^2 + \dots x_N^2 \leq 1.$ The coordinates $x_1,\dots , x_N$ may be rotated into each other by orthogonal transformations.
You may represent the rotations by linear transformations$\pmatrix{x_1\\x_2\\ \vdots\\ x_N} \to M \cdot \pmatrix{x_1\\x_2\\ \vdots\\ x_N}$ where the dot represents the matrix multiplication and $M$ is an $N\times N$ matrix. If the matrix is orthogonal i.e. if it obeys$M M^T = M^T M = {\bf 1},$ then it is easy to see that the Pythagorean norm of a vector $\vec x$ is preserved when $\vec x$ is replaced by $M\vec x$:$ds^2 = \vec x^T\cdot \vec x = (M \vec x)^T \cdot (M \vec x) = \vec x^T\cdot M^T\cdot M \cdot \vec x$ because the matrix multiplication $(\cdot)$ is associative and because the product $M^TM$ may be replaced by the identity matrix that may be erased because ${\bf 1}\cdot \vec x = \vec x$. Fine. One may consider transformations that are very modest, very close to the identity. There are $N(N-1)/2$ of independent generators that produce such transformations because each pair of the $(x_i,x_j)$ coordinates may be rotated into each other.
Because $M^T M = {\bf 1}$ and because$\det(M^T M ) = \det M^T \det M = (\det M)^2 = +1,$ we see that $\det M=\pm 1$. That means that the group manifold $O(N)$ – the set of all possible matrices that satisfy the orthogonality condition – is composed of two independent continuous pieces. The determinants of the matrix in these two pieces are $\pm 1$, respectively. If we only consider the matrices that obey$M^T M = {\bf 1},\qquad \det M = {\bf 1},$ then we define the so-called group $SO(N)$ which is also $N(N-1)/2$-dimensional and only has one component (every element of the group is continuously connected to every other element of the group).
Unitary groups
There exists a straightforward generalization of the orthogonal groups $O(N)$ and $SO(N)$ for complex linear spaces. Just replace $x_i$ by complex variables $z_i$ and require that the following norm$|z_1|^2+|z_2|^2+\dots+|z_N|^2 = 1$ is preserved by the linear transformations acting on the complex $N$-dimensional space. The complex matrix $M$ that acts on $\vec z$ has to obey$M^\dagger M = {\bf 1}, \qquad M^\dagger \equiv (M^T)^*$ for the bilinear expression constructed from the absolute values of $z$ to stay constant. The condition above defines the so-called unitary group, $U(N)$. One may see that in this case, the determinant satisfies$\abs{\det M}^2 = 1$ which means that $\det M$ is any number whose absolute value is equal to one. In other words, the determinant has to belong to the one-dimensional manifold of the complex numbers of the type $\exp(i\delta)$ for $\delta\in\RR$. If one demands $\det M = +1$ again, we obtain the so-called special unitary group, $SU(N)$. Note that the real dimension of the $U(N)$ group is $N^2$ while one more dimension is removed for $SU(N)$; its real dimension is therefore $N^2-1$.
Symplectic groups
Those were simple enough things but there actually exists one more infinite class of similar groups. The quaternions are associative so one may construct nice associatively-multiplicative matrices with quaternionic entries. The group $USp(2N)$ is defined as nothing else than $U(N,{\mathbb H})$. Recall that quaternions are numbers of the type$q = a+bi+cj+dk$ where the most important defining property is the multiplication table for the unit imaginary quaternions,$\eq{ i^2&=j^2=k^2=-1,\\ ij=-ji=k, \quad jk&=-kj=i,\quad ki=-ik=j }$ If you consider vectors with quaternionic entries $q_1,q_2,\dots, q_N$, the group of quaternionic-valued matrices $M$ obeying $M^\dagger M={\bf 1}$ will be defined as the symplectic group $USp(2N)$. Note that the dagger $\dagger$ involves both transposition and the complex conjugation which changes the sign of the three coefficients in front of all the $i,j,k$ imaginary units.
One may offer an equivalent, quaternion-free definition of the group. Write the quaternions as $q=z_1+j z_2$ where $z_1,z_2$ are complex numbers. That's possible because $z_1$ takes care of the real multiples of $1$ and $i$ while $z_2$ takes care of the $j$ and $k=ij$ terms. Now, replace each entry $q_i$ of the quaternionic vector as the column with two entries $z_1,z_2$ above each other. The quaternionic multiplication may be mimicked by a procedure acting on the complex $2\times 2$ matrices and the quaternionic matrix $M$ may be emulated by a larger complex matrix $M_\CC$ of the size $2N\times 2N$ which obeys$M_\CC^\dagger M_\CC = {\bf 1}, \qquad M_\CC^T \cdot a \cdot M_\CC = a$ where $a$ is a specific antisymmetric matrix, namely$a = {\rm diag} \zav{ \pmatrix{0&-1\\+1&0}, \,\, \pmatrix{0&-1\\+1&0}, \cdots \pmatrix{0&-1\\+1&0} }$ where the $2\times 2$ basic block is repeated $N$ times on the diagonal. This is another way to define the symplectic group. Note that the real dimension of $USp(2N)$ is the same as the real dimension of the $SO(2N+1)$ group, namely $(2N+1)N$.
The orthogonal, unitary, and symplectic groups are all the possible examples of "infinite families" of simple compact Lie groups. The adjective "simple" means that the group can't be written as $G_1\times G_2$ where both $G_1,G_2$ are nontrivial groups. The adjective "Lie" named after Sophus Lie (picture at the top) means that the groups are continuous, i.e. their elements are parameterized by continuous parameters. The word "compact" means that if you imagine the set of all the elements, it is a manifold, and with some natural "measure" on that manifold that is invariant under the group operation itself, the volume of the group manifold is finite. There are no directions in which you may "escape to infinity" away from the identity.
Off-topic: David Gross gave a talk at Chapman University a month ago (95 minutes) on the state of physics.
Isomorphisms
Now, there are lots of important isomorphisms between the elements of the three infinite classes of the compact simple Lie groups, especially for small values of $N$. I will use the symbol $=$ for the isomorphism which will tolerate the fact that one of the groups that are "identified" may be factored to be really isomorphic to the other one. Equivalently, the isomorphisms below hold at the level of the Lie algebras. We have$\eq{ U(N) &= SU(N)\times U(1)\\ U(1) &= SO(2) \\ SO(3) &= SU(2) = USp(2)\\ SO(4) &= SU(2) \times SU(2) = USp(2)\times USp(2)\\ SO(5) &= USp(4)\\ SO(6) &= SU(4) }$ where the most complicated isomorphisms require you to learn about the spinor representations. I don't want to expand this blog entry into a huge book so I must be very brief. The orthogonal groups $SO(N)$, more precisely $Spin(N)$ which are "refined" versions of $SO(N)$ that, unlike $SO(N)$, distinguish the rotation by 0 degrees (identity) from the rotation by 360 degrees (while the rotation by 720 degrees is the identity again), i.e. groups obeying$SO(N) \approx Spin(N)/\ZZ_2$ have a nice new type of a representation whose dimension is something like $2^{[N/2]}$, the so-called spinors. The existence of such representations is linked to our ability to define the Dirac gamma matrices whose width as well as height is the same power of two. These gamma matrices obey$\gamma_i \gamma_i + \gamma_j \gamma_i = 2\cdot \delta_{ij}\cdot{\bf 1}, \qquad i,j=1,2,\dots, N.$ Such matrices may be defined as tensor products of the Pauli matrices (and the identity matrix). And if the key identities above hold, we may also see that $\gamma_{ij}/2=\gamma_i\gamma_j/2$ for $i\neq j$ obey commutation relations that coincide with the commutation relations for the usual generators of $SO(N)$. The commutator of $G_{ij}$ and $G_{jk}$ is equal to $i=\sqrt{-1}$ times $G_{ik}$ for $i\neq j\neq k\neq i$. This isomorphism between the commutation relations guarantees that one may construct matrices of the same size as the gamma matrices whose products are related by the same dictionary as the products of the usual $SO(N)$ matrices we started with.
The spinors – which change the sign if you rotate the coordinates by 360 degrees – are the new and unexpected kind of representations of the orthogonal groups you have to master when you really want to understand the compact Lie groups. In some sense, the spinor index which takes one of the $2^{[N/2]}$ possible values may be interpreted as "one-half" of the vector index of $SO(N)$. Otherwise, you may build the most general representations as tensors with many vector indices – which may be constrained by various symmetric or antisymmetric rules as well – but you must also allow the spinor indices.
Going exceptional
So far, the text was unexceptional. We haven't gotten anywhere yet. ;-) I want you to understand the exceptional groups. They're five more simple compact Lie groups that don't belong to the orthogonal, unitary, and symplectic infinite families listed above. It turns out that these groups are $G_2,F_4,E_6,E_7,E_8$ where the subscript indicates the rank, i.e. the maximum number of independent $U(1)$ generators of these groups that commute with each other (any pair commutes).
As you can see, the largest group seems to be $E_8$, so we will start with it. It's the "master example" of an exceptional group; all other exceptional groups may be identified with some subgroups of $E_8$. We will construct $E_8$ as a particular "extension" of $SO(16)$, one of the ordinary orthogonal groups.
Start with the generators $J_{IJ}$ where $I,J=1,2,\dots 16$. They have the commutators of the type$-i[J_{IJ},J_{KL}] = [\delta_{IL} J_{JK} - (K\leftrightarrow L)] - (I\leftrightarrow J)$ where I added the factor of $(-i)$ to agree with some conventions and to allow all the generators to be Hermitian (rather than antihermitian). The "exchange arrow" terms guarantee the two antisymmetry conditions. These matrices $J_{IJ}$ may be obtained as $16\times 16$ matrices whose only nonzero entries are at positions $IJ$ and $JI$ and they are equal to $+1$ and $-1$, respectively. Alternatively, as I suggested, you may also define $J_{IJ}$ as $\gamma_{IJ}/2$, a product of two gamma matrices (the matrix is defined to vanish for $I=J$ i.e. it is antisymmetrized in $IJ$.
We have $16\times 15/ 2\times 1 = 120$ generators like that.
Now, we add $128$ generators $Q_\alpha$ that transform as a real chiral spinor of $Spin(16)$ i.e. "essentially" $SO(16)$. According to the general rule, the Dirac spinor for $Spin(16)$ has $2^{16/2}=2^8=256$ components. One may show that because $16$ is a multiple of eight, it is a real spinor: all the rotation matrices may be expressed as real matrices in an appropriate basis.
However, because $16$ is even, the rotation matrices decompose into a block-diagonal form in a nicely chosen basis. So the $256\times 256$ matrices for the rotations in the "Dirac spinor representation" may be decomposed into two $128\times 128$ matrices acting on the "chiral", i.e. Weyl spinor; one of the spinors is left-handed and one of them is right-handed. Let's take only one of the spinors which has, as I said, $128$ real components.
What does it mean for $128$ bonus generators $Q_\alpha$ to transform as spinors? It means that$[J_{IJ},Q_\alpha] = \frac {i}2 (\gamma_{IJ})_{\alpha\beta} Q_\beta$ because the commutator is how the symmetry generators $J_{IJ}$ act on the objects $Q_\alpha$ if these objects are meant to be operators themselves and not just "vectors". We obtain some linear combinations of the same $Q_\beta$ generators and the coefficients are the matrix entries of the $Spin(16)$ generators in the spinor matrix.
To prove that we have defined a Lie algebra, one must verify the Jacobi identities$[A,[B,C]]+ [B,[C,A]] + [C,[A,B]] = 0$ where the three terms differ by cyclic permutations. This identity – which is tautologically satisfied whenever $[A,B]$ may be expressed as $AB-BA$: just expand the damn commutators – has to be obeyed for every triplet of generators $A,B,C$ if we actually want some representation(s) of the commutator in terms of the product differences to exist. Now, each of the generators $A,B,C$ may be either of the $J_{IJ}$ type, i.e. the generator of $Spin(16)$, or the $Q_\alpha$ type, the bonus spinor generator. The $JJJ$ Jacobi identity holds because it holds in $SO(16)$, a nice group. The $JJQ$ Jacobi identity also holds because it just says that the commutator of two $J$'s acts on a $Q$ in the same way as if you compute the commutator of the two actions.
One must also verify the $JQQ$ and $QQQ$ Jacobi identity. Only the latter calculation turns out to be hard enough; one has to use various identities for the spinors. Both $JQQ$ and $QQQ$ identities require you to know the commutator of two $Q_\alpha$ generators:$[Q_\alpha,Q_\beta] = J_{IJ}\cdot (\gamma_{IJ})_{\alpha\beta}.$ The right hand side is actually the only candidate you may construct – a linear combination of generators that has the $\alpha,\beta$ indices ($I,J$ are contracted) which preserves the $SO(16)$ symmetry. We are obliged to use nice $SO(16)$-preserving defining relations because we want to keep this orthogonal group as our symmetry (a subgroup of the emerging $E_8$ group). The only freedom is the overall normalization. The coefficient has to be real. If it were rescaled by a positive number, the change might be absorbed to the normalization of $Q_\alpha$.
The only freedom that is left is the sign. If you change the sign of the commutator above, you switch between $E_8$ and its noncompact version $E_{8(8)}$ which also contains the $SO(16)$ subgroup.
Now, if we have the Lie algebra, the Lie group may be defined as the set of the exponentials of the Lie algebra matrices$E_8^{\rm group} = \left\{ \exp\left(i \sum_a G_a \omega_a\right),\quad \omega_{a}\in \RR \right \}$ where $G_a$ with $a=1\dots 248$ stands for $J_{IJ}$ as well as $Q_\alpha$ – or, equivalently, as the automorphism group of the Lie algebra. We have added a 128-dimensional real spinor to the 120-dimensional real Lie algebra so we have created a 248-dimensional Lie algebra, the $E_8$. The decomposition of the adjoint representation of $E_8$ under the $SO(16)$ subgroup is${\bf 248} = {\bf 120}_{\rm adj} \oplus {\bf 128}_{\rm spinor}.$ The strategy to construct a new Lie algebra by adding a spinor to an adjoint of an orthogonal group works in 3 cases; only in 3 cases, all the Jacobi identities, especially the hard $QQQ$ identity, may be satisfied. Aside from the $E_8$ case above, one may also construct the 52-dimensional $F_4$ exceptional Lie algebra by adding the 16-dimensional real spinor of $SO(9)$ to the 36-dimensional $SO(9)$ adjoint. Note that there is no chirality in 9 dimensions which is an odd number.
The final, third example is all about the addition of the 8-dimensional real chiral spinor to the 28-dimensional $SO(8)$ adjoint.
The triality of $SO(8)$ may be seen as the permutation symmetry between the three "branches" of the Dynkin diagram of this group.
In this way, we obtain a 36-dimensional group. What is it? Well, it's an $SO(9)$. It may remind you of a simple way to construct $SO(9)$ out of $SO(8)$: add an 8-vector of new generators $J_{i9}$ to the $SO(8)$ generators $J_{ij}$. The addition of an 8-dimensional left-handed spinor is related to the addition of the 8-dimensional vector by an operation called "triality": the third sibling is the addition of the right-handed spinor (which is just a left-right reflection of the addition of the left-handed one). The triality symmetry is a special symmetry of the $SO(8)$ group and no other compact Lie group. It permutes its representations, too. In particular, the two spinor representations and the vector representation (each of them has 8 dimensions) are permuted.
One may say that we have defined $E_8$ and $F_4$ by the "adjoint plus spinor" strategy above. We may define all the remaining exceptional groups by constructions that only depend on ordinary Lie groups discussed at the beginning.
First, what is a $G_2$? It is the centralizer of $F_4$ embedded into $E_8$. Of course, there are other ways to define it but this is one of them. We constructed $F_4$ out of $SO(9)$. But $SO(9)\times SO(7)$ may be embedded in the obvious "additive" way to $SO(16)$ which is inside $E_8$. So a centralizer or $F_4$ is a centralizer of an "extension of $SO(9)$" which is inevitably a "subgroup of $SO(7)$". Note that a similar statement with the opposite terms "extension" and "subgroup" omitted is also true: the centralizer of $SO(9)$ in $SO(16)$ and even in $E_8$ is nothing else than $SO(7)$.
The decomposition of the fundamental representation of $E_8$ (which is the same as the adjoint for this group and only this group) under the $F_4\times G_2$ maximal subgroup ("maximal" means that one can't embed any other embedding that works on both sides in between) is${\bf 248} = ({\bf 52},{\bf 1}) \oplus ({\bf 1}, {\bf 14}) \oplus ({\bf 26},{\bf 7})$ which is the direct sum of the two adjoint representations and the tensor product of the fundamental representations.
However, one may also find the groups $E_6,E_7$ inside $E_8$. They're centralizers of $SU(3)$ and $SU(2)$ inside $E_8$, respectively. These special unitary groups have to be embedded into $SU(4)=SO(6)$ where $SO(6)$ is a part of $SO(6)\times SO(10)$ embedded into $SO(16)$ and $E_8$ in the obvious way. The decompositions of the representations may be rather easily derived to be$\eq{ {\bf 248} &= ({\bf 78},{\bf 1}) \oplus ({\bf 1}, {\bf 8}) \oplus ({\bf 27},{\bf 3})\oplus {\rm c.c.}\\ {\bf 248} &= ({\bf 133},{\bf 1}) \oplus ({\bf 1}, {\bf 3}) \oplus ({\bf 56},{\bf 2}) }$ under $E_6\times SU(3)$ and $E_7\times SU(3)$, respectively. The abbreviation ${\rm c.c.}$ stands for the complex conjugate representation to the previous one, i.e. $(\bar{\bf 27},\bar{\bf 3})$.
There are tons of other basic things to be said about the five exceptional groups and the compact simple Lie groups in general but I suspect that some of the readers may already be tired at this point so let me stop.
Any questions? ;-)
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2015-05-30 08:24:11
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https://www.physicsforums.com/threads/ground-state-in-peskin-and-schroeder.994738/
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# Ground State in Peskin and Schroeder
Summary:
I am having a little confusion regarding the limit taken in Peskin and Schroeders quantum field theory book in ch. 4.
In P&S, it is shown that $$e^{-iHT}\ket{0}=e^{-iH_{0}T}\ket{\Omega}\bra{\Omega}\ket{0}+\sum_{n\neq 0}e^{-iE_nT}\ket{n}\bra{n}\ket{0}$$.
It is then claimed that by letting $$T\to (\infty(1-i\epsilon))$$ that the other terms die off much quicker than $$e^{-iE_0T}$$, but my question is why is this the case? For example, why wouldn't the other terms also die off quicker if we simply sent $$T\to \infty$$ instead? Perhaps there is something about the limit of complex numbers I'm not understanding. Any insight would be appreciated. Thanks.
Last edited:
Nevermind, I think I figured it our. I mistakenly assumed the $$e^{-iE_n T}\to 0$$ as $$T\to \infty$$, but that is not the case, which is why the substitution is needed.
mfb
Mentor
Yes, it's only the small imaginary term in T that leads to an e-x behavior. The real term just leads to oscillation.
vanhees71
Gold Member
It becomes much clearer by "renormalizing" the Hamiltonian such that ##E_0=0##, i.e., the ground state energy eigenvalue is set to 0 by shifting the total energy of the system (represented by the Hamilton operator) by adding a constant such that ##E_0=0##. Then all ##E_n>0##.
Now it is utmost important to add an infinitesimal imaginary part to ##T##, i.e., substituting ##T \rightarrow T-\mathrm{i} \epsilon##. This is crucial for all further developments of the theory to get the correct propagator (in vacuum QFT perturbation theory the time-ordered free-field propagator) and the correct "adiabatic switching" for the LSZ reduction. This is a pretty subtle point and should be very well studied!
It becomes much clearer by "renormalizing" the Hamiltonian such that ##E_0=0##, i.e., the ground state energy eigenvalue is set to 0 by shifting the total energy of the system (represented by the Hamilton operator) by adding a constant such that ##E_0=0##. Then all ##E_n>0##.
Now it is utmost important to add an infinitesimal imaginary part to ##T##, i.e., substituting ##T \rightarrow T-\mathrm{i} \epsilon##. This is crucial for all further developments of the theory to get the correct propagator (in vacuum QFT perturbation theory the time-ordered free-field propagator) and the correct "adiabatic switching" for the LSZ reduction. This is a pretty subtle point and should be very well studied!
Can you perhaps say a bit more about vacuum QFT and adiabatic switching. Recently I saw these terms more frequently and would like to know more about them. Do you know a good reference to read about this subject?
vanhees71
Gold Member
That's one of the points which are very nicely and carefully presented in the classic textbook by Bjorken and Drell (of course the 2nd volume on quantum field theory; the 1st volume is not so much my favorite ;-)).
M91
Do you also happen to know which chapter? I think its either: chapter 16 Vacuum Expectation Values and S-Matrix or, chapter 17 Perturbation Theory
vanhees71
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2021-10-28 01:59:40
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http://mathhelpforum.com/differential-equations/94447-find-implicit-form.html
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1. ## find implicit form
hi
find in implicit form the general solution of differential equation
$\frac{dy}{dx}=3y^2e^{-2x}\sqrt{8+e^{-2x}}$
= $\int{3y^2}=\int{e^{-2x}}\sqrt{8+e^{-2x}}$
= $\int{3y^2}=\int{e^{-2x}}+e^{\frac{-3}{2}}+8^{\frac{1}{2}}$
$y^3=\frac{-1}{2}e^{-2x}-\frac{2}{3}e^{\frac{3}{2}}+\frac{2}{3}8x^\frac{3}{ 2}$
i did try but as you can see not sure what i am doing could do with some expert help please
2. Originally Posted by smartcar29
hi
find in implicit form the general solution of differential equation
$\frac{dy}{dx}=3y^2e^{-2x}\sqrt{8+e^{-2x}}$
= $\int{3y^2}=\int{e^{-2x}}\sqrt{8+e^{-2x}}$
= $\int{3y^2}=\int{e^{-2x}}+e^{\frac{-3}{2}}+8^{\frac{1}{2}}$
$y^3=\frac{-1}{2}e^{-2x}-\frac{2}{3}e^{\frac{3}{2}}+\frac{2}{3}8x^\frac{3}{ 2}$.
i did try but as you can see not sure what i am doing could do with some expert help please
When you separate the variables, you get $\frac{\,dy}{y^2}=3e^{-2x}\sqrt{8+e^{-2x}}\,dx$. So when you integrate the LHS, you get $-\frac{1}{y}+C$. When you integrate the RHS, you apply the substitution $u=8+e^{-2x}\implies \,du=-2e^{-2x}\,dx$. So it follows that $3\int e^{-2x}\sqrt{8+e^{-2x}}\,dx=-\tfrac{3}{2}\int\sqrt{u}\,du=-u^{3/2}+C=-\left(8+e^{-2x}\right)^{3/2}+C$
So it follows now that $\frac{1}{y}+C=-\left(8+e^{-2x}\right)^{3/2}+C\implies \frac{1}{y}=-\left(8+e^{-2x}\right)^{3/2}+K$. This would be the implicit solution.
The explicit solution would be $y=\frac{1}{K-\left(8+e^{-2x}\right)^{3/2}}$
3. thank you for your much needed help i really suck at math, going to spend less time playing xbox and more time doing revision.
thanks again.
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2016-08-30 22:58:25
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https://mathhelpforum.com/threads/asymptotes-and-intercepts.196247/
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# asymptotes and intercepts
#### vaironxxrd
I need to find the y,x intercepts and asymptotes of the following function
$$\displaystyle f(x)=\frac{2}{x^2-3}$$
x intercepts = No x intercents
y intercepts = $$\displaystyle \sqrt{3}$$
horizontal asymptote = 0
vertical asymptote = $$\displaystyle \sqrt{3}$$
Last edited:
#### skeeter
MHF Helper
I need to find the y,x intercepts and asymptotes of the following function
$$\displaystyle f(x)=\frac{2}{3x-3}$$
x intercepts = No x intercents correct
y intercepts = $$\displaystyle \sqrt{3}$$ no
horizontal asymptote = 0 y = 0 is the horizontal asymptote (it's the equation of a line)
vertical asymptote = $$\displaystyle \sqrt{3}$$ no
... where did $$\displaystyle \sqrt{3}$$ come from?
#### Prove It
MHF Helper
I need to find the y,x intercepts and asymptotes of the following function
$$\displaystyle f(x)=\frac{2}{3x-3}$$
x intercepts = No x intercents
y intercepts = $$\displaystyle \sqrt{3}$$
horizontal asymptote = 0
vertical asymptote = $$\displaystyle \sqrt{3}$$
y intercepts are where x = 0. This does not give $$\displaystyle \displaystyle \sqrt{3}$$.
The vertical asymptote is at the same x value as the horizontal translation. Maybe try rewriting the function as $$\displaystyle \displaystyle \frac{2}{3}\left(\frac{1}{x - 1}\right)$$. Does that help?
#### vaironxxrd
... where did $$\displaystyle \sqrt{3}$$ come from?
y intercepts are where x = 0. This does not give $$\displaystyle \displaystyle \sqrt{3}$$.
The vertical asymptote is at the same x value as the horizontal translation. Maybe try rewriting the function as $$\displaystyle \displaystyle \frac{2}{3}\left(\frac{1}{x - 1}\right)$$. Does that help?
Sorry i made a big mistake, while looking at my paper i stated the numerator correctly but the denominator incorrectly, I have fixed it.
#### skeeter
MHF Helper
$$\displaystyle f(x) = \frac{2}{x^2-3}$$
the y-intercept is $$\displaystyle f(0)$$ , which is not $$\displaystyle \sqrt{3}$$
note that the denominator will factor ... $$\displaystyle (x - \sqrt{3})(x + \sqrt{3})$$
now, what are the equations of the vertical asymptotes?
#### vaironxxrd
... where did $$\displaystyle \sqrt{3}$$ come from?
y intercepts are where x = 0. This does not give $$\displaystyle \displaystyle \sqrt{3}$$.
The vertical asymptote is at the same x value as the horizontal translation. Maybe try rewriting the function as $$\displaystyle \displaystyle \frac{2}{3}\left(\frac{1}{x - 1}\right)$$. Does that help?
$$\displaystyle f(x) = \frac{2}{x^2-3}$$
the y-intercept is $$\displaystyle f(0)$$ , which is not $$\displaystyle \sqrt{3}$$
note that the denominator will factor ... $$\displaystyle (x - \sqrt{3})(x + \sqrt{3})$$
now, what are the equations of the vertical asymptotes?
Let me see if I understand because my review paper is getting me a bit confused.
x and y intercepts are found by f(0)?
horizontals asymptotes are determined by the first degrees m & n
vertical asymptotes is found by setting the denomenator = 0?
#### skeeter
MHF Helper
Let me see if I understand because my review paper is getting me a bit confused.
x and y intercepts are found by f(0)?
horizontals asymptotes are determined by the first degrees m & n
vertical asymptotes is found by setting the denomenator = 0?
f(0) is the y-intercept (only one)
values of x where f(x) = 0 are the x-intercepts (could be one, more than one ... or none at all)
horizontal asymptote is y = 0 when degree of numerator < degree of denominator, or y = ratio of the leading coefficients if degrees are equal.
vertical asymptotes are x = the value(s) where the function is undefined (and the discontinuities are non-removable) ... so yes, x-values where the denominator equals 0.
#### vaironxxrd
f(0) is the y-intercept (only one)
values of x where f(x) = 0 are the x-intercepts (could be one, more than one ... or none at all)
horizontal asymptote is y = 0 when degree of numerator < degree of denominator, or y = ratio of the leading coefficients if degrees are equal.
vertical asymptotes are x = the value(s) where the function is undefined (and the discontinuities are non-removable) ... so yes, x-values where the denominator equals 0.
In that case...
Y intercept = g(0)= (0)^2-3 = -3
vertical asymptote =$$\displaystyle x^2-3=0$$
$$\displaystyle x^2=3$$
$$\displaystyle x=\sqrt(3)$$
#### skeeter
MHF Helper
In that case...
Y intercept = g(0)= (0)^2-3 = -3
vertical asymptote =$$\displaystyle x^2-3=0$$
$$\displaystyle x^2=3$$
$$\displaystyle x=\sqrt(3)$$
$$\displaystyle f(0) = \frac{2}{0^2 - 3} = -\frac{2}{3}$$
vertical asymptotes are $$\displaystyle x = \sqrt{3}$$ and $$\displaystyle x = -\sqrt{3}$$
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2019-12-11 23:03:50
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https://www.mersenneforum.org/showthread.php?s=2026dc166dec840d2d711e58cbead800&t=23042&page=3
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mersenneforum.org mtsieve
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2018-02-22, 16:08 #23
rogue
"Mark"
Apr 2003
Between here and the
22·32·11·17 Posts
Quote:
Originally Posted by pepi37 View Post fbncsieve -p50000000000000 -P 100000000000000 -i 500.npg -fN -W4 -O fact.txt fbncsieve v1.3.1, a program to find factors of k*b^n+c numbers for fixed b, n, and c and variable k Sieve started: 5e13 < p < 1e14 with 159895 terms p=50055611474419, 28.92M p/sec, 4 factors found at 15298 sec per factor, 0.1% done. ETA 2018-02-22 16:51
Can you e-mail me your input file?
Last fiddled with by rogue on 2018-02-22 at 16:09
2018-02-22, 17:37 #24
pepi37
Dec 2011
After milion nines:)
32·52·7 Posts
Quote:
Originally Posted by rogue How was the newpgen file created? Was it created by newpgen or fbnciseve? Is there a reason that you choose that format over the ABC or ABCD format? Getting sr1sieve into mtsieve is one of my goals, but it is behind the GPU options.
Newpgen file is created using merge function in that program (sieved across few machines) and then join into one.
Quote:
Is there a reason that you choose that format over the ABC or ABCD format?
Of course there is reason: and reason is called removing factors. I dont know why you now make some new formats when we have srfile that can make miracles , and make remove factors and convert from one to another formats. If you make some new formats, then please make utility to
-allow removing factors
-allow change from one format to another ( and also support to convert to npg format)
How I will remove factors: if I have three computers and every computer make own range?
2018-02-22, 17:39 #25
pepi37
Dec 2011
After milion nines:)
32·52·7 Posts
Quote:
Originally Posted by rogue Can you e-mail me your input file?
Here it is
https://www.dropbox.com/s/hmmyd6lqrwgamf6/500.zip?dl=0
P.S sieve depth is lower, so header in file is not OK, but this is just for experiment so dont care about it
2018-02-22, 18:42 #26 wombatman I moo ablest echo power! May 2013 22×11×41 Posts Pepi, just as a heads-up. The linux version of sr1sieve is multithreaded. If you have Windows 10, you can compile the source within the Ubuntu shell and it will work fine.
2018-02-22, 18:51 #27
rogue
"Mark"
Apr 2003
Between here and the
22×32×11×17 Posts
Quote:
Originally Posted by pepi37 Newpgen file is created using merge function in that program (sieved across few machines) and then join into one. Of course there is reason: and reason is called removing factors. I dont know why you now make some new formats when we have srfile that can make miracles , and make remove factors and convert from one to another formats. If you make some new formats, then please make utility to -allow removing factors -allow change from one format to another ( and also support to convert to npg format) How I will remove factors: if I have three computers and every computer make own range?
Use the -I argument to pass a file of factors in the form "p | candidate" into fbncsieve. If you have multiple factor files, concatenate before using as input. fbncsieve can be used to convert input files from ABCD/ABC/NPG formats into ABCD/ABC/NPG format using the -f switch. What it doesn't do is "convert then exit immediately", but I could probably add a switch for that.
fbncsieve does not output "invalid header", so I don't know what issue it had with your file. I will d/l that file later today and see what is tripping it up as I don't see anything obviously wrong with what you pasted.
BTW, one of my long term goals is to remove the need for srfile. None of the programs in this framework rely on srfile for any processing. If they do, then please let me know what I can do to move you away from using it.
Last fiddled with by rogue on 2018-02-22 at 19:25
2018-02-22, 19:37 #28
pepi37
Dec 2011
After milion nines:)
32·52·7 Posts
Quote:
Originally Posted by rogue Use the -I argument to pass a file of factors in the form "p | candidate" into fbncsieve. If you have multiple factor files, concatenate before using as input. fbncsieve can be used to convert input files from ABCD/ABC/NPG formats into ABCD/ABC/NPG format using the -f switch. What it doesn't do is "convert then exit immediately", but I could probably add a switch for that. fbncsieve does not output "invalid header", so I don't know what issue it had with your file. I will d/l that file later today and see what is tripping it up as I don't see anything obviously wrong with what you pasted. BTW, one of my long term goals is to remove the need for srfile. None of the programs in this framework rely on srfile for any processing. If they do, then please let me know what I can do to move you away from using it.
What it doesn't do is "convert then exit immediately", but I could probably add a switch for that.- that will be great , so at the end I can get one or more npg files.
Then I dont need srfile anymore ( agree with you)
2018-02-22, 21:36 #29
rogue
"Mark"
Apr 2003
Between here and the
22·32·11·17 Posts
Quote:
Originally Posted by pepi37 What it doesn't do is "convert then exit immediately", but I could probably add a switch for that.- that will be great , so at the end I can get one or more npg files. Then I dont need srfile anymore ( agree with you)
How many terms are remaining in the output from fbncsieve that require you to split it? In the case of CRUS, you would be using that output as input to srbsieve. In other cases is it possible for you to set up a PRPNet server to hand out the work? If not, I can look into adding a switch (similar to what is in gfndsieve) for splitting the remaining terms from fbncsieve into multiple files.
I recommend avoiding NPG file formats. The ABC format is similar, but the difference between the headers of the two formats makes the ABC format easier to comprehend.
2018-02-22, 22:17 #30
pepi37
Dec 2011
After milion nines:)
32×52×7 Posts
one more cosmetic bug
if you use option -h then you got text
look at last line ( in the red)
Quote:
C:\Users\Alpha-I7\Desktop\mtsieve>gfndsieve -h gfndsieve v1.3, a CPU program to find factors of k*2^n+1 numbers for variable k and n -h --help prints this help -p --pmin=P0 sieve start: P0 < p (default 1) -P --pmax=P1 sieve end: p < P1 (default 2^62) -w --worksize=w primes per chunk of work (default 1000000) -W --workers=W start W workers (default 1) -i --inputterms=i input file of remaining candidates -I --inputfactors=I input file with factors -o --outputterms=o output file of remaining candidates -O --outputfactors=O output file with new factors -k --kmin=k minimum k to search -K --kmax=K maximum k to search -n --nmin=N minimum n to search -N --nmax=N maximum n to search -T --nsperfile=T number of n per output file Fatal Error: kmin must be specified
2018-02-22, 22:24 #31
rogue
"Mark"
Apr 2003
Between here and the
11010010011002 Posts
Quote:
Originally Posted by pepi37 one more cosmetic bug if you use option -h then you got text look at last line ( in the red)
The question is whether or not it should exit immediately after printing the help.
2018-02-23, 00:39 #32 rogue "Mark" Apr 2003 Between here and the 22·32·11·17 Posts pepi37 reported a bug with the newpgen format output by fbncsieve. It is with the cryptic details buried in that pesky first line. I hope to fix it this weekend.
2018-02-23, 02:31 #33 Dylan14 "Dylan" Mar 2017 2×33×11 Posts I found a bug with the cksieve executable: when the -n flag is supplied with 1 as its argument it gives a fatal error saying that 1 is out of range. The output is given below: Code: C:\Users\Dylan_000\Desktop\mtsieve\mtsieve>cksieve -P1e9 -n1 -N50000 -b50 cksieve v1.2, a program to find factors of (b^n+/-1)^2-2 numbers Fatal Error: cksieve: out of range argument -n 1 whereas if the -n flag is supplied with a 2 instead it works fine: Code: C:\Users\Dylan_000\Desktop\mtsieve\mtsieve>cksieve -P1e9 -n2 -N50000 -b50 cksieve v1.2, a program to find factors of (b^n+/-1)^2-2 numbers Sieve started: 1 < p < 1e9 with 99998 terms In cksieve v1.1.4 -n 1 works fine: Code: cksieve -P1e9 -n1 -N50000 -b50 cksieve 1.1.4 -- A sieve for Carol (b^n-1)^2-2 and Kynea (b^n+1)^2-2 numbers. Started with 100000 terms for (50^n+/-c)^2-2 from command line. cksieve 1.1.4 started: 1 <= n <= 50000, 3 <= p <= 1000000000 I can confirm that the multithreading works in cksieve: with one thread on a otherwise idle i5-5200U it takes 261.82 seconds to process the sieve in the second code snippet, whereas with 4 threads the time taken is 88.82 sec. Last fiddled with by Dylan14 on 2018-02-23 at 02:34
All times are UTC. The time now is 18:00.
Tue Sep 27 18:00:07 UTC 2022 up 40 days, 15:28, 0 users, load averages: 0.93, 1.02, 1.21
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2022-09-27 18:00:07
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http://www.mathematics2.com/Geometry/WordsAndSymbols?action=diff&source=y&minor=n
|
Subject: Geometry
# Words & Symbols In Geometry
## Geometry.WordsAndSymbols History
December 26, 2010 by bobdoleorama -
Changed line 92 from:
(:cellnr:)'+{$\begin{matrix}\rightarrow\\ AB \end{matrix}$}+'
to:
(:cellnr:)'+{$\overset{\rightarrow}{AB}$}+'
Changed line 94 from:
(:cell:){$\begin{matrix}\rightarrow\\ AB \end{matrix}$}
to:
(:cell:){$\overset{\rightarrow}{AB}$}
December 26, 2010 by bobdoleorama -
Changed line 124 from:
(:cell:){$\overrightarrow{AB} = \infty$}
to:
(:cell:){$\overleftrightarrow{AB} = \infty$}
December 26, 2010 by bobdoleorama -
Changed line 92 from:
(:cellnr:)'+{$\overrightarrow{ab}$}+'
to:
(:cellnr:)'+{$\begin{matrix}\rightarrow\\ AB \end{matrix}$}+'
Changed line 94 from:
(:cell:){$\overrightarrow{ab}$}
to:
(:cell:){$\begin{matrix}\rightarrow\\ AB \end{matrix}$}
Changed lines 102-104 from:
(:cell:)a – b = c ∴ c + b = a
to:
(:cell:)a – b = c
∴ c + b = a
Changed line 114 from:
(:cellnr:)'+≈+'
to:
(:cellnr:)'+{$\approx$}+'
Changed line 116 from:
(:cell:)a ≈ x
to:
(:cell:)'+{$a \approx x$}+'
December 26, 2010 by bobdoleorama -
Added lines 75-144:
(:table border=1 cellpadding=5 cellspacing=0:)
(:cellnr align=center bgcolor=#d4d7ba:)'''Symbol'''
(:cell align=center bgcolor=#d4d7ba:)'''Name'''
(:cell align=center bgcolor=#d4d7ba:)'''Example'''
(:cell align=center bgcolor=#d4d7ba:)'''Meaning'''
(:cellnr:)'+{$\Delta$}+'
(:cell:)Triangle
(:cell:)'+{$\Delta ABC$}+'
(:cell:)The triangle with points A, B and C.
(:cellnr:)'+{$\angle$}+'
(:cell:)Angle
(:cell:){$\angle A$}= 23°
(:cell:)Angle A is 23 degrees.
(:cellnr:)'+{$\overline{AB}$}+'
(:cell:)Line Segment
(:cell:){$\overline{AB}$} = 2.4 ft
(:cell:)The distance between points A and B is 2.4 feet.
(:cellnr:)'+{$\overrightarrow{ab}$}+'
(:cell:)Ray
(:cell:){$\overrightarrow{ab}$}
(:cell:)The line that begins at point A and continues through point B to infinity.
(:cellnr:)'+{$\overleftrightarrow{AB}$}+'
(:cell:)Line
(:cell:){$\overleftrightarrow{AB}$}
(:cell:)The infinite line that passes through or has points A and B.
(:cellnr:)'+∴+'
(:cell:)Therefore
(:cell:)a – b = c ∴ c + b = a
(:cell:)a minus b is c, therefore c plus b is equal to a.
(:cellnr:)'+< and >+'
(:cell:)Less/Greater Than
(:cell:)a < b
(:cell:)Length a is smaller than length b.
(:cellnr:)'+≤ and ≥+'
(:cell:)Less/Greater Than Or Equal To
(:cell:)a < b ≥ c
(:cell:)a is smaller than b, b is larger than or equal to c.
(:cellnr:)'+≈+'
(:cell:)Is Almost Equal To
(:cell:)a ≈ x
(:cell:)The difference between a and x is negligible.
(:cellnr:)'+{$\perp$}+'
(:cell:)Perpendicular To
(:cell:){$AB \perp BC$}
(:cell:)Line AB is perpendicular to Line BC.
(:cellnr:)'+{$\infty$}+'
(:cell:)Infinity
(:cell:){$\overrightarrow{AB} = \infty$}
(:cell:)Line AB has infinite length.
(:cellnr:)'+r+'
(:cell:)Radius
(:cell:)r = 16.5
(:cell:)The radius is 16.5 units in length
(:cellnr:)'+d+'
(:cell:)Diameter
(:cell:)d = 50
(:cell:)The diameter is 50 units in length
(:cellnr:)'+⊙+'
(:cell:)Circle
(:cell:)⊙A<⊙B
(:cell:)Circle named A is smaller than Circle named B.
(:cellnr:)'+π+'
(:cell:)Pi (Pronounced ‘pie’)
(:cell:)π = C/D
(:cell:)Pi equals the circumference of a circle divided by that circles diameter.
(:cellnr:)'+{$\cong$}+'
(:cell:)Congruent To
(:cell:){$AB \cong BC$}
(:cell:)Lines AB and BC are of equal length, they are congruent to each other.
(:tableend:)
December 23, 2010 by bobdoleorama -
December 23, 2010 by bobdoleorama -
Added lines 1-75:
(:title Words & Symbols In Geometry:)
This section will serve as a basic introduction to some of the things you’ll see in geometry – you’ll no doubt have heard many of these terms and seen many of these symbols before, but it’s good to see what we’ll be using, and it’s handy to have a quick guide.
!!! A List of Words Used in Geometry
''(More in depth explanations and examples will be covered in subsequent pages!)''
(:table border=1 cellpadding=5 cellspacing=0:)
(:cellnr bgcolor=#d4d7ba colspan=14 align=center:) '+'''The Breakdown'''+'
(:cellnr:)
*[[#t1 | Definitions for Points, Lines & Planes]]
*[[#t2 | Definitions for Angles]]
*[[#t3 | Definitions for Circles]]
*[[#t4 | Definitions for Other Geometric Words]]
*[[#t5 | Geometric Symbols]]
(:tableend:)
[[#t1]]
!!POINTS, LINES AND PLANES
:Line: In geometry, the definition of line must be more precise than our everyday use of the word because we also have to deal with rays and segments. A line may be straight or curved, and straight lines continue on past any points upon them. Lines are usually named after their two furthest points.
:Ray: A portion of a straight line that begins at a point and continues indefinitely.
:Segment: A portion of a straight line with two distinct endpoints, a line with finite length.
:Point: A symbol used to show position. Represented by a small dot or a letter.
:Endpoint: A point at one end of a segment, or the beginning of a ray.
:Midpoint: A point exactly halfway along a segment.
:Axis / Axes: A fixed, numbered reference line used in coordinate geometry.
:Origin: The point at which two or more axes meet in coordinate geometry.
:Vertex: The point where two straight lines, segments or rays meet creating an angle.
:Bisect: To bisect is to cross a line or segment at it’s midpoint.
:Plane: A two-dimensional form with length and width but no depth or thickness.
:Collinear: A set of two or more points is said to be collinear if they lie upon the same line.
:Congruent: Segments that have the same length are congruent.
\\
[[#t2]]
!!ANGLES
:Measure: A word used when talking about the value of an angle. An example might be “The measure of angle X is 120°”.
:Right Angle: An angle with a measure of exactly 90°.
:Perpendicular: Two lines are perpendicular if the angle which their intersection or meeting point forms is exactly 90°.
:Obtuse: An angle with measure greater than 90° and less than 180°.
:Acute: An angle with a measure of less than 90°.
:Adjacent Angles: Two angles are adjacent if they share the same vertex.
:Vertical Angles: Two non-adjacent angles formed by intersecting lines.
:Complementary: Angles whose combined total value add to exactly 90°.
:Supplementary: Angles whose combined total value add to exactly 180°.
\\
[[#t3]]
!!CIRCLES
:Chord: A straight line joining any two points on a circle.
:Secant: A straight line joining any two points on a circle and continuing beyond those points in both directions.
:Arc: A portion of the circumference of a circle between and including two points.
:Diameter: A chord that passes through the middle of the circle.
:Radius: The straight length or distance between the centre of a circle and a point on its circumference. The radius of a circle is exactly half of that circles diameter. The plural of radius is radii (pronounced ‘ray-dee-eye’).
:Circumference: The length around a circle.
:Semicircle: An arc measuring exactly half the circumference of a circle. Literally half a circle.
:Central Angle: An angle formed by two radii of the same circle (not two overlapping circles).
:Pi: A constant, irrational number derived from the ratio of the circumference of a circle to its diameter. Pi always has the same value, regardless of the size of a circle. This value has been calculated to many trillions of digits, but 3.141 is fine for our purposes.
\\
[[#t4]]
!!OTHER
:Parallelogram: An enclosed two-dimensional shape wherein all opposing sides are parallel.
:Polygon: An enclosed two-dimensional shape with a finite number of straight sides.
:Quadrilateral: A polygon with exactly four sides.
[[#t5]]
!! A TABLE OF GEOMETRIC SYMBOLS
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2019-08-23 18:30:22
|
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https://blog.flyingcoloursmaths.co.uk/secrets-of-the-mathematical-ninja-squares-near-50/
|
Difficulty: ** Impressiveness: ****
(Many thanks to Swar for pointing me at this one - and challenging me to explain it well!)
It’s surprisingly easy to square numbers near 50. Here’s the recipe:
1. Find the difference between your number and 50. (If you’re looking at 46, it’d be -4. If you’re looking at 59, it’d be 9).
2. Add this to 25. This would give you 21 (for 46) or 34 (for 59). This is your ‘hundreds’ number - so you really have 2100 or 3400.
3. Square the number from step 1. For these examples, it’s 16 or 81.
4. Add this on to your answer in step 2. $46^2 = 2116$; $59^2 = 3481$.
Easy peasy! You can go further with it, if you like: to work out $65^2$, you could do $25 + 15 = 40$ for the hundreds (4000) and $15^2 = 225$ to get $65^2 = 4,225$ - exactly what you get from the squaring fives routine.
### Why does it work?
Good question. It all comes down to algebra again. Consider $(50 + x) (50 + x)$. That multiplies out to: $2500 + 100x + x^2$ - which is exactly what the recipe works out, one step at a time!
|
2022-05-22 22:37:59
|
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|
http://math.stackexchange.com/questions/108547/why-are-log-and-ln-being-used-interchangeably
|
# Why are $\log$ and $\ln$ being used interchangeably?
A definition for complex logarithm that I am looking at in a book is as follows -
$\log z = \ln r + i(\theta + 2n\pi)$
Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the log will make a difference to the answer.
It also says a few lines later $e^{\log z} = z$.
Yet again I don't see how this makes sense. Why isn't $\ln$ used instead of $\log$?
-
Abstract Duplicate: math.stackexchange.com/questions/90594/… – user21436 Feb 12 '12 at 17:07
As far as I can surmise, when speaking of the logarithm of a complex variable (inverse of the complex exponential), you write "$\log$", and when speaking of the (natural) logarithm of a real variable, you write "$\ln$" – David Mitra Feb 12 '12 at 17:14
$$ln = log_e$$ $$log = log_{10}$$ (unless the context implies e) $$log = log_2$$ (Algorithms and Computer Science) – Inquest Feb 12 '12 at 17:34
@David Mitra - Normally log on its own means "$\log_{10}$" so when speaking of the logarithm of a complex variable you write "log" and it is implied that you are actually using "$\log_{e}$"...is that correct? – Jim_CS Feb 12 '12 at 18:33
This is fairly standard notation in complex analysis texts. The reason is that you have to define a branch cut of the logarithm, which needs to be distinguished from the real valued logarithm used implicitly in the construction of the branch cut. The issue here is not that the base of the logarithm has changed, but that it is important to specify which branch of the complex logarithm you are talking about and to differentiate that object from the real natural logarithm which is unambiguously defined. – Chris Janjigian Feb 12 '12 at 18:39
Often in math books the base of $\log$ is just assumed to be $e$.
In this case it looks like the reason they are using $\log z$ instead of $\ln$ is to differentiate between when it is a complex function versus when it is a real function.
http://en.wikipedia.org/wiki/Complex_logarithm
-
"$\log$" with no base generally means base the base is $e$, when the topic is mathematics, just as "$\exp$" with no base means the base is $e$. In computer programming languages, "$\log$" also generally means base-$e$ log.
On calculators, "$\log$" with no base means the base is $10$ because calculators are designed by engineers. Ironically, the reasons for frequently using base-$10$ logarithms were made obsolete by calculators, which became prevalent in the early 1970s.
-
Why the down-vote here? If something is objectionable about what I wrote, then telling me what it is would correct the problem. – Michael Hardy Feb 13 '12 at 17:57
I didn't downvote, but I disagree with you position. The issue at hand is the difference between real vs. complex logarithm. Also your claim that base-10 logs are obsolete is ... detached from reality. Have you ever done any signal processing or acoustics? Have you heard of decibels? Ok, I learned to use slide rules before calculators became commonplace, but even so :-). – Jyrki Lahtonen Feb 13 '12 at 22:29
Ironically, you seek support from the conventions adopted by programming language designers, which is more or less the same set of people that designed the calculators. And you denounce the conventions adopted there. – Jyrki Lahtonen Feb 13 '12 at 22:32
It would be exaggerated to say that I "denounced" anything. And I did say "when the topic is mathematics". – Michael Hardy Feb 13 '12 at 23:38
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2016-07-24 07:00:24
|
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|
https://codereview.stackexchange.com/questions/165980/unicode-friendly-password-generation
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The following algorithm is meant to generate a password from a given set of characters, and a length. Is it biased or predictable in any way? And are graphemes the appropriate Unicode thing to use here, or should I be using Unicode Scalar Values, or just using raw bytes?
// cargo-deps: ring="0.9"
extern crate ring;
use ring::rand::{SystemRandom, SecureRandom};
fn pwgen(
length: usize,
chars: &[&str]
) -> String {
if chars.len() > 256 {
panic!("Alphabet too large.");
}
if chars.len() == 0 {
panic!("Alphabet too small.");
}
let mut result = String::with_capacity(length);
let mut bytes = vec![0; length];
let mut complete = 0;
// The largest multiple of chars.len() <= 256.
// This is necessary to maintain uniformity.
let bound = 256 - (256 % chars.len());
while complete < length {
if SystemRandom.fill(&mut bytes).is_err() {
panic!("RNG error.");
}
for &byte in &bytes {
let byte = byte as usize;
if byte < bound {
result.push_str(chars[byte % chars.len()]);
complete += 1;
if complete == length {
return result;
}
}
}
}
result
}
fn main() {
println!("{}", pwgen(24, &["←", "→", "↑", "↓"]))
}
Note: The code can be compiled with cargo-script and all error handling code has been replaced with panics to keep it short.
• You haven't added all of the required code. Trying to compile this errors with "cannot find type Error in this scope". I can also see that SystemRandom isn't defined. – Shepmaster Jun 17 '17 at 14:47
• Sorry! I didn't know that the code had to actually compile, just that all of the relevant code had to be added in. :P – quadrupleslap Jun 17 '17 at 15:23
• I can't find the right meta question now, but you are correct: the question doesn't have to include the support code, but at the least you should link to the support code. As one of the few rust reviewers, I prefer to have the entire compilable code because I compile the code to make sure that any suggestions I make actually work. – Shepmaster Jun 17 '17 at 15:28
• The rule is that enough code has to be included for the question to make sense on its own, even if the links die. An excerpt from real code is fine (Rev 2). You don't have to post a self-contained runnable example, and you certainly don't have to simplify your code for the sake of the question (as you did in Rev 3) — especially if you have to add a disclaimer that it is no longer your real code. – 200_success Jun 17 '17 at 15:54
• Why do you limit the character set to 256 characters? That's way too few for Chinese, Japanese and Korean. – Roland Illig Jul 10 '17 at 6:22
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2020-01-27 17:05:34
|
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|
https://www.statsmodels.org/stable/generated/statsmodels.tsa.forecasting.stl.STLForecastResults.get_prediction.html
|
# statsmodels.tsa.forecasting.stl.STLForecastResults.get_prediction¶
STLForecastResults.get_prediction(start: Optional[Union[int, str, datetime.datetime, pandas._libs.tslibs.timestamps.Timestamp, numpy.datetime64]] = None, end: Optional[Union[int, str, datetime.datetime, pandas._libs.tslibs.timestamps.Timestamp, numpy.datetime64]] = None, dynamic: Union[bool, int, str, datetime.datetime, pandas._libs.tslibs.timestamps.Timestamp, numpy.datetime64] = False, **kwargs: Dict[str, Any])[source]
In-sample prediction and out-of-sample forecasting
Parameters
startint, str, or datetime, optional
Zero-indexed observation number at which to start forecasting, i.e., the first forecast is start. Can also be a date string to parse or a datetime type. Default is the the zeroth observation.
endint, str, or datetime, optional
Zero-indexed observation number at which to end forecasting, i.e., the last forecast is end. Can also be a date string to parse or a datetime type. However, if the dates index does not have a fixed frequency, end must be an integer index if you want out of sample prediction. Default is the last observation in the sample.
dynamicbool, int, str, or datetime, optional
Integer offset relative to start at which to begin dynamic prediction. Can also be an absolute date string to parse or a datetime type (these are not interpreted as offsets). Prior to this observation, true endogenous values will be used for prediction; starting with this observation and continuing through the end of prediction, forecasted endogenous values will be used instead.
**kwargs
Additional arguments may required for forecasting beyond the end of the sample. These arguments are passed into the time series model results’ get_prediction method.
Returns
PredictionResults
PredictionResults instance containing in-sample predictions, out-of-sample forecasts, and prediction intervals.
|
2021-06-18 08:17:25
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http://physics.stackexchange.com/questions/57944/has-unnaturalness-motivated-new-ideas-in-physics
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Has unnaturalness motivated new ideas in physics?
The cosmological constant problem arises because the natural scale of the cosmological constant is $10^{120}$ times larger than what we observe. This implies that the dimensionless constant is much smaller than one and that is considered to be unnatural (though that's a perfect logical possibility). The hierarchy problem is of a similar nature. My question is whether a similar kind of reasoning motivated by unnaturalness has been useful to physics in the past. I.e., is there any example where some constant was deemed unnatural and to resolve the issue new physics was discovered? I would like examples from settled physics rather than from speculative theories. (So supersymmetry doesn't count.)
-
I don't think there are good examples of "unnatural physics" in the past, because of all established physics, except for the CC and EWSB, is "natural". – Vibert Mar 25 '13 at 7:41
That's what I suspected. I don't know how much faith to put into this sort of naturalness arguments as they are more philosophical than physical. I was trying to find out if there have been any precedents. From the long silence to my answer, apparently there is not. – Sudip Paul Mar 28 '13 at 7:23
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2015-03-05 06:55:41
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http://www.formuladirectory.com/user/formula/112
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HOSTING A TOTAL OF 318 FORMULAS WITH CALCULATORS
## $baCOS\left(g\right)$
The variables needed for this formula are : b = magnetic flux, a = area of loop and g = angle between b and the perpendicular to the plane of the loop.
ENTER THE VARIABLES TO BE USED IN THE FORMULA
Similar formulas which you may find interesting.
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2018-10-16 04:56:57
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https://quantumcomputing.stackexchange.com/questions/9091/how-to-get-specific-state-applying-e-i-phi-sigma-2-2-to-0-rangle-or-1
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# How to get specific state applying $e^{-i\phi \sigma_2/2}$ to $|0\rangle$ or $|1\rangle$?
I try to solve problems from Problems in Quantum Computing.
I stuck with problem #3:
I do the following:
Because: $$\sigma_2 = \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix}$$ Then: $$-i \frac{\phi}{2}\sigma_2 = \begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix}$$
$$\exp\begin{pmatrix} 0 & -\frac{\phi}{2}\\ \frac{\phi}{2} & 0 \end{pmatrix} = \begin{pmatrix} 1 & \exp(-\frac{\phi}{2})\\ \exp(\frac{\phi}{2}) & 1 \end{pmatrix}$$
If I multiply the result of the last calculation with $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$ or $$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$$ I can't get $$\psi_1(\phi)$$ or $$\psi_2(\phi)$$. I get some unnormalised state like:
$$\begin{pmatrix} 1 \\ \exp(\frac{\phi}{2}) \end{pmatrix}$$
Does it mean that the definition of the problem is not correct?
Your mistake is computing exponent of matrix; use the formula
$$\exp(i\theta\sigma_2)=\cos(\theta)\cdot I+i\cdot \sin(\theta)\cdot\sigma_2$$
• This "hack" should be in any good QM or Quantum Information course; see wikipedia; I substituted $\hat{n}=(0,1,0)$ – kludg Dec 6 '19 at 21:19
A matrix function $$f(A)$$ for normal matrix $$A$$ is defined as follows $$$$f(A)=\sum_{i=1}^{n}f(\lambda_i)v_iv_i^T$$$$ where $$\lambda_{i}$$ is an eigenvalue and $$v_{i}$$ is coresponding eigenvector (note: transposed vector $$v_{i}$$ is a row vector).
In your case: $$f(A) = \mathrm{e}^A$$ and $$A = -i\frac{\phi}{2}\sigma_{2}$$.
• please use comments for additional questions. $v_{i}$ is ith eigenvector of matrix $A$. – Martin Vesely Jan 16 at 12:16
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2020-02-22 13:30:16
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https://brilliant.org/problems/lewis-dot-diagram-2/
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# Lewis dot diagram
Chemistry Level pending
The above are the Lewis dot diagrams of molecules (A), (B) and (C). Which of the following explanations are correct? (The atomic weights are H=$$1,$$ C=$$12$$ and N=$$14.$$)
a) The boiling point of (A) is lower than that of (B).
b) The bond angle of (A) is larger than that of (C).
c) The solubility of (A) into liquid (C) is larger than that of (B).
×
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2016-10-24 03:34:41
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http://experiment-ufa.ru/Factors-of-625
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# Factors of 625
If it's not what You are looking for type in the field below your own integer, and You will get the solution.
Factors of 625:
By prime factorization of 625 we follow 5 simple steps:
1. We write number 625 above a 2-column table
2. We divide 625 by the smallest possible prime factor
3. We write down on the left side of the table the prime factor and next number to factorize on the ride side
4. We continue to factor in this fashion (we deal with odd numbers by trying small prime factors)
5. We continue until we reach 1 on the ride side of the table
625 prime factors number to factorize 5 125 5 25 5 5 5 1
Factors of 625 = 1×5×5×5×5= $1 × 5^4$
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2018-05-25 18:24:04
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https://math.stackexchange.com/questions/4359423/question-about-summation-in-calculus
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When I have a sum in the type of $$\sum _{n=0}^{\infty }\:c_nx^n$$ and for a given value of $$x$$ the sum is infinity (or converges to infinity? I am not sure if this is correct) does the Sum converges in general?
For example, the $$\sum _{n=0}^{\infty }\:1^n= 1+1+1+\dots=+\infty$$ does this sum converges and if yes the terminology is, it converges to infinity?
• We usually say the latter sum "diverges" as opposed to converges to infinity. Jan 17 at 22:09
The sum diverges to infinity - we wouldn't usually say it converges to infinity, and it would certainly be misleading to say it's convergent without further clarification.
The idea of converging is that the sum approaches a limit with smaller and smaller fluctuations: for instance, in $$1+1/2+1/4+1/8+\cdots$$, the sum is eventually always within $$0.1$$ of $$2$$ (after adding five terms), and then eventually always within $$0.000000001$$, and so on (for any positive $$\epsilon$$ you choose). The same is true of convergent series that fluctuate above and below the limit, like $$1/2-1/4+1/8-1/16+\cdots=1/3$$.
In the case of $$1+1+1+\cdots$$, the sum is never within $$0.1$$ of $$\infty$$: each partial sum ($$1,2,3,4,\dots$$) is infinitely far away from $$\infty$$. So it doesn't fit the reason we define limits the way we do, nor that defining property of a convergent series.
• That was what I thought as well however in one of my book when I have the sum; Σ(x^n)= 1/(1-x) for -1<x<1, it says that the sum converges for -1<x<1. He then tests for x=-1 it says it diverges and then for x=1 it says it converges. It says that for x=1 the 1/(x-1) there is a vertical asymptote. It then concludes that the sum converges for -1<x<1, 1. I do not get it. Jan 17 at 22:23
• The geometric series converges if and only if $-1<x<1$, and not for $x=1$, so I'm not too sure what the textbook's point is. Can you post the full context to see if there is something lost in translation by your description?
– A.M.
Jan 17 at 22:29
• It is in greek so I don't think you will understand. Do you want me to post the question I mentioned above for Σ(x^n), with its answer that is on the book and try to translate what the question asks, to see if I misunderstood the question? Jan 17 at 22:32
• Ah, I see. I may not be able to help in this case, other than to reassert that we wouldn't say in English that the series converges when $x = 1$. I couldn't tell you if the Greek language uses different terminology.
– A.M.
Jan 17 at 22:38
• I read that answer as "The radius of convergence is $1$. The convergence domain is $-1<x<1$." Jan 17 at 23:15
Before discussing convergence itself, it is profitable to remember the definition of series.
A numerical sequence $$s_{n} = a_{1} + a_{2} + \ldots a_{n}$$ (when $$a_{k}\in\mathbb{K}\in\{\mathbb{R},\mathbb{C}\})$$ converges to $$s\in\mathbb{K}$$ iff
\begin{align*} (\forall\varepsilon > 0)(\exists n_{\varepsilon}\in\mathbb{N})(\forall n\in\mathbb{N})(n\geq n_{\varepsilon} \Rightarrow |s_{n} - s| < \varepsilon) \end{align*}
In order to conclude if it converges or not, there are several approaches which may help to conclude so.
The most known (as far as I know) are the absolute convergence, comparison test, the Leibniz test, the ratio test, the root test, the Cauchy test, the condensation test and so on.
Now consider the following (formal) power series \begin{align*} \sum_{n=0}^{\infty}a_{n}x^{n} := a_{0} + a_{1}x + a_{2}x^{2} + \ldots \end{align*}
Depending on the values of $$x$$, it may converge or not.
With the purpose of discovering so, you can apply one of the above criteria to study convergence.
Here we present some examples.
Consider the following power series and its corresponding closed form \begin{align*} \sum_{n=0}^{\infty}x^{n} = 1 + x + x^{2} + x^{3} + \ldots = \frac{1}{1-x} \end{align*}
If we are interested in knowing when it converges, we can apply the root test, for instance.
What about the following series? \begin{align*} \sum_{n=0}^{\infty}\frac{x^{n}}{n!} = 1 + \frac{x}{1!} + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \ldots \end{align*}
Based on the ratio test, one concludes it converges for every possible real number.
Hence the convergence of a power series strictly depends on the value $$x$$ you choose to plugin.
Hopefully this helps !
EDIT
With respect to the geometric series, observe the partial sums are given by \begin{align*} s_{n}(x) = 1 + x + x^{2} + x^{3} + \ldots + x^{n-1} \end{align*}
Hence if we multiply both sides by $$x$$, one arrives at \begin{align*} xs_{n}(x) = x + x^{2} + x^{3} + x^{4} + \ldots + x^{n} \end{align*}
Finally, subtracting the second equation from the first, one gets that \begin{align*} (1-x)s_{n}(x) = 1 - x^{n} \Rightarrow s_{n}(x) = \frac{1 - x^{n}}{1 - x} \end{align*}
So, in order to study the limit of $$s_{n}$$, it suffices to study the convergence of $$x^{n}$$.
Let us consider that $$|x| < 1$$. Then it can be proven (by induction) that $$|x|^{n} < |x|^{n-1}$$.
Moreover, we do also know that $$|x|^{n}\geq 0$$. Since $$|x|^{n}$$ is strictly decreasing and bounded, it converges.
Let us denote such limit by $$L$$. The next procedure makes us conclude that $$L = 0$$: \begin{align*} L & = \lim_{n\to\infty}|x|^{n+1}\\\\ & = \lim_{n\to\infty}|x|\times |x|^{n}\\\\ & = |x|\times\lim_{n\to\infty}|x|^{n}\\\\ & = |x|L\\\\ & \Rightarrow L(1 - |x|) = 0\\\\ & \Rightarrow L = 0 \end{align*}
Before answering your question, it is profitable to remind that convergence implies the general term goes to zero as $$n$$ approaches infinity. Since $$|x|^{n}\not\to 0$$ whenever $$|x|\geq 1$$, the convergence of the last series happens iff $$|x| < 1$$, and we are done.
• Could you explain why the downvote? Jan 17 at 22:29
• I did not downvote and I find it extremely helpful. For the second sum that you mention why use the root test when theory says that it converges for -1<x<1? And does it converges to 1? If so why? Because there is a question in my book about this sum that says it converges to 1. Jan 17 at 22:36
• @Vaggelis I have edited my answer. Hopefully it helps ! Jan 18 at 2:45
• Yes it did thank you very much! Jan 18 at 8:17
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2022-05-17 20:52:46
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https://guitarknights.com/guitar-hero-online-guitar-app.html
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There are two basic types of electric guitars: solidbody and hollowbody. Today, the electric guitar still features in all types of music Ð rock, blues, jazz and big bands Ð and is played by men and women, young and old, throughout the world. Some well-known electric guitarists include Muddy Waters, Jimi Hendrix, Eric Clapton, Jimmy Page, Eddie Van Halen, Steve Vai, Joe Satriani, Pat Metheny, Wes Montgomery, Chrissie Hynde and Liz Phair.
"Open" chords get their name from the fact that they generally include strings played open. This means that the strings are played without being pushed down at a fret, which makes chords including them easier to play for beginners. When you start to learn chords, you have to focus on using the right fingers to press down each note and make sure you're pressing the strings down firmly enough.
Whether you just started guitar lessons or you've been playing for a while, you may be itching to learn some new songs and take on some new challenges. You might be wondering: where can I go from here? That's where alternate guitar tunings come in! With this guide from Michael L., you'll learn how alternate guitar tunings can take your playing to the next level... One of the amazing things about the guitar is its versatility. Not only can you play rhythm and/or melody in different genres,
WE ARE THE OLDEST MUSIC SCHOOL IN LEE COUNTY!!!!!!! STUDENTS COME TO OUR STUDIO IN NORTH FORT MYERS, BUT PIANO IS AVAILABLE VIA SKYPE, OR WINDOWS LIVE LESSONS ARE AVAILABLE IN OUR STUDIOS ONLY. Huffmaster's Centre of Music began 59 years in North Fort Myers and remains in the same area. Brooke Huffmaster, has taught 20 years in her grandmother's, Patricia Huffmaster&...
The musical theory of chords is reviewed, to provide terminology for a discussion of guitar chords. Three kinds of chords, which are emphasized in introductions to guitar-playing,[10][11] are discussed. These basic chords arise in chord-triples that are conventional in Western music, triples that are called three-chord progressions. After each type of chord is introduced, its role in three-chord progressions is noted.
Open tuning refers to a guitar tuned so that strumming the open strings produces a chord, typically a major chord. The base chord consists of at least 3 notes and may include all the strings or a subset. The tuning is named for the open chord, Open D, open G, and open A are popular tunings. All similar chords in the chromatic scale can then be played by barring a single fret.[16] Open tunings are common in blues and folk music,[17] and they are used in the playing of slide and bottleneck guitars.[16][18] Many musicians use open tunings when playing slide guitar.[17]
The guitar is a type of chordophone, traditionally constructed from wood and strung with either gut, nylon or steel strings and distinguished from other chordophones by its construction and tuning. The modern guitar was preceded by the gittern, the vihuela, the four-course Renaissance guitar, and the five-course baroque guitar, all of which contributed to the development of the modern six-string instrument.
###### As stated above, construction has just as much of an impact on a guitar’s tone as material. The factors that make up construction are as follows: gauge, string core, winding type, and string coating. And while these factors are all important, keep in mind that different companies use different approaches to all of them. So never be afraid to try out a variety brands, because while the strings may look the same you will get a different response.
As you start practicing, your fingers may be sore for a while, but that will pass with four to six weeks. One thing I want to warn you about is that new guitar players can get frustrated when they can’t play clean chords because they try to switch between chords too soon. Often, they try to switch chords before they’ve really learned and memorized each chord shape. For now, don’t worry about switching chords and just work on each shape, getting them down, and going right to them.
The sound of an acoustic guitar is arguably the most well known in all of modern popular music. Without electronic components, the guitar relies solely on the interaction between the strings and the sound box to produce every note. Because of this, the strings of your acoustic guitar can significantly influence its sound. To get the most out of your guitar, keep an eye on the condition of the strings. You don't need to wait until your strings break to replace them. When they get old enough that their tone starts to change, it's time to re-string. Depending on the guitar, you may choose either nylon or metal strings. Nylon strings are the modern substitute for gut strings, so they're usually found on older styles of guitar such as baroque or flamenco. If you play the classical guitar, take care to fit it with the correct strings: since classical guitars are sized and tensioned differently from other varieties of acoustic guitar, they can be damaged by standard strings. Similarly, classical guitar strings won't work with other guitars. For metal-stringed guitars, there are several options available, each with its own acoustic character. The three most common types of metal strings are bronze, phosphor-bronze and silk-and-steel. The staple string for many guitarists, bronze produces a bright, quickly-fading tone that's lively and well-suited to any style of music. Phosphor-bronze is similar, but with added warmth and longer sustain. For a completely different sound, consider silk-and-steel. These strings create a tone that's gentle and mellow. Lower in tension and available in lighter gauges, silk-and-steel strings are easier to play and are great for vintage guitars that need special strings. It's also important to keep in mind the gauge of the strings you're choosing. Higher gauges are thicker, producing increased volume and extended sustain with an overall warmer tone. The trade-off of the rich overtones of heavy-gauge strings is that they are more challenging to play, requiring more force to fret, pick and strum. If you are an experienced guitarist, you likely have a preferred gauge already. If you're a beginner, it's a good idea to start with a lighter gauge to make the learning curve more forgiving. In the end, the right combination of material and gauge is a matter of personal preference. You may need to try several different acoustic guitar strings before you find the perfect ones for you, but the results will certainly be worth it.
Another class of alternative tunings are called drop tunings, because the tuning drops down the lowest string. Dropping down the lowest string a whole tone results in the "drop-D" (or "dropped D") tuning. Its open-string notes DADGBE (from low to high) allow for a deep bass D note, which can be used in keys such as D major, d minor and G major. It simplifies the playing of simple fifths (powerchords). Many contemporary rock bands re-tune all strings down, making, for example, Drop-C or Drop-B tunings.
Unlike a piano or the voices of a choir, the guitar (in standard tuning) has difficulty playing the chords as stacks of thirds, which would require the left hand to span too many frets,[40] particularly for dominant seventh chords, as explained below. If in a particular tuning chords cannot be played in closed position, then they often can be played in open position; similarly, if in a particular tuning chords cannot be played in root position, they can often be played in inverted positions. A chord is inverted when the bass note is not the root note. Additional chords can be generated with drop-2 (or drop-3) voicing, which are discussed for standard tuning's implementation of dominant seventh chords (below).
###### Another class of alternative tunings are called drop tunings, because the tuning drops down the lowest string. Dropping down the lowest string a whole tone results in the "drop-D" (or "dropped D") tuning. Its open-string notes DADGBE (from low to high) allow for a deep bass D note, which can be used in keys such as D major, d minor and G major. It simplifies the playing of simple fifths (powerchords). Many contemporary rock bands re-tune all strings down, making, for example, Drop-C or Drop-B tunings.
Many influences are cited as antecedents to the modern guitar. Although the development of the earliest "guitars" is lost in the history of medieval Spain, two instruments are commonly cited as their most influential predecessors, the European lute and its cousin, the four-string oud; the latter was brought to Iberia by the Moors in the 8th century.[7]
Solo, lead, and rhythm guitarists everywhere can now access the best selection of instantly downloadable digital sheet music and guitar tab on the internet. Put down the pick for just a moment and put your fingers to work browsing through Musicnotes.com's vast archives of guitar tabs ready to be enjoyed by musicians of all ages. Our collection features a weekly updated catalogue of some of guitars greatest compilations.
Do you play a warm-up exercise when you practice guitar? Guitar teacher Kirk R. shares three guitar exercises that are perfect for players at all levels... There are literally thousands of exercises and studies for the guitar. There are some that are great for beginners who are just getting used to having their fingers on the guitar, and some that are designed to challenge and grow the technique of seasoned players. But who has time to learn thousands of guitar exercises, even over man
Getting to grips with how chords are formed gives you a basic introduction to music theory and helps you understand the ways you can alter them to create more interesting sounds. All chords are built from certain notes in scales. The C major scale is the easiest, because it just runs C, D, E, F, G, A and B. These notes are numbered (usually using Roman numerals) in that order, from one (I) to seven (VII).
Are you stuck in a musical rut? New tunings and tricks can help you keep learning guitar in fresh, fun ways. Try one of these great tips from guitar teacher Samuel B. to breathe new life into your guitar playing... One of the first things I tell any new student is that I don't specialize in a formal discipline. If jazz or classical training is your objective, then I'm not your guy. Instead, I specialize primarily in American roots music (that which we tend to casually lump together as "folk"
The ratio of the spacing of two consecutive frets is {\displaystyle {\sqrt[{12}]{2}}} (twelfth root of two). In practice, luthiers determine fret positions using the constant 17.817—an approximation to 1/(1-1/ {\displaystyle {\sqrt[{12}]{2}}} ). If the nth fret is a distance x from the bridge, then the distance from the (n+1)th fret to the bridge is x-(x/17.817).[15] Frets are available in several different gauges and can be fitted according to player preference. Among these are "jumbo" frets, which have much thicker gauge, allowing for use of a slight vibrato technique from pushing the string down harder and softer. "Scalloped" fretboards, where the wood of the fretboard itself is "scooped out" between the frets, allow a dramatic vibrato effect. Fine frets, much flatter, allow a very low string-action, but require that other conditions, such as curvature of the neck, be well-maintained to prevent buzz.
If you're looking for some of our older videos - you can browse the guitar lessons archive. You may also be interested browsing our guitar lessons on YouTube. There you can learn how to play guitar with all of our most popular beginner guitar lessons, rhythm guitar lessons, lead guitar lessons, and blues guitar lessons. We have something for guitar students of all skill levels.
UPDATE: SEPTEMBER 3rd, 2010 -- Good evening, and hi everybody! I get requests to add tabs once in a while, and for years one of the most common requests has been 'Psychic Hearts', and more recently 'Trees Outside the Academy'. I resisted for years, but boredom and the need to please has a funny way of making things happen, so I'm proud to bring you tabs for the entirety of "Psychic Hearts" and its related tracks, as well as the majority of "Trees Outside the Academy". I was originally planning on providing bass tabs as well as the Mascis solos, but I decided I wasn't that desperate for accolade. With all this attention on Thurston, I felt bad for Lee, so I've updated my outdated tab for his excellent solo acoustic piece "Here" (located under "Other Tabs") with the proper tuning, which also happens to be the tuning for the equally excellent "Lee #2", so I've updated that one too!
The BMus in performance with a concentration in guitar is a program that focuses on the study of classical guitar literature and techniques. Goals include enabling students to express themselves musically while emphasizing the skills necessary to pursue careers as professional musicians. The course of study includes extensive performance experiences.
#### We were privileged to have worked with hundreds of top artists and educators who authored the original magazine articles and then recorded audio lessons in their own studios. While video is the medium of choice today, these 1,200+ audio lessons still deliver amazing instruction from some of the best in the biz — perfect for those long drives and commutes!
As their categorical name suggests, extended chords indeed extend seventh chords by stacking one or more additional third-intervals, successively constructing ninth, eleventh, and finally thirteenth chords; thirteenth chords contain all seven notes of the diatonic scale. In closed position, extended chords contain dissonant intervals or may sound supersaturated, particularly thirteenth chords with their seven notes. Consequently, extended chords are often played with the omission of one or more tones, especially the fifth and often the third,[92][93] as already noted for seventh chords; similarly, eleventh chords often omit the ninth, and thirteenth chords the ninth or eleventh. Often, the third is raised an octave, mimicking its position in the root's sequence of harmonics.[92]
Extending the tunings of violins and cellos, all-fifths tuning offers an expanded range CGDAEB,[25] which however has been impossible to implement on a conventional guitar. All-fifths tuning is used for the lowest five strings of the new standard tuning of Robert Fripp and his former students in Guitar Craft courses; new standard tuning has a high G on its last string CGDAE-G.[26][27]
The Beatles668 tabs 637 visualizations1 Elvis Presley542 tabs 410 visualizations2 Nirvana513 tabs 360 visualizations3 Eagles139 tabs 349 visualizations4 Frank Sinatra387 tabs 348 visualizations5 Misc Soundtrack1160 tabs 279 visualizations6 Misc Cartoons897 tabs 265 visualizations7 Green Day650 tabs 254 visualizations8 Metallica382 tabs 247 visualizations9 Johnny Cash441 tabs 221 visualizations10
Extending the tunings of violins and cellos, all-fifths tuning offers an expanded range CGDAEB,[25] which however has been impossible to implement on a conventional guitar. All-fifths tuning is used for the lowest five strings of the new standard tuning of Robert Fripp and his former students in Guitar Craft courses; new standard tuning has a high G on its last string CGDAE-G.[26][27]
At least two instruments called "guitars" were in use in Spain by 1200: the guitarra latina (Latin guitar) and the so-called guitarra morisca (Moorish guitar). The guitarra morisca had a rounded back, wide fingerboard, and several sound holes. The guitarra Latina had a single sound hole and a narrower neck. By the 14th century the qualifiers "moresca" or "morisca" and "latina" had been dropped, and these two cordophones were simply referred to as guitars.[8]
The history of stringed instruments dates back thousands of years, while the modern guitar can be traced back to the oud and the lute of the 8th and 9th centuries. Today's guitars come in countless variations and at JustStrings.com, we have a dazzling variety of guitar strings and other types of strings for many different instruments. Advanced technology combined with traditional craftsmanship produces some of the finest strings that boast long-lasting durability and exceptional tonal quality. Whether you're looking for rugged, great-sounding electric guitar strings to take on tour with you, or the perfect set of strings for your lap steel, we're bound to have exactly what you need. We take great pride in bringing you the best quality strings from the world's leading innovators. Our acoustic guitar strings, from manufacturers like Ernie Ball, GHS, Thomastik-Infeld and Fender, are available in many varieties, from silver-plated copper flat wound to 80/20 bronze round-wound. You'll find plain uncoated strings and coated strings that improve longevity. Classical guitar strings, traditionally made from gut, are now strung with nylon or advanced synthetic materials for better performance. Electric guitar strings are available in nickel-plated steel, pure nickel or stainless steel. We also have strings for resophonic guitars (also known as Dobros), Steinberger guitars and pedal steel guitars. Browse our bass guitar strings, and find 7-string, 8-string or 12-string guitar strings. We also carry banjo strings, harp strings, and strings for octave, tenor, baritone and Renaissance guitars. We are the leading source of premium strings for nearly every type of instrument.
# Jump up ^ "We know from literary sources that the five course guitar was immensely popular in Spain in the early seventeenth century and was also widely played in France and Italy...Yet almost all the surviving guitars were built in Italy...This apparent disparity between the documentary and instrumental evidence can be explained by the fact that, in general, only the more expensively made guitars have been kept as collectors' pieces. During the early seventeenth century the guitar was an instrument of the people of Spain, but was widely played by the Italian aristocracy." Tom and Mary Anne Evans. Guitars: From the Renaissance to Rock. Paddington Press Ltd, 1977, p. 24.
For the longest time I have "messed around" with my guitar. I bought a book on Jazz guitar chords many years ago and found that those fingerings were way above my skill level. I decided to get a basic chord book to reinitiate myself. This is that book. Going through these pages has re-ignited my lust to learn more about playing my guitar. One chord at a time. Simple. Skip to another key just as simple. Actually, I haven't realized how many chords I really know how to play. I just need to learn the name of those chords. This book will help with that too. Give it a try.
#### Many influences are cited as antecedents to the modern guitar. Although the development of the earliest "guitars" is lost in the history of medieval Spain, two instruments are commonly cited as their most influential predecessors, the European lute and its cousin, the four-string oud; the latter was brought to Iberia by the Moors in the 8th century.[7]
There's an abundance of guitar information out there on the web, some good, some not. I stumbled across Justin Sandercoe's site a year ago and now tell everyone about it. The lessons are conveyed so clearly, concisely and in the most congenial way. The site is laid out logically as well so you can to go straight to your area of interest... beginner, blues, rock, folk, jazz, rhythm, fingerpicking... it's all there and more. Spend ten minutes with Justin and you'll not only play better but feel better too. From novice to know-it-all, everyone will learn something from Sandercoe.
Welcome to video eight in the Beginner Guitar Quick-Start Series. In this lesson, we’re going to go through your first two chords. You’ll learn A minor 7 and C major. These two guitar chords will be useful for you because you’ll be using them often through your guitar career. A minor 7 is good to start with because it is fairly easy, and C major is great chord to learn how to play clean sounding chords.
#### Every songwriter runs into writers' block at some point in their career. To help you dig your way out of the dreaded doldrums of songwriters' block, we put together 25 songwriting tips and prompts plus great songs to inspire you. Check out these songwriting tips and find your muse today! Bonus: Take the quiz to find out what you should write your next song about! Write about your day. Think your life is boring and you have nothing to say? Check out the lyrics to thi
Once you know the basic chords, it might be easier to think of them the way the function inside a key. For example, when in the key of E, the E (I) is called the Tonic. It's what all the other chords want to get to—which is what helps give western music its sense of motion. The A (IV) in the key of E functions as the Subdominant—it's sort of a passive in-between, just as happy to continue forward, as to relax back to the Tonic. The Dominant is just what it sounds like: it leads you where it wants to go. In the key of E, that role is filled by the B (V), and will definitely make your brain want to get back to the Tonic! When you get more familiar with the chords, and want to sketch out a tune, try writing it as I-IV-V (or variations of that) instead of E-A-B. It will make it much easier to transpose when you find out your singer cannot sing in the original key!
A string’s gauge is how thick it is. As a general rule, the thicker a string is the warmer its response will be and the more volume it will produce. However, thicker strings are also stiffer. This makes it harder to fret the string and makes it more difficult to execute heavy string bends. Thinner strings are generally brighter and easier to play, but on some instruments they can sound thin and tinny.
Play each note of the chord one after another (known as playing an arpeggio) to check if any are being accidently muted or need pressing down harder. If you find that you're accidently muting a string with one of your fingers, try lowering your thumb so the tip reaches around half way up the back of the guitar neck. This gives your fingers a better angle to approach the fretboard.
I would especially like to stress the gentle approach Justin takes with two key aspects that contributed to my development as a musician - music theory and ear training. Justin has succeeded in conveying the importance and profoundness of understanding music both theoretically and through your ears while maintaining a simple and accessible approach to them, all while sticking to what is ultimately the most important motto: 'If it sounds good, it is good'.
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2019-06-16 21:20:40
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https://mlpro.io/blog/learn/sobel-edge-detection/
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Blogs/Sobel Edge Detection
# Sobel Edge Detection
mesakarghm May 21 2021 1 min read 41 views
Computer Vision
Edge Detection includes a variety of mathematical methods that helps us to identify points in an image where the image has some sort of discontinuities. Edge Detection is an important tool which is used extensively for feature detection and feature extraction in the field of digital image processing.
Sobel Edge Detectior is a edge detection implementation used in image processing where it creates an image emphasising edges. In Sobel Edge Detection, the image is processed in the X and Y directions seperately which is later combined together to form a new image. The new array formed after combination of the first two arrays represents the sum of the X and Y edges of the image. If A is the original image, the following operation can be used for sobel edge detection in the X-axis.
$$G_x = 1/4\begin{bmatrix} -1 & 0 & 1\\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix} * A$$
Similarly, we can compute it in the Y-direction as:
$$G_y = 1/4 \begin{bmatrix} 1 & 1& 1\\ 0 & 0 & 0 \\ -1 & -2 & -1 \end{bmatrix} * A$$
Here Gx and Gy represent the vertical and horizontal derrivative approximations of the input Image respectively. We can combine the gradient approximation to give the gradient magnitude using:
$$G = \sqrt{G_x ^{2} + G_y^{2} }$$
Below I provide an implementation of the Sobel Edge Detection in Python. The two images show the effects on image before and after the application of Sobel Edge Detection.
Learn and practice this concept here:
https://mlpro.io/problems/edge_detection/
def edge_detection(image):
kernel1 = np.array([[-1/4,0,1/4],[-1/2,0,1/2],[-1/4,0,1/4]])
kernel2 = np.array([[1/4,1/4,1/4],[0,0,0],[-1/4,-1/2,-1/4]])
# Flip the kernel
kernel1 = np.flipud(np.fliplr(kernel1))
kernel2 = np.flipud(np.fliplr(kernel2))
# convolution output
output1 = np.zeros_like(image)
output2 = np.zeros_like(image)
return np.asarray(g,dtype = 'uint8')
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2021-12-07 23:37:07
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http://electronics.stackexchange.com/questions/27195/applets-illustrating-the-operation-of-diodes-and-transistors
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# Applets illustrating the operation of diodes and transistors
Anyone know a site with interactive applets illustrating the operation of diodes and transistors as well?
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Here you can find a simulator with all the main components, and some demos about diodes, like this, and transistors, like this.
Take a look, but I would suggest you to try matching the examples with the theory, to understand better what is the transconductance $\beta _F$ and so on.
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+1 for falstad. It is an excellent learning tool. – sptrks Feb 26 '12 at 15:43
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2014-03-10 01:17:44
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https://undergroundmathematics.org/calculus-meets-functions/thinking-constantly/warm-up-solution
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Building blocks
# Thinking constantly Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
## Warm-up solution
You are moving along a straight line, with your acceleration given by $a(t) = 6t - 2.$
• How fast will you be travelling at $t=2$?
You can think about the relationship between acceleration, velocity and displacement in two ways.
$a(t) = \dfrac{d}{dt}\big(v(t)\big)$
$v(t) = \dfrac{d}{dt}\big(s(t)\big)$
$s(t) = \int{v(t)}\, dt$
$v(t) = \int{a(t)}\,dt$
So we need to integrate the acceleration function to find your velocity function.
\begin{align*} v(t) &= \int{(6t - 2)}\,dt\\ v(t) &= 3t^2 - 2t + c \end{align*}
What does the constant $c$ represent in terms of velocity?
We know that $t=2$. Do we have enough information to say how fast we are going?
If we calculate $v(2)$ we get
\begin{align*} v(2) &= 12 - 4 + c\\ &= 8+c, \end{align*}
so we could say our velocity is $8 + c$ units.
Or we could calculate the integral
\begin{align*} \int_{0}^{2}{(6t - 2)}\,dt &= \left[3t^2 - 2t +c \right]^2_0\\ &= v(2) - v(0) \\ &= 8 \end{align*}
and say our velocity is $8$ units faster than it was at $t = 0$.
In both cases we have accounted for the constant of integration which represents our initial velocity. This is important as we weren’t given any information about our velocity at $t=0$, so we cannot assume that it was zero.
• Where will you be on the line at $t=2$?
To then find your displacement function we need to integrate the velocity function.
\begin{align*} s(t) &= \int{(3t^2 - 2t + c)}\,dt\\ s(t) &= t^3 - t^2 + ct + d \end{align*}
Why is it important not to forget the constant of integration from the velocity function?
What does the constant $d$ represent in terms of displacement?
With the displacement function we could calculate
\begin{align*} s(2) &= 2^{3} - t^{2} + 2c + d \\ &=4 + 2c +d \end{align*}
or
\begin{align*} \int_{0}^{2}{(3t^2 - 2t + c)}\,dt &= \left[t^3 - t^2 + ct + d \right]^2_0 \\ &= s(2) - s(0) \\ &= 4 + 2c. \end{align*}
One of these gives us your displacement from your starting point, and one gives us your displacement relative to a fixed origin. Which is which?
Under what circumstances would the two answers be the same?
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2018-01-17 10:52:42
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http://www.zora.uzh.ch/22710/
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Quick Search:
Browse by:
Zurich Open Repository and Archive
Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-22710
# Arratia, R; Barbour, A D; Tavaré, S (1992). Poisson process approximations for the Ewens sampling formula. Annals of Applied Probability, 2(3):519-535.
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## Abstract
The Ewens sampling formula is a family of measures on permutations, that arises in population genetics, Bayesian statistics and many other applications. This family is indexed by a parameter ; the usual uniform measure is included as the special case . Under the Ewens sampling formula with parameter , the process of cycle counts converges to a Poisson process with independent coordinates and . Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of and . This Poisson approximation can be used to give simple proofs of limit theorems with bounds for a wide variety of functionals of such random permutations.
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2014-10-31 18:45:38
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https://forum.bebac.at/forum_entry.php?id=22337
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Additional Group in Bioequivalence Study [Study Performance]
» I am assuming you meant you had intended one group, but now have two. In your post your mentioned "instead of four groups" which did not compute for me if we talk dosing groups. But perhaps I got you wrong?
Hi Elmaestro,
What i mean is 36 subjects were divided in to four groups for the study. However an additional group was added due to that the study not completed in four groups. I assume that this will not have an effect on the study? But would like to understand what is actually the group effect?
PVRC
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2021-09-27 03:33:49
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https://proofwiki.org/wiki/Definition:Integrating_Factor
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# Definition:Integrating Factor
## Definition
Consider the first order ordinary differential equation:
$(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
such that $M$ and $N$ are real functions of two variables which are not homogeneous functions of the same degree.
Suppose also that:
$\dfrac {\partial M} {\partial y} \ne \dfrac {\partial N} {\partial x}$
Then from Solution to Exact Differential Equation, $(1)$ is not exact, and that method can not be used to solve it.
However, suppose we can find a real function of two variables $\map \mu {x, y}$ such that:
$\map \mu {x, y} \paren {\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} } = 0$
is exact.
Then the solution of $(1)$ can be found by the technique defined in Solution to Exact Differential Equation.
The function $\map \mu {x, y}$ is called an integrating factor.
## Also known as
An integrating factor is sometimes known as an Euler multiplier, after Leonhard Paul Euler, who first introduced such a notion.
## Also see
• Existence of Integrating Factor, in which it is shown that if an equation in the form of $(1)$ has a general solution, then it always has an integrating factor.
## Historical Note
The technique of using an integrating factor to solve a differential equation was invented by Leonhard Paul Euler.
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2019-08-24 18:51:01
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https://socratic.org/questions/the-oxidation-and-reduction-equations-for-when-zinc-is-placed-in-water
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The oxidation and reduction equations for when zinc is placed in water?
Jul 10, 2017
$\text{Oxidation:}$ $Z n \left(s\right) \rightarrow Z {n}^{2 +} + 2 {e}^{-}$ $\left(i\right)$
Explanation:
Water COULD be reduced.........
${H}_{2} O + {e}^{-} \rightarrow \frac{1}{2} {H}_{2} \left(g\right) \uparrow + H {O}^{-}$ $\left(i i\right)$
We add (i) and (ii) such that the electrons are eliminated from the reaction.......
$\left(i\right) + 2 \times \left(i i\right)$
$Z n \left(s\right) + 2 {H}_{2} O \rightarrow Z {n}^{2 +} + {H}_{2} \left(g\right) \uparrow + 2 H {O}^{-}$
We would need exceptionally clean and pure zinc metal for this reaction to occur, but in the presence of acid.......we could simply write.....
$Z n \left(s\right) + 2 H C l \left(a q\right) \rightarrow Z n C {l}_{2} \left(a q\right) + {H}_{2} \left(g\right) \uparrow$
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2020-02-18 17:21:13
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http://mathhelpforum.com/calculus/104067-how-do-you-find-horizontal-asymptotes.html
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# Thread: How do you find horizontal asymptotes?
1. ## How do you find horizontal asymptotes?
How do I find the horizontal asymptotes in this (or any) limit? I attached the question as well as my work but I am lost with horizontal asymptotes so my work might be meaningless and I would appreciate any detailed explanation.
2. Originally Posted by s3a
How do I find the horizontal asymptotes in this (or any) limit? I attached the question as well as my work but I am lost with horizontal asymptotes so my work might be meaningless and I would appreciate any detailed explanation.
The horizontal asymptotes of this function will be the vertical asymptotes of the inverse function.
So, the inverse function is defined by
$x = \frac{17y}{(y^4 + 1)^{\frac{1}{4}}}$
$x^4 = \frac{83521y^4}{y^4 + 1}$
$x^4 = 83521 - \frac{83521}{y^4 + 1}$
$x^4 - 83521 = - \frac{83521}{y^4 + 1}$
$\frac{1}{x^4 - 83521} = -\frac{y^4 + 1}{83521}$
$-\frac{83521}{x^4 - 83521} = y^4 + 1$
$-\frac{83521}{x^4 - 83521} - 1 = y^4$
$\left(-\frac{83521}{x^4 - 83521} - 1\right)^{\frac{1}{4}} = y$.
So the vertical asymptotes of the inverse function are $x = -17$ and $x = 17$.
Thus, the horizontal asymptotes of the original function are $y = -17$ and $y = 17$.
3. What am I doing wrong? (Work attached)
(Ignore the black handwriting)
4. Oh I get it! x^4 = +/- x^(1/4) that's why x = 17 and x = -17 as well.
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2017-07-22 01:05:48
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https://math.meta.stackexchange.com/questions/31998/rationale-and-need-for-rule-barring-low-reputation-users-from-inserting-images/31999
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# Rationale and need for rule barring low-reputation users from inserting images
Occasionally, a new user will ask a question for which a picture is required. Sometimes it's because they don't have the time, energy, or knowledge to convert a handwritten or typeset equation (often their own substantial work) into MathJax/LaTeX, but sometimes it's because an actual diagram is vital to understanding the question.
Because they're new, the user will have insufficient reputation to insert the actual picture into the question. They are forced to simply link to the picture (as, for example, here). As a result, the question seems incomplete, it gets downvotes, etc. Sometimes, it's warranted, but not always. Someone often comes by to do the image insertion for them (or to convert into MathJax), but this seems like a kluge, making someone wait around for that.
I assume that the restriction for importing an image is for some reason such as resource consumption, but I've never seen this reason stated. It might be somewhere in the site rules I haven't looked, but for example, searching "insert image reputation" here or on the main Math SE site didn't turn up anything. Whatever the reason is, is it worth hamstringing new users in this way? Perhaps a limit on image size could be used, instead of a flat out ban?
• I'm chagrined to realise that it was the first thing I thought of. I thought I was innocent. $\ \ddot\smile$ – Calum Gilhooley Jun 26 '20 at 13:51
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2021-04-23 18:00:33
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https://kr.mathworks.com/help/pde/ug/heat-equation-for-metal-block-with-cavity.html
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# Heat Transfer in Block with Cavity
This example shows how to solve for the heat distribution in a block with cavity.
Consider a block containing a rectangular crack or cavity. The left side of the block is heated to 100 degrees centigrade. At the right side of the block, heat flows from the block to the surrounding air at a constant rate, for example $-10W/{m}^{2}$. All the other boundaries are insulated. The temperature in the block at the starting time ${t}_{0}=0$ is 0 degrees. The goal is to model the heat distribution during the first five seconds.
### Create Thermal Analysis Model
The first step in solving a heat transfer problem is to create a thermal analysis model. This is a container that holds the geometry, thermal material properties, internal heat sources, temperature on the boundaries, heat fluxes through the boundaries, mesh, and initial conditions.
thermalmodel = createpde('thermal','transient');
### Import Geometry
Add the block geometry to the thermal model by using the geometryFromEdges function. The geometry description file for this problem is called crackg.m.
geometryFromEdges(thermalmodel,@crackg);
Plot the geometry, displaying edge labels.
pdegplot(thermalmodel,'EdgeLabels','on')
ylim([-1,1])
axis equal
### Specify Thermal Properties of Material
Specify the thermal conductivity, mass density, and specific heat of the material.
thermalProperties(thermalmodel,'ThermalConductivity',1,...
'MassDensity',1,...
'SpecificHeat',1);
### Apply Boundary Conditions
Specify the temperature on the left edge as 100, and constant heat flow to the exterior through the right edge as -10. The toolbox uses the default insulating boundary condition for all other boundaries.
thermalBC(thermalmodel,'Edge',6,'Temperature',100);
thermalBC(thermalmodel,'Edge',1,'HeatFlux',-10);
### Set Initial Conditions
Set an initial value of 0 for the temperature.
thermalIC(thermalmodel,0);
### Generate Mesh
Create and plot a mesh.
generateMesh(thermalmodel);
figure
pdemesh(thermalmodel)
### Specify Solution Times
Set solution times to be 0 to 5 seconds in steps of 1/2.
tlist = 0:0.5:5;
### Calculate Solution
Use the solve function to calculate the solution.
thermalresults = solve(thermalmodel,tlist)
thermalresults =
TransientThermalResults with properties:
Temperature: [1320x11 double]
SolutionTimes: [0 0.5000 1 1.5000 2 2.5000 3 3.5000 4 4.5000 5]
Mesh: [1x1 FEMesh]
### Evaluate Heat Flux
Compute the heat flux density.
[qx,qy] = evaluateHeatFlux(thermalresults);
### Plot Temperature Distribution and Heat Flux
Plot the solution at the final time step, t = 5.0 seconds, with isothermal lines using a contour plot, and plot the heat flux vector field using arrows.
pdeplot(thermalmodel,'XYData',thermalresults.Temperature(:,end), ...
'Contour','on',...
'FlowData',[qx(:,end),qy(:,end)], ...
'ColorMap','hot')
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2022-05-19 03:45:20
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http://www.chegg.com/homework-help/questions-and-answers/intravenous-infusions-often-made-gravity-shown-infig-10-55-assuming-fluid-density-100g-cm3-q408715
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Intravenous infusions are often made under gravity, as shown inFig. 10-55. Assuming the fluid has a density of 1.00g/cm3, at what height h should the bottle beplaced so the liquid has the following pressure?
Figure 10-55
(a) 73 mm Hg
m
(b) 595 mm H2O
m
(c) If the blood pressure is 22 mm Hgabove atmospheric pressure, how high should the bottle be placed sothat the fluid just barely enters the vein?
m
Show transcribed image text
## Want an answer?
### Get this answer with Chegg Study
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Q:
Intravenous infusions are often made under gravity, as shown inFig. 10-55. Assuming the fluid has a density of 1.00g/cm3, at what height h should the bottle beplaced so the liquid has the following pressure?Figure 10-55(a) 53 mm Hg m(b) 585 mm H2O m(c) If the blood pressure is 23 mm Hgabove atmospheric pressure, how high should the bottle be placed sothat the fluid just barely enters the vein? m
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2016-07-29 02:18:20
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https://homework.cpm.org/category/CON_FOUND/textbook/mc2/chapter/11/lesson/11.1.1/problem/11-6
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### Home > MC2 > Chapter 11 > Lesson 11.1.1 > Problem11-6
11-6.
Anna kept track of the height of her fast-growing corn. Her records are shown in the table below.
Day Height (cm) $0$ $2$ $4$ $6$ $8$ $x$ $2$ $7.5$ $13$ $18.5$ $24$
1. What is the growth rate of Anna's corn? Is this number always the same?
Find the difference in height of Anna's corn from day $0$ to day $2$ to find
how much her corn has grown in $2$ days.
$7.5 − 2 = 5.5$
Anna's corn grows $5.5$ cm every $2$ days.
To find how much Anna's corn grows in $1$ day, divide that number by $2$.
$\frac{5.5}{2} = 2.75$
The growth rate of Anna's corn is $2.75$ cm per day.
2. If you graph the points in this table, will they form a line or a curve? Explain how you know.
If the growth rate of Anna's corn is constant, as concluded from (a),
what would a graph with constant growth look like?
If you need more help, graph the points on graphing paper.
Let the number of days be on the x-axis and the height in centimeters be on the $y$-axis.
Since the rate of change is constant, the graph will form a line.
3. How would the growth rate show in the graph?
What would the graph look like if the number of days was on the
$x$-axis, while the height in centimeters was on the $y$-axis?
If the y-coordinate increases by $2.75$ for every day, the growth
rate will be represented by the slope of the line.
4. If Anna harvests the corn when it reaches $3$ meters, when will she harvest it?
$3$ meters is equivalent to how many centimeters?
$3$ meters is $300$ centimeters. When will the
height (the $y$-coordinate) reach $300$?
Find and use the rule to solve.
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2022-06-27 23:54:31
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https://www.gamedev.net/blogs/entry/2261445-woa-ii-daywho-knows-now/
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• entries
14
28
• views
25809
# WOA II - Day....who know's now.
960 views
The day's are blurring together....
I have been drowning in AI pathing problems that have crippled the game. Every night I feel like I have them solved and the game appears to work and then all of sudden everything that could go wrong with the pathing is. At the start of day 6, I thought we were going to be okay and get something playable - and then it all broke again.
About 2 hours ago I think we FINALLY (I'm afraid to say it at this point) have it working properly. Since this time I have been slaying a massive amount of simple bugs which is making the whole game feel a lot better. We are still far behind schedule, and full of programmer art but we are actually making progress now.
First playable prototype is attached here, please report any bugs you find.
Some things to note;
-Controls are explained in the tutorial
-Skils are all over the place currently (mix of cool downs, damagae etc), no balancing has been taken
-At the end of the 2nd level, the game will crash as level 3 isn't there.
-The levels are test levels from the prototyping and are not meant to be filled or hard yet.
-The build is a debug build so you may see things that you will not in the end product
** EDIT** Removed the download as it doesn't work if you do not have XNA installed. I will be opting to use a ClickOnce installer for the actual release. Thanks for reporting the issue though
And I jinxed it....
i get the following crash:
Unhandled Exception: System.IO.FileNotFoundException: Could not load file or ass
embly 'Microsoft.Xna.Framework.Input.Touch, Version=4.0.0.0, Culture=neutral, Pu
blicKeyToken=842cf8be1de50553' or one of its dependencies. The system cannot fin
d the file specified.
at Microsoft.Xna.Framework.WindowsGameHost..ctor(Game game)
at Microsoft.Xna.Framework.Game.EnsureHost()
at Microsoft.Xna.Framework.Game..ctor()
at GWNorthEngine.Engine.BaseRenderer..ctor()
at WOA3.Engine.Renderer..ctor() in F:\Team Work\Week Of Awesome III\WOA3\WOA3
\Engine\Renderer.cs:line 35
at WOA3.Engine.Program.Main(String[] args) in F:\Team Work\Week Of Awesome II
I\WOA3\WOA3\Engine\Program.cs:line 11
That's not good, appears to be hard coded to my path. This is why I use installers, why aren't we allowed to!?
Thanks for the input though, I'll look into it.
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2018-02-20 17:55:16
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https://vulcanhammer.info/category/vulcan-the-offshore-experience/
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## The Pile Buck Ads 1: Vulcan 3100 Assembled — vulcanhammer.net
This site has never had an “advertising budget” but in the last decade the publisher Pile Buck gave it the opportunity to advertise itself in its books Sheet Pile Design by Pile Buck and Pile Driving by Pile Buck. There were five in the series, and this is the first, using the assembly of the […]
## From “Deal Yourself a Winner” to “A Pile Driver Talks About God”
The ad above is another Offshore Technology Conference ad from the early 1970’s. It was aimed at its industry: the oilfield was well endowed with hard-drinking, card-playing people, a simple fact that doesn’t fit into some peoples’ idealisation of the past. The onshore construction industry wasn’t much different, although the higher risks–and rewards–of the oilfield made everything more intense.
Contrast this with the following forty years later from Rusty Signor, then President of the Pile Driving Contractors Association and President of TX Pile, LLC, in an issue of Pile Driver:
In my last message, I ended with a different, more positive view on the news in our current world situation. This time, I am going to do another first: a book review. The book is Seven Men and The Secrets of Their Greatness by Eric Metaxas.
Certainly advice on engineering techniques, safety practices and legal tips are very important for our pile driving business; however, personal character development is also something to consider for most. You may or may not know of all the seven men in this book, but the ones you thought you knew are viewed from a very different standpoint than how you probably learned about them in school. The book focuses on their complete reliance on their spiritual calling. Since this is not a government publication, I can use the word God.
For instance, everyone knows about George Washington and the story of the cherry tree. However, did you know that he was a deeply religious man and that he relied on his faith in helping him make decisions? He prayed on his knees several times a day with a Bible before him. Washington believed that God had a special purpose for his life and that providence saved him from being killed. In one battle alone, three horses were shot out from under him and he had bullet holes through his hat and clothing. He empowered his men with God-filled inspiration and they would follow him anywhere. I bet you never read that in grade school.
Another man mentioned is Jackie Robinson, who broke the color barrier in Major League Baseball. I recently watched the movie about his story, 42. Again, the movie didn’t really focus on Robinson’s critical reliance on his faith in God to be able put up with and finally put down all the Jim Crow nonsense. He had extraordinary athletic talent in basketball, football, baseball, tennis and track and field. Robinson also had a tendency for anger explosions dealing with racial injustices. His mother and preacher led to a deeper faith that controlled his anger and justice allowed would him only to be see won that with the restraint path to and love. The manager for the Brooklyn Dodgers was an extremely religious person who was looking for this sort of man: someone talented in baseball, but who also had a strong, Bible-based character. Everyone knows the rest of the story, but generally not the one centered on God.
In the business world, sometimes we get too caught up in our challenges with competition, problems with equipment, governmental codes, etc. We just need to stop and look up like these men did – to result in your success and happiness.
## Vulcan 3100 Hammer: Specifications and Information
Like the 060 and even more the 040, the 3100 was a major step up for the company. Even though it became the “gateway” to the company’s largest hammers, itself it was a dead end offshore for reasons that weren’t fully appreciated at the time, at least not by Vulcan or some of its end users.
The specifications:
The first 3100 was built for McDermott. Even though the 560 had been introduced earlier and was lighter for the same energy, McDermott felt that the traditional “heavy ram-low striking velocity” approach was better, and also had the crane capacity to handle this size of hammer. The hammer was ordered in the fall of 1973.
The road to completing the hammer was a rough one. That fall was the occasion of the first oil shock, which was great news and bad news at the same time. It was great news because the oil price spikes made the oil industry very active during that decade and early into the next one. It was bad news because the demands on the supply chain of foundries and forge shops, coupled with the energy shortages that resulted from the oil shock itself, made lead times immensely long. And, of course, patterns had to be built for all of the major castings.
The hammer was finally completed on 11 June 1975, but there was another twist: it was assembled on the deck of McDermott’s Derrick Barge 8 in Bayou Boeuf, Louisiana. Vulcan traditionally preferred to ship their hammers assembled, but freight and delivery issues forced this method. It was successful, not only making it simpler to ship the heavy hammer parts in pieces, but also to familiarize the end user’s personnel with the hammer itself. By the 1990’s it became the standard method of delivery for hammers going to the Gulf of Mexico.
In spite of its difficult production road, the 3100 was successful from the beginning, with fewer of the “growing pains” that some of the earlier hammers had experienced.
As was the case with the 040, Vulcan used the hammer for advertising purposes, both then and many years later.
The general assembly is below (the hammer was so large, it required a two-sheet drawing.)
In spite of its success the 3100’s main claim to fame was to be the basis for the 5100. Why was this so?
The first was obvious: the 560, virtually the same energy, was lighter and more economical to produce and operate. The second was that, with offshore high-impedance steel piling, the higher impact velocity, problematic with concrete and wood piles, was actually preferable, albeit harder on the hammer. The hammer never went much past its origin, in spite of the celebration that surrounded its inception.
## Vulcan 040 and 340 Hammers: Specifications and Information
Vulcan’s personnel brought back many colourful stories from the field. One of those came from Jesse Perry, Vulcan’s senior field service representative. Offshore pile driving is a brutal, unforgiving business; offshore piles are tip elevation piles, and the expediency of “beating the pile to death” to get done in the high hourly barge rates was hard on hammers, especially those new in the product line. One of those end users vented his frustration on Jesse, who responded by throwing his wallet on the table and telling the customer that he’d bet its contents that the hammer would work.
I never knew that Jesse ever lost his wallet in that way.
In a sense, however, Vulcan itself “threw its wallet on the table” with the 040 and 060 hammers; the 040, more than any other hammer, brought it in to the “big leagues” of offshore pile driving and, through its growing pains, made Vulcan the “stamp of quality offshore everywhere.
First, the basics: the 040 specifications.
The first 040 was sold to Ingram in August 1965; below are some photos from their barge.
Many other offshore construction concerns joined Ingram in using the 040, including McDermott, Dragados, DeLong, Santa Fe, Movible Offshore (soon Teledyne Movible Offshore,) Fluor, Brown & Root, AGIP, Creole Petroleum (now PDVSA,) and Humble Oil.
Offshore wasn’t the only place where the 040 could be found. One of the most significant projects it was involved with was the long I-10 bridge across the Atchafalaya from Lafayette to Breaux Bridge, LA, built in 1969.
The 040 underwent many changes as it went along; early 040’s have many versions, as is evidenced by the general assemblies below.
Being the seminal hammer that it was, the 040 was useful for advertising, a usefulness that went past the Vulcan Iron Works itself.
### 340 Hammer
In 1972, with the introduction of the 560, Vulcan decided to rename the 040 the 340 hammer. Vulcan also made some other important changes, such as moving to an iron (as opposed to a steel) ram. The first 340 was delivered to McDermott in early 1973. Specifications, a general arrangement and a photo are shown below. It turned out to be the last hammer the Vulcan Iron Works produced, sold to PDVSA in 2000.
## The Stamp of Quality…Everywhere
Vulcan frequently produced an ad for the Offshore Technology Conference. Probably the best one was the “Stamp Ad.” The “stamp of quality” theme had appeared in Vulcan’s literature for many years before that, but Vulcan’s graphic artist Carol Carr took it to a new level with this one in the early 1970’s. It was unusual in many respects; it was in colour (colour wouldn’t become standard in Vulcan literature until late in the decade) and it was an 11″ x 17″ fold-out.
Snippets of Carol’s artwork have been on this site since its beginning in 2007, such as the masthead below:
It’s also in the current masthead as well.
The back of this ad is here:
## Compressible Flow Through Nozzles, and the Vulcan 06 Valve
Most of our fluid mechanics offerings are on our companion site, Chet Aero Marine. This topic, and the way we plan to treat it, is so intertwined with the history of Vulcan’s product line that we’re posting it here. Hopefully it will be useful in understanding both. It’s a offshoot of Vulcan’s valve loss study in the late 1970’s and early 1980’s, and it led to an important decision in that effort. I am indebted to Bob Daniel at Georgia Tech for this presentation.
## Basics of Compressible Flow Through Nozzles and Other Orifices
The basics of incompressible flow through nozzles, and the losses that take place, is discussed here in detail. The first complicating factor when adding compressibility is the density change in the fluid. For this study we will consider only ideal gases.
Consider a simple orifice configuration such as is shown below.
The mass flow through this system for an ideal gas is given by the equation
$\dot{ m }=A'_{{o}}\rho_{{1}}\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{{k}^{-1}}\sqrt {2}\sqrt {g_{{c}}kRT_{{1}}\left (1-\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{{\frac {k-1}{k}}}\right )\left (k-1\right )^{-1}}{\frac {1}{\sqrt {1-{A_{{o}}}^{2}\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{2\,{k}^{-1}}{A_{{1}}}^{-2}}}}$
where
• $\dot{m} =$ mass flow rate, $\frac{lb_m}{sec}$
• $A_o =$ throat area of orifice, $ft^2$
• $A'_o =$ adjusted throat area of orifice (see below,) $ft^2$
• $\rho_1 =$ upstream density, $\frac{lb_m}{ft^3}$
• $p_1 =$ upstream pressure, psfa
• $p_2 =$ downstream pressure, psfa
• $g_c =$ gravitational constant $= 32.2 \frac{lb_m-ft}{lb_f-sec^2}$
• $k =$ ideal gas constant or ratio of specific heats $= 1.4$ for air
• $R =$ gas constant $= 53.35 \frac{ft-lb_f}{lb_m\,^\circ R}$
• $T_1 =$ upstream absolute temperature $\,^\circ R$
At this point we need to state two modifications for this equation.
First, we need to eliminate the density, which we can do using the ideal gas equation
$\rho_1 = {\frac {p_{{1}}}{RT_{{1}}}}$
Second, we should like to convert the mass flow rate into the equivalent volumetric flow rate for free air. Most air compressors (and our goal is to determine the size of an air compressor needed to run a test through this valve) are rated in volumetric flow of free air in cubic feet per minute (SCFM.) This is also the basis for the air consumption ratings for Vulcan hammers as well, both adiabatic and isothermal. This is accomplished by using the equation
$\dot{m} = {\frac {1}{60}}\,{\it SCFM}\,\rho_{{{\it std}}}$
Making these substitutions (with a little algebra) yields
$SCFM = 60\,A'_{{o}}p_{{1}}\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{{k}^{-1}}\sqrt {2}\sqrt {-g_{{c}}kRT_{{1}}\left (-1+\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{{\frac{k-1}{k}}}\right )\left (k-1\right )^{-1}}{\rho_{{{\it std}}}}^{-1}{R}^{-1}{T_{{1}}}^{-1}{\frac {1}{\sqrt {-\left(-{A_{{1}}}^{2}+{A_{{o}}}^{2}\left ({\frac {p_{{2}}}{p_{{1}}}}\right )^{2 \,{k}^{-1}}\right){A_{{1}}}^{-2}}}}$
In this article the coefficient of discharge $C_D$ is discussed. It is also the ratio of the effective throat area to the total throat area, or
$A'_o = C_DA_o$
We are basically considering the energy losses due to friction as an additional geometric constriction in the system.
One final–and very important–restriction on these equations is the critical pressure, given by the equation
$p_c =p_{{1}}\left (2\,\left (k+1\right )^{-1}\right )^{{\frac {k}{k-1}}}$
The critical pressure is the downstream pressure for a given upstream pressure below which the flow is “choked,” i.e., the mass or volumetric flow rate will not increase no matter how much you either increase the upstream pressure or decrease the downstream pressure. This limitation, which was observed by Saint-Venant, is due to achieving the velocity of sound with the flow through the nozzle or valve. A more common way of expressing this is to consider the critical pressure ratio, or
$p_{cr} = \frac{p_c}{p_{{1}}} = \left (2\,\left (k+1\right )^{-1}\right )^{{\frac {k}{k-1}}}$
As you can see, this is strictly a function of the ideal gas constant. It’s certainly possible to get around this using a converging-diverging nozzle, but most nozzles, valves or orifices are not like this, and certainly not a Vulcan 06 valve. We now turn to the analysis of this valve as an example of these calculations.
## Application: the Vulcan 06 Valve
The first thing we should note is that pile driving equipment (except that which is used underwater) is designed to operate at sea level. Using this calculator and the standard day, free air has the following properties:
• Temperature: 518.67 °R
• Density: $\rho_{std} = 0.00237 \frac{slugs}{ft^3} = 0.0763 \frac{lb_m}{ft^3}$
• Pressure: $2116.22 \frac{lb}{ft^2}$ (or psfa)
Now let’s consider the valve for the 06 hammer (which is identical to the #1 hammer.) A valve setting diagram (with basic flow lines to show the flow) is shown below.
Note the references to steam. Until before World War II most of these hammers (along with most construction equipment) was run on steam. With its highly variable gas constant and ability to condense back to liquid, steam presented significant analysis challenges for the designers of heavy equipment during the last part of the nineteenth century and the early part of the twentieth. For our purposes we’ll stick with air.
There are two cases of interest:
• The left panel shows the air entering the hammer and passing through the valve to the cylinder. Pressurising the cylinder induces upward pressure on the piston and raises the ram. The valve position (which shows the inlet port barely cracked) is shown for setting purposes; in operation the valve was rotated more anti-clockwise, opening the inlet port.
• The centre panel shows exhaust, where air is allowed to escape from the cylinder. The piston is no longer pressurised and the ram falls to impact.
According to the vulcanhammer.info Guide to Pile Driving Equipment, the rated operating pressure for the Vulcan 06 at the hammer is 100 psig = 14,400 psfg = 16,516.22 psfa = 114.7 psia. For simplicity’s sake, we can consider the two cases as mirror images of each other. In other words, the upstream pressure in both cases is the rated operating pressure. This should certainly be the case during air admission into the hammer. For the exhaust, it should be true at the beginning of exhaust. Conversely, at the beginning of intake the downstream pressure should be atmospheric (or nearly so) and always so for exhaust.
From this and the physical characteristics of the system, we can state the following properties:
• Upstream pressure = 114.7 psia
• Downstream pressure = 14.7 psia
• Upstream area (from hammer geometry, approximate) $A_1 = 0.00705 ft^2$
• Throat area $A_o = 0.00407 ft^2$
• Coefficient of Discharge, assuming sharp-edge orifice conditions $C_D = 0.6$
• Adjusted throat area $A'_o = 0.00407 \times 0.6 = 0.002442 ft^2$
At this point calculating the flow in the valve should be a straightforward application of the flow equations, but there is one complicating factor: choked flow, which is predicted using the critical pressure ratio. For the case where $k = 1.4$, the critical pressure ratio $p_{cr} = .528$. Obviously the ratio of the upstream pressure and the downstream pressure is greater than that. There are two ways of considering this problem.
The first is to fix the downstream pressure and then compute the upstream pressure with the maximum flow. In this case $p_1 = \frac{p_{atm}}{p_{cr}} =$ 27.84 psia = 13.14 psig. This isn’t very high; it means that it doesn’t take much pressure feeding into the atmosphere to induce critical flow. It is why, for example, during the “crack of the exhaust,” the flow starts out as constant and then shortly begins to dissipate. The smaller the orifice, the longer the time to “blow down” the interior of the hammer or to fill the cylinder with pressurised air.
The reverse is to fix the upstream pressure and then to vary the downstream pressure. The critical downstream pressure is now $p_2 = p_1 \times p_{cr} = 114.7 \times 0.528 =$ 60.59 psia = 45.89 psig. This means that, when the cylinder is pressurising at the beginning of the upstroke, the cylinder pressure needs to rise to the critical pressure before the flow rate begins to decrease.
We will concentrate on the latter case. If we substitute everything except the downstream pressure (expressed in psia,) we have
$SCFM = 0.05464605129\,{\frac {{{\it p_2}}^{ 0.7142857143}\sqrt { 3126523.400-806519.7237\,{{\it p_2}}^{ 0.2857142857}}}{\sqrt { 0.9999999996-0.0003806949619\,{{\it p_2}}^{ 1.428571429}}}}$
If $p_2$ falls below the critical pressure, the flow is unaffected by the further drop and is constant. In this case the critical flow is 795 CFM. For downstream pressures above the critical pressure, the flow varies as shown below.
As noted earlier, when air is first admitted into the cylinder the flow is constant. Once the critical pressure ratio is passed, the flow drops until the two pressures are equal.
It was this large volume of flow which prevented the use of the 06 valve (which could have been separated from the cylinder using a valve liner) in the valve loss study. The smaller DGH-100 valve was used instead.
It is interesting to note that the rated air consumption of the hammer is 625 cfm. This is lower than the instantaneous critical flow. Although on the surface it seems inevitable that the hammer will “outrun” the compressor, as a further complication the hammer does not receive air on a continuous basis but on an intermittent one. For much of the stroke the compressor is “dead headed” and no air is admitted into the cylinder from the compressor. To properly operate such a device, a large receiver tank is needed to provide the flow when it is needed. The lack of such large tanks on modern compressors is a major challenge to the proper operation of air pile hammers.
## The Valve Loss Study
All fluid flow in Vulcan hammers is regulated and directed by a valve. For most Vulcan hammers (the California series being a notable exception, the #5 is another) the valve is a Corliss type valve modified from those used in steam engines. Simple and reliable, it, like any other valve, is subject to losses as the air or steam passes through it. These are reflected in the mechanical efficiency of the hammer.
The losses due to air or steam flowing through the valve are generally not the most significant source of energy losses in a pile hammer. In the late 1970’s and early 1980’s, with the increase in sheer size of the hammers, these losses became of more concern. It was necessary to at least attempt to quantify these losses instead of using a “standard” back pressure value.
In May 1979 Vulcan contacted the Georgia Institute of Technology in Atlanta about using a Vulcan #1 series valve (like used in the #1, 06, etc.) in a test to determine the losses of air flowing through these valves. At this point a major problem was encountered: the air flow required to properly test the valve was too large for Georgia Tech’s equipment. Reaching out to Lockheed didn’t help either; they couldn’t do it. At this point Vulcan came up with an alternative: use the DGH-100 valve, which was a Corliss valve albeit much smaller, for the test. Making things easier was the fact that the DGH-100 used a small aluminium valve chest, which made the valve mounting simpler.
This proved feasible and Vulcan received a proposal from Brady R. Daniel at Georgia Tech for these tests. The valve was tested in two “configurations”:
The tests were run and the report was presented in October 1980. The immediate results were as follows:
1. The report showed that the valve could be modelled essentially as a sharp-edge orifice. In the context of incompressible fluids, this is explained here.
2. A numerical method was developed to analyse the hammer cycle, as opposed to the closed-form solutions that had been used since the beginning of Vulcan pile hammers. This led to some design changes, and was also adapted for the Single-Compound hammer design.
The report also contained some suggestions for “streamlining” the design of the valve. These were not adopted, and the reason should be noted.
With the Corliss type valve, the Valve Port 1 is continuously pressurised, and this in turn forces the valve against the valve chest (or liner in the case of most newer Vulcan hammers.) With proper lubricant this seals the valve and further sealing (rings, seals, etc.) are unnecessary. This is a major reason why Vulcan hammers are as reliable as they are under the dire circumstances many operate. But that comes with a price. As with any design, there are trade-offs, and in this case the simplicity of the valve is traded off for efficiency. The simplest way to deal with this is to properly size the valve, and this was the main reason for the Valve Loss Study.
The Valve Loss Study is an interesting example of design analysis (others are here) which even an old product line like Vulcan’s can benefit from.
## General Arrangements and Assemblies
One of the typical information items Vulcan would send out would be the “general arrangement” (or assembly, to use the Raymond terminology) of a hammer, or a sub-assembly such as a capblock follower. These were also included in the offshore field service manuals. Sometimes they would feature the specifications of the hammer. They are useful for basic clearance and other dimensions or to understand the basic layout of the machine.
Some of these were put in data format. We feature for download some collections of these as follows:
• Vulcan 020 Offshore Hammer Specification Sheet. Not a general arrangement per se, but a specification sheet (in US and SI units) along with parts of the general arrangement on the back. These were issued in the 1970’s and were very popular for many years.
• Vulcan 040 Offshore Hammer Specification Sheet.
• Vulcan Offshore Hammers
• Auto-Jack Cable Tensioning Device for most Vulcan offshore hammers
• Vulcan 535 Hammer, 54″ and 80″ Jaws (similar to the 530)
• Vulcan 530/535 Capblock Follower Assembly (80″ Jaws)
• Vulcan 560 Hammer
• Vulcan 5110 Hammer
• Vulcan 5100 Capblock Follower Assembly
• Vulcan/Raymond Hammers
• Vulcan 513 Hammer
• Vulcan 515 Hammer
• Vulcan 517 Hammer
• Vulcan 525 Hammer
• Vulcan/Raymond 60X Hammer, with and without Vari-Cycle II
• Vulcan/Foster Vibratory Hammer. Vulcan manufactured L.B. Foster vibratory hammers during the 1990’s on a “private label” basis. These are the general assemblies for the 1050 and 4200.
Some of our general arrangements are in image format; we present some of them below.
We also have an extensive collection of these (including the specification sheets) in other “traditional formats.” If you would like to contact us about obtaining these, click here. We also have extensive information in our Vulcan Data Manual.
## Decelflo Pile Hammer Muffer, and the Thruflo Muffler
One thing most people notice first about pile driving jobs is that they generate an elevated level of noise. Until the 1960’s, most people simply put up with this and many other aspects of industrialisation and development. In the early 1970’s, Vulcan and other pile driving equipment manufacturers were confronted with new regulations–both at the federal and local level–which sought to regulate the noise output of construction equipment. Needless to say, pile driving equipment was high on the list.
Vulcan’s first reaction was to study the issue. It retained the services of United Acoustical Consultants in Glastonbury, CT, and its principal, Stannard Potter, to study the nature of noise output of Vulcan hammers. In December 1972 they conducted a study of a Vulcan #1 at the Chattanooga facility, and the report on the test is shown here.
Vulcan hammers, although durable and simple, suffered from two specific difficulties for noise abatement: a) their open construction gave little natural noise attentuation, and b) their lack of a recoil dampener increased the impact load on the frame, thus making it difficult to attach shrouds and other devices to attenuate sound. However, one of the results of the study was that a large proportion of the sound emission from an operating Vulcan hammer came from the exhaust. Since muffling the exhaust was simpler than doing same with the impact, Vulcan commissioned Potter to design an exhaust muffler, which it called the Decelfo Muffler.
Below: a diagram of the Decelflo Muffer concept, from Potter’s U.S. Patent 3,981,378.
The muffer was simple, a box which directed the air or steam output of the exhaust through perforated pipe surrounded by acoustical foam. The drawing shows a stacked arrangement for the muffler, but Vulcan never employed this arrangement.
The first test of the Decelflo took place in October 1973 in the Alameda yard of Santa Fe construction in Alameda, California. It involved muffling a Vulcan 020 hammer.
As shown below, the test was successful; the muffer performed as anticipated and its used resulted in reduction of hammer noise.
Flush with this success, Vulcan continued in its development of the muffler. In July 1974 it had another opportunity to demonstrate (and verify) the Decelflo’s capabilities, this time in Chicago at a sheet piling project. Below: the Decelflo mounted on top of the Vulcan hammer, in this case a 50C. The hose connection from the exhaust port to the muffler can be clearly seen, along with its connection to the hammer via the sheave pin. For a photo of the muffer in action during pile driving, click here.
We have an audio clip from this test which compares the hammer sound with and without the muffer; you can click here to listen to it and compare for yourself.
Vulcan had great plans for the Decelflo; at this time it was working on a method to mount the muffler directly on the hammer, as shown below.
But then things took a strange twist.
To begin with, there was considerable contractor resistance to the concept of having to add another device to the hammer assembly. Mounting it above the hammer lengthened the leaders required to operate the hammer, and the large installed base of Vulcan hammers dictated that this would be the normal way the muffler would be mounted.
Beyond that, the level of noise emissions, and how people perceive them, vary widely from one jobsite to another. This variation is a function of the location of the job (urban, remote, etc.), the presence or absence of neighbouring buildings to reflect the sound, and whatever ambient noise is in proximity to the jobsite. For example, driving piling next to an existing interstate, with the road noise already present, may not be very perceptible.
Finally, as far as those working on the jobsite are concerned, contractors (and OSHA) found it simpler to deal with noise emissions from pile drivers and other equipment on site by providing hearing protection to the workers, which of course is standard on jobsites today.
In any case, the Decelflo muffler was never very popular, “noise pollution” never achieved the notoriety of air and water pollution, and both Vulcan and its customer base moved on to other concerns. For his part Stan Potter moved on to patent the Decelflo concept independently of Vulcan.
### Thruflo (Geothermal) Muffer
The need to attenuate noise combined with another concern of the era, the need for alternative energy resources, with the geothermal muffer. An experimental product, it nevertheless touched on issues that are still important today.
Geothermal energy is possible when the hot magma which exists in the earth is close enough to the surface and the underground water to turn the latter into steam, which can be used to drive the turbines and generators to produce electric power. The means that the source of geothermal energy is not only free economically, but also that carbon dioxide (greenhouse gas) is not emitted in the production of electricity.
In the course of producing energy, the steam is vented to the atmosphere, and unmuffled this can produce a high noise level.
Vulcan built a prototype and tested it in its own facility in August 1974.
Below: measuring the sound as the steam passes through the straight pipe (left) and the muffler (right.) You can hear the difference by clicking here to hear the audio clip of the test.
Unfortunately the Thruflo Muffer did not get past the prototype shown above. Some of the mufflers that did make it to The Geysers had a difficult time of it, as this report attests.
## Vulcan Hammer Noise Study
Note: This study was commissioned by Vulcan and conducted in early December 1972 by United Acoustical Consultants of Glastonbury, CT, and dated 23rd June 1973. The report was submitted by the President of UAC, Mr. Stannard M. Potter. It has been released publicly by Vulcan at various times since its completion. This is the text for “Volume II” to the entire study. The graphs and other external references given in the report are not available. The fine print for this document applies.
## A. DESCRIPTION OF TESTS
During the week of December 4th through December 9th, 1972, tests were conducted on a standard Vulcan Hammer, Size 1, at the Pile Hammer Test Stand in Chattanooga. Tennessee.
### 1. Test Site
The normal structural steel test rig was removed and a special wooden support was provided to support the hammer in the test pit. The wooden beam structure prevented the secondary noise source from the steel structure from intervening with the hammer noise.
Unfortunately, the test stand site is within 75′ of the main manufacturing area of the Vulcan Plant. This provides a serious reflection of the noise from the hammer. There are other reflecting surfaces nearby. though they are smaller and at greater distances (250′). To the east of the test stand, the ground surface is hard pavement for all of the microphone locations. Although the site had acoustic shortcomings, it was felt that the proximity of the Vulcan Plant and its personnel was an obvious advantage over a possible “free field” at some distance from the plant. UAC’s Instrument Van was located in a supply shed next to the boiler room for the test stand at a distance of about 60′.
For the noise tests, the boiler and steam supply for the hammer were replaced by a truck-mounted air compressor unit. The truck was located in the front parking lot, as far away as the hoses would reach. A special muffler was designed to augment the exhaust noise reduction of the air compressor’s muffler. During the first two tests, this muffler proved inadequate and was replaced by a second unit, starting with Test 3 of the Muffled Auxiliary Exhaust.
The weather throughout the testing period was frequently cold and rainy. Wind protection was provided for the microphones and a plastic shield was built of thin vinyl in tent form over the hammer test rig to facilitate working during the rain during Test 11.
### 2. Instrumentation
Four microphone outputs were recorded on magnetic tape simultaneously. Three were data positions and one was for general announcements.
Microphone Location A Close to The Hammer – generally 6″ away from the radiating surface. (See Vulcan Drawing D-10249) 7 different microphone placements. B 25′ from the centreline of the Hammer 3 positions: Northeast, East, Southeast C 50′ from the centerline of the Hammer 3 positions: Northeast, East, Southeast
Direct readings of Peak Impact Noise were made on an oscilloscope for later verification of laboratory playback.
For most of the tests. a direct field chart was made of the A-weighted output for use in direct assessment of the signal character during each of the tests.
A detailed list of the instrumentation is at the end of this section.
### 3. Hammer Configurations
There were eleven basic changes made to the hammer and its cushion, as listed below:
Test No. Hammer Configuration 1 Bare Hammer – Steel Cushion 2 Muffled Exhaust 3 Mufflex Auxiliary Exhaust and Air Compressor 4 Nylon Slide Bar 5 Ascon Cushion 6 Micarta Cushion 7 Wood Cushion 8 Wrapped Base + Double Auxiliary Exhaust 9 Wrapped Cylinder 10 Damped Ram – 1″ Plate and EAR on Ram 11 Ram Cover – Armaplate Steel Strike Plate Micarta Strike Plate
#### Test No. 1
The Bare Hammer – Steel Cushion was a standard Vulcan Hammer, Size 1.
#### Test No. 2
During the second test. the exhaust noise us muffled by coupling a flexible hose (heavy duty rubber) to the hammer exhaust port and piping the noise about 40′ away behind an embankment. The hose terminated in a wooden box lined with fibreglass. Unfortunately the seal between the rubber hose and the exhaust discharge duct leaked and it was not until Test No. 10 – Damped Ram – that the leak was properly sealed.
#### Test No. 3
A single chamber plywood box stuffed with fiberglass was affixed to the Auxiliary Exhaust Port with an opening transverse to the normal auxiliary exhaust airflow.
#### Test No. 5
The base of the hammer was covered so that as much of the exterior radiating surface as possible was enclosed with 2″ of polyurethane foam. This was covered with a second layer of foam attached to a lead vinyl sheet manufactured by Ferro Composites. Norwalk, Conn. At the same time the Auxiliary Exhaust Muffler was rebuilt to contain a second chamber. Again, the plywood box was lined with flberglass and protected with netting and the gas flow exhausted to atmosphere through a side port transverse to the normal auxiliary exhaust air flow.
#### Test No. 9
The cylinder was treated in the same manner as the base.
#### Test No. 10
A sheet of rubber-like damping material (nominally 1/4″ thick) with the trade name EAR, manufactured by the National Research Company, a Division of Cabot Industries Cambridge. Mass., was clamped between the outer surfaces of the ram and a 1″ steel plate. The plate was attached to the ram with nuts and lock washers on previously installed ram studs. Unfortunately, due to the non-uniformity of the ram castings. only a small percentage of the radiating area was actually damped even after machining some surfaces of the ram.
#### Test No. 11
A cover comprising a steel plate bonded to rubber (about 1/4″ thick) with the trade name ARMAPLATE, manufactured by Goodyear, was fashioned to cover the entire exterior part of the ram. Tests were run with a) a Steel Strike Plate and b) a Micarta Strike Plate.
## B. METHODS OF ANALYSIS
### 1. instrumentation for Impact Analysis
Three different types of instruments were compared to evaluate the Peak Level of impact. Since this part of the noise cycle is a sharp transient the response characteristics of instruments are bound to give different values.
#### a. General Radio Impact Analyser
This Is an instrument which is designed to retain the Peak Sound Level on a meter so that the meter can he read easily. Though it has several different settings such as Time Average, Peak and Quasi Peak, we only tested the Peak.
#### b. Dumont Oscilloscope with a compressed time base
The vertical deflection of the scope trace was read from a persistent screen with a recticle which had been calibrated previously. With the exception of the inaccuracy in the visual readout the scope has the fastest and most accurate response of all the instruments.
#### c. Sanborn Graphic Level Recorder
The electronic circuitry controlling the writing pen ballistics was redesigned to give a nominal 0.005 seconds rise time. In any such display device with transient stimuli, the pen tends to over ride the actual peak level due to inertia. For the analysis used in this report, an actual writing speed of 6000 millimetres/sec. without overshoot was obtained. By moving the paper fast enough under the deflecting pen, a very clear display of the noise amplitude variation (Sound Pressure level in dB vs. time) is obtained. A comparison of all three methods indicated similar spectra. Though the Peak Impact levels from the Sanborn charts were lower (between 3 and 9 dB) than the Impact Analyser and Scope, it appeared to be consistent. The Sanborn Graphic Level Recorder has the very important advantage that it shows all the detail in the complete noise cycle while the Impact Analyser and the Scope show only Peak Impact level. This allowed us to make a detailed analysis of the various parts of the cycle affecting the total noise emanations. It was decided that the knowledge of the detailed parts of the cycle was more important than the absolute value of the Peak Impact level. Correction factors were added to all Band Pressure Levels affected by the high writing speed circuitry to provide a flat response. Accordingly. the method of analysis used for tape playbacks on tile Sanborn Graphic level Recorder
### 2. Sanborn Charts
#### a. Noise Signatures
To determine the uniformity of the noise spectra and the reliability of using a single cycle for analysis, copies of the charts showing the variation in Noise Signatures were made for linear, dB(A), 63 Hz and 2000 Hz. These data will be found in Volume III of this report. To obtain the dB Level on the ordinate (vertical direction) for any of the charts, reference should be made to the digital information presented for the Peak Impact Level in dB re 20 u N/m&Mac178; given in Tables IA thru F in Volume II. The Noise Signature charts are identified by a GR73 No. in the upper right-hand corner of each page. Turning to the tabulated digital data, the Noise Signature identifying numbers will be found under the Column “Graph 73-” in the right-hand column of each table. The noise levels in dB are listed under the appropriate column for each of the Octave Bands, Lin and dB(A).
To obtain the Sound Pressure Level of any portion of any cycle, relate the digital data to the Impact Peak. For example, GR73-024 is the analogue output for three successive cycles of the Bare hammer – Test 1, for the Microphone A located at the Base of the Hammer with a steel cushion. The synchronous readout on four channels, top to bottom, is for –
1. LIN short for Linear Weighting Network: includes all frequencies with a flat frequency response for engineering purposes and some contract specifications.
2. dB(A) short for “A” Weighting Network: it weighs the frequency response of the spectra closely with human auditory response and is frequently used in legislation.
3. 63Hz short for 63 Hertz Octave Band Pressure level: the lower practical band for noise evaluation both for engineering and subjective purposes.
4. 2000 Hz same as above except the Octave Band Center frequency is 2000 Hertz instead of 63 Hz.
Thus, in studying these charts. one can see
1. how repetitive the noise signatures are, and
2. roughly whether the peaks comprise low frequencies (costly to fix) or high frequencies (easier to reduce).
The smallest divisions are 1 dB for the ordinate and 0.02 seconds for the abscissa. The heavy lines are 5 dB and 0.1 seconds, respectively. A “blip” at one second intervals is given at the bottom or 5th trace on each chart.
On certain charts the 4th channel is the “Ticker” signal from the trip switch installed to insure identification of the ram position. It replaces the 2000Hz trace. Another variation of chart (GR73-165 thru 179), the dbA(A) channel has moved from channel 2 to 4.
For discussion or the noise signatures, we have arbitrarily divided the cycle of GR73-024 as follows:
Seconds Identification 0-0.14 Impact – the vertical rise of the trace indicating the instant of impact is used as the time reference. 0.14-0.48 Rise – the period of lifting the ram. 0.48-0.64 Exhaust – that part of the cycle normally dominated by the exhaust. 0-64-1.06 Fall – the remainder of the cycle before the next impact.
Obviously, the time periods will vary slightly with the variation of pressure and ram weight (especially with the EAR, Test 10). but the four basic parts of the cycle will be useful for reference.
To determine the sound pressure level of any part of the analogue trace, say the exhaust peak of 024 LIN, refer to the digital tables in Volume II for the appropriate reference Line GR73-024, find the Peak Impact Sound Pressure Level = 137 dB. On GR73-024 in Volume III, we have written in these values above the peaks. Again. referring to the LIN trace, note the dB level reduces after impact to 113 dB at the beginning of the Rise period and reaches as low as 103 dB before Exhaust. The Peak Exhaust of this cycle is about 123.5 or 124 dB. The next Exhaust Peak is 123 dB.
#### b.Noise Cycle Spectra
The Noise Cycle Spectra are a composite of the Linear, db(A) and eight Octave Bands from 63 Hz – 8000 Hz Noise Signatures for the same hammer noise cycle. These are published for each test and source microphone location as well as 50′ East. The number above the impact part of the cycle is the Sound Pressure level for the Octave Band or Weighting Network at Peak Impact. It is from these spectra that the digital data were obtained. For any part of the noise cycle, the Sound Pressure level in dB re 20 u N/m&Mac178; can be determined referencing the Sanborn Chart grid system and file units described above. Since the subjective reaction to the hammer noise Is a function of frequency, it is important to know how the noise varies with frequency. This spectral information also is important as to the control measures that are required to reduce the noise.
### 3. Digital Data
When there is a large body of steady state analysis to be done, it is most efficiently done with a Real Time Analyser. In the case of the many events during the pile hammer noise cycle, the Real Time Analyser falls short. The analyser needs to be started at the precise instant the event of interest occurs. The synchronization problem associated with the event time and the analyser starting time is, at present, unsolved. If the hammer cycle was constant to within a fraction of the impact duration, one could then program a computer to start the analyser at the instant desired. Of course, the hammer cycles are not that constant, particularly during a test program involving changes to the ram weight. For this reason, the digital data, determined in this study, have all been obtained manually from the analogue readouts on the Sanborn charts.
### 4. Statistical Distribution Analysis
Though major events such as ram impact, exhaust and impacts from the valve tripping mechanism are discernible from the noise signature, the affect of different hammer configurations is not readily determined for the entire noise signature. To assess the changes in the total signature, Statistical Distribution Analyses were made for certain microphone locations and hammer configurations.
These data are determined from an inspection of the noise signature at each 50th/sec interval. If the cycles are dissimilar, more than one cycle is analysed. The results are then tabulated for each level and the cumulative percentages are determined. The data are plotted on probability graph paper to show the deviation from Gaussian distribution. From each individual random source. the curve should be a straight line when plotted on the probability paper. Thus, sharp discontinuities in the curvature of the Statistical distribution indicate the dominance of another type of random source. For the most part, the curves show few straight line portions other than ambient. We interpret this to mean that several of the sources are intermingled in the noise signature at any one time.
The data are very useful because, at any individual percentage of the time, the effectiveness of each hammer configuration can be assessed by merely subtracting the differences between the Statistical Distribution Levels.
One will notice on each Statistical Distribution plot that there are two different abscissas. At the bottom of the graph, the percentages are Indicated as the “% Time Noise is Lower Than the Indicated Level”. At the top of the graph, the reverse percentage in plotted as the “‘% Time Noise is Higher Than the Indicated Level”. It is the latter scale which is finding increasing acceptance by legislative bodies and standards organizations.
These Sound Pressure Levels in dB are simply referred to as L followed by the percentage. For example, “L40” means that 40% of the time the noise is above the level quoted for “L40”.
## C. RESULTS OF TESTS
The results of file tests are contained in Volumes II and III as follows:
VOLUME II Section E UAC Drawing No. DWG. NO. 730V2 Section E is actually a graphical summary of file tests which are on a drawing labelled “Peak Impact”. The drawing is folded and tucked in the pocket in the front of Volume II’s binder. This summary shows the variation of the Linear Sound Pressure Level as a function of time for each test and each microphone location. These are representative cycles taken from the noise signatures. This drawing affords an overview of the entire test. F Tables IA thru F G Noise Cycle Spectra – Graphs No. 188 thru 277 H Statistical Distribution Analyses – Tables II, III, IV Graphs No. 398 thru 402e, 435 thru 475 VOLUME III Noise Signatures Graphs No- 024 thru 179B
For access to any particular data. use the GR73 number cross referenced in the Data Index.
## D. ANALYSIS OF DATA
### a. Noise Signatures and Cycle Spectra
#### a. Test 1 – BARE
At each of the microphone positions. there are two dominant noise sources and their Peak Sound Levels are listed below:
Mike Impact Exhaust 1- Base 138 124 2 – Ram 136 126 3 – Exhaust 134 144 4 – Trip 136 125 5 – Top Cyl. 132 124 6 – Bot. Cyl. 133 119 7 – Aux. 134 118 25′ 112 106 50′ 104 101
Except when the mike is at the Exhaust the Impact noise dominates.
As the mike is moved away from the base, the level is lower except at the Trip and the Auxiliary Exhaust. It may well be that the piston at impact transmits to the cylinder wall and certainly the columns do. The Auxiliary Exhaust ports open directly to the interior of the cylinder which acts like a reverberation chamber. These differences are minor but do show clues to the sources. Generally, the whole hammer radiates during Impact as one would expect with such rigid connections between the supporting structure. The levels range around 135 dB and project to 25′ at 112 dB and 104 dB at 50′. It is recognized that the area is reverberant around the Test Stand and, if anything, levels at 25′ and 50′ are probably high. From this data, we calculate an equivalent spherical radius equal to roughly 1.5′. Calculated radius is 2.9′. Obviously. the whole hammer is not radiating or it has directional characteristics.
Certainly the Exhaust appears to be directional as one might expect from examination of the above table. Moving from Position 3 to Position 2, a distance of only 4′, changed the Impact noise by only2 dB while the Exhaust noise changed 18 dB. Measuring perpendicular to the Exhaust axis at distances of 25′ and 50′, the levels drop by 38 and 43 dB, respectively, showing the source radius to be only 0.315′ in contrast to the estimated physical radius of 0.47′. Obviously. the Exhaust is not only small but directional. The Exhaust source, though small in size and directional, is a very potent 144 dB or 6 dB higher than the apparent peak Impact noise.
See Table A for the estimated Equivalent Radiating Area Spherical Radii of other hammer parts.
The Exhaust directionality is partially attested, as the far field mikes are moved to the Southeast Positions (more in line with the Exhaust). Though the increase is a modest 2 dB for 50′, it is a drop of 2 dB for the 25′ mike distance. The reason for this is not clear except for the less reverberant condition (no direct reflections – See Vulcan drawing P168). Note that the Trip Mike Position 4 is only 3′ away but the level is lower than the Ram at 4.5′. This shows how easy the shadowing of the valve chest can hide the dominantly short wave lengths of the Exhaust which peaks at 2000 Hertz as shown in the Noise Cycle Spectra GR73-190. All of these factors must be considered in designing the control measures.
VULCAN HAMMER NOISE SOURCE RADIATING AREAS Total Area = 104.8 Square Feet Source % of Total Mech. Area Square Feet Equivalent Spherical Radius Feet A. Mechanical 1. Ram 32.2 33.7 1.63 2. Cylinder 26.5 27.8 1.49 3. Base 17.3 18.1 1.20 4. Columns 14.3 15.0 1.10 5. Valve Chest 3.4 3.6 0.54 6. Sheave 3.3 3.5 0.53 7. Piston Rod 3.0 3.1 0.50 100.0 104.8 B. Aerodynamic 1. Exhaust 2.74 0.47 2. Auxiliary Exhaust 4.74 0.61
#### b. Other tests
Other observations of interest are:
1. The trip noise shows up before and after the Exhaust.
2. The Bottom Rear Cylinder radiates a similar shaped bulge after impact as the Auxiliary Exhaust.
3. An unexplained increasing level just prior to Impact as seen principally at the Exhaust which disappears in the Damped Ram, Test 10 – possibly piston rod seal leakage.
4. Exhaust Port seal leakage clearly disappears in Test 10.
5. The Nylon Slide Bar allows a possibly puzzling increase in its impact noise over the steel bar. Subjectively, the steel bar has a ringing that the Nylon Slide Bar lacks. The data, however, do not bear this out. Though the presumed Exhaust leak between the Trip Impacts disappears, it does so when the cylinder is wrapped in Test 9 instead of Test 10 when the leak was fixed. The inconsistency of this signature may well be due to the flexibility of the Nylon. particularly after the Cylinder wrapping.
6. One of the most dramatic changes in the Impact noise occurs when the Peak level drops from 134 dB to 123 dB at the Top Cylinder after it was wrapped.
7. Most of the change in the Peak Impact level was very gradual and actually negative at times. It shows the difficulties attendant to a noise reduction program of this type. When the overall reduction is dependent on a lot of small changes, much control must be exercised over the test technique and accuracy.
8. It is quite obvious in studying the 25 and 50′ signature that much of the significant source noise sort of “washes out” as one gets further from the hammer. This is particularly true of high frequencies and one should be cautious in expecting this to happen with low frequencies. If the low frequencies are high enough in level, they will still control the measurement even though the hammer may seem a lot quieter.
9. By similar observation and calculation. one comes to the conclusion that the principal sources are listed below in approximate order of importance.
1. Ram
2. Cylinder
3. Exhaust
4. Auxiliary Exhaust
5. Trip
6. Base
7. Columns
8. Valve Chest
9. Sheave
10. Piston Rod
### 2. Statistical Distribution Analysis
Though the Peak Impact Level dominates the undetailed information of the enforcer’s meter, it is not a significant measure of the quietness quotient. For example. the difference in Peak Impact Levels between Tests 1 and 10 is only 1 dB and 4 dB at 25′ and 50′, respectively. Obviously, some other measure is required to evaluate the substantial reduction in Loudness observed by the listener. Statistical Distribution Analysis helps fill this need since it measures the entire signature every 0.02 seconds.
#### a. Linear Probability Graphs
From the Probability Graphs. data has been summarized in Tables II, III and IV showing the differences between incremental changes in hammer configuration an well as the overall. These overall differences have been plotted as Probability Graph Noise Reductions, GR73-447 thru 453. showing the maximum reduction from Test 1 to 10 as 32 dB at L20 and the Auxiliary Exhaust Position. The maximum source noise reductions are tabulated below:
Mike Percentile dB Reduction 1 – Base L50-20 16 2 – Ram L50 18 3 – Exhaust L30 25 4 – Trip L40 21 5 – Top Cyl. L30 27 6 – Bot. Cyl. L40 27 7 – Aux. Ex. L20 32
#### b. Histograms
Comparative Noise Reductions are shown at the L40, L30, L20 and L3 Percentiles for each change in hammer configuration at 50′ East. The pattern clearly speaks well for the Damped Ram, though we must caution that the Exhaust leak was also fixed during this test. Still it shows as the best at most of the Percentiles and we have noted before that only a small percentage of the damping plate was actually constraining the EAR to the Ram. Even so, a substantial reduction was obtained. This, coupled with the fact that the Ram has the largest Radiating Area, makes it the number one control measure to be applied in the hammer redesign.
In Graph GR73-454, the overall differences are given for each test change while Graph GR73-455 more clearly shows the contribution of each change by itself.
#### c. Octave Probability
From these graphs. the data in Table V has been derived which in turn has lead to the Statistical Spectra Summary graphs.
#### d. Statistical Spectra Summary Graph
Finally, the “proof of the pudding” is in the actual reductions achieved at likely observer distances. Graphs 472 thru 475 show the overall reduction in noise at a distance of 50′ for the L40, L30, L20 and L3 Percentiles. Levels of Noise Reduction for the annoying frequencies of 500 Hz and above ranged from 14 – 20 dB for all except the brief Percentile of L3 where the Noise Reduction ranged from 4 to 12 dB.
With the above evidence. it is proven that a concerted design and evaluation effort will pay substantial dividends in quieter pile hammers. Recommendations for appropriate control measures are given in Volume I of this report.
## INSTRUMENTATION LIST
### Field
• Microphones
B&K 4135 S/N 125108
B&K 413S S/N 125107
AKG C60 S/N 382
GR 1560-P40
• Power Supplies
B&K 280L S/N 144144, 144736
AKG N60EA S/N 148
GR 1560-4100
CR Octave Band Analyser 1558AP, S/N 203
• Attenuators
UAC S/N’s 60, 61, 62
• Recorders
Ampex 351 S/N 4042057
Preamp Channel A S/N 27688
Preamp Channel S S/N 28677
Ampex 351 S/N 0140176
Preamp Channel A S/N 34053
Preamp Channel B S/N 29042
Sanborn Strip Chart Recorder 954B-1OO
Log Preamps 350-1400
Channel A S/N 922, Channel B S/N 928, Channel C S/N 867, Channel D S/N 625
• Other Equipment
Hewlett Packard Scope 120B S/N 601-06957
B&K Octave Filter Set
KLH Speaker Model 6 S/N 35950
B&K Microphone cable set
Taylor weather Station
### LABORATORY
• Ampex Playback Deck 351A S/N 4840140
• Marantz Power Amp S/N 5042
• B&K 1612 1/3 Octave Band filter
• B&K Octave Filters
• Sanborn Strip Chart Recorder 954B-100
Log Preamps 350-1400
Channel A S/N 922, Channel B S/N 928, Channel C S/N 867, Channel D S/N 625
• Dumont Scope 401B S/N 160
• GR Octave Band Analyser 1558AP S/N 203
GR Impact Noise Analyser 1556A S/N 767
• Dynaco Monitor Preamp
• CM labs Power Amp S/N 0266
• KLH Speakers Model 6 S/N 35950, 35839
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2019-07-18 01:00:47
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https://codereview.stackexchange.com/questions/193036/multi-threaded-in-place-mergesort-on-java
|
# Multi-threaded in-place Mergesort on Java
What is an efficient way to implement the Mergesort algorithm in Java such that it meets the following criteria:
• Must retain the Mergesort time complexities.
• Must be in-place implementation, no copying arrays.
• Must use Java Collection classes, not an array.
• Must work on any Java Comparable.
My solution involves using an ArrayList and fork-join to create the in-place Mergesort. I've tested this on multiple inputs and they all work.
I'm curious if anyone can find a way I can improve my code. Either a more efficient solution or perhaps an edge case I didn't address.
import java.util.*;
import java.util.concurrent.*;
public class MergeSort<N extends Comparable<N>> extends RecursiveTask<List<N>> {
private List<N> elements;
public MergeSort(List<N> elements) {
this.elements = elements;
}
@Override
protected List<N> compute() {
if(this.elements.size() <= 1)
return this.elements;
else {
final int pivot = this.elements.size() / 2;
MergeSort<N> leftTask = new MergeSort<N>(this.elements.subList(0, pivot));
MergeSort<N> rightTask = new MergeSort<N>(this.elements.subList(pivot, this.elements.size()));
merge(left, right);
return this.elements;
}
}
private void merge(List<N> left, List<N> right) {
int leftIndex = 0;
int rightIndex = 0;
while(leftIndex < left.size()) {
if(rightIndex == 0) {
if( left.get(leftIndex).compareTo(right.get(rightIndex)) > 0 ) {
swap(left, leftIndex++, right, rightIndex++);
} else {
leftIndex++;
}
} else {
if( right.get(0).compareTo(right.get(rightIndex)) < 0 ) {
swap(left, leftIndex++, right, 0);
} else {
swap(left, leftIndex++, right, rightIndex++);
}
}
}
if(rightIndex < right.size() && rightIndex != 0)
merge(right.subList(0, rightIndex), right.subList(rightIndex, right.size()));
}
private void swap(List<N> left, int leftIndex, List<N> right, int rightIndex) {
//N leftElement = left.get(leftIndex);
left.set(leftIndex, right.set(rightIndex, left.get(leftIndex)));
}
public static void main(String[] args) {
ForkJoinPool forkJoinPool = ForkJoinPool.commonPool();
List<Integer> result = forkJoinPool.invoke(new MergeSort<Integer>(new ArrayList<>(Arrays.asList(1,3,5,7,2,4,6))));
System.out.println("result: " + result);
}
}
Update: The code will fail on the edge case where the following input integers are used:
{5,9,8,7,6,1,2,3,4}
This occurs because the recursive call to merge only works successfully when the left and right lists are of equal length or differ by only one in length. This call however will result in the comparison of two lists that breaks that expectation. I have yet to find a solution which handles all edge cases properly.
• You need to rethink your merge() function, because it doesn't work properly even on cases where the left and right sides are of equal length. The problem is that you swap elements from the left side to the right side, but other than right[0], you don't revisit those elements and put them in the correct position. For example, once you swap some left element into right[1], you will never move that element again. As far as I know, there is no simple in-place merge algorithm that can be done the way you are trying to do it. – JS1 Apr 26 '18 at 22:41
• You may want to look at this which explains how to do a proper in-place mergesort. – JS1 Apr 26 '18 at 22:49
• @JS1 Yea i figured that much out myself. I think i can do it with a third index as a pointer. – Jeffrey Phillips Freeman Apr 26 '18 at 23:42
• @JS1 Also thanks for the link. I did see that. It is fairly similar to my approach though im trying a slightly different approach. Not sure if it will be workable. – Jeffrey Phillips Freeman Apr 27 '18 at 0:13
### Quick-Glance nitpicks
• elements can (and should) be final. This reinforces the point that you're not changing that reference.
• Always place braces around blocks. That's especially true for single-line if-blocks like the element size guard. Somewhat interestingly inside merge you placed the braces in all if-blocks except the last.
• You're inconsistent in the spacing for conditionals. In merge the inner if-statement's conditionals are surrounded by spaces, everywhere else, they are not. I personally prefer putting a space before the opening parenthesis of any condition, if only to distinguish it from a method invocation.
### Density of semantics
The swap calls in merge are rather heavy in what they do. There's a lot going on at once, relying on a conscious and full understanding of unary operators and how the method works.
If I were you, I'd consider extracting the incrementing of the indices to separate statements. That can also make the reasoning somewhat easier to understand, seeing that leftIndex++ is in every branch of the tree:
while (leftIndex < left.size()) {
if (rightIndex == 0) {
if (left.get(leftIndex).compareTo(right.get(rightIndex)) > 0) {
swap(left, leftIndex, right, rightIndex);
rightIndex++;
}
} else if (right.get(0).compareTo(right.get(rightIndex)) > 0) {
swap(left, leftIndex, right, 0);
} else {
swap(left, leftIndex, right, rightIndex);
rightIndex++;
}
leftIndex++;
}
At this point it should also be clear that the first if-statements can be collapsed into a single statement. JS1 mentions that the in-place merge you are doing does not work for all cases. Unfortunately I can't tell you quite exactly why, but making the code easier to grasp should help troubleshooting and fixing this :)
### Optional List features
You're making use of the set operation on Lists. It's important to be aware that set is an optional operation. Not all implementations of List provide an implementation and may throw UnsupportedOperationExcpetion instead. Then again all the other features you could use to replace this are also optional (what with modification of Lists being optional)...
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2019-05-25 20:16:40
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Tr. Mosk. Mat. Obs., 2011, Volume 72, Issue 1, Pages 127–188 (Mi mmo14)
Topological applications of graded Frobenius $n$-homomorphisms
D. V. Gugnin
M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract: This paper generalizes the theory of Frobenius $n$-homomorphisms, as expounded by V. M. Buchstaber and E. G. Rees, to graded algebras, and applies the new algebraic technique of graded Frobenius $n$-homomorphisms to two topological problems. The first problem is to find estimates on the cohomological length of the base and of the total space of a wide class of branched coverings of topological spaces, called the Smith-Dold branched coverings. This class of branched coverings contains, in particular, unbranched finite-sheeted coverings and the usual finite-sheeted branched coverings from the theory of smooth manifolds. The second problem concerns a description of cohomology and fundamental groups of $n$-valued topological groups. The main tool there is a generalization of the notion of a graded Hopf algebra, based on the notion of a graded Frobenius $n$-homomorphism.
Key words and phrases: graded algebra, graded $n$-homomorphism, Frobenius, Smith-Dold branched covering, cohomological length, $n$-valued topological group.
Full text: PDF file (562 kB)
References: PDF file HTML file
English version:
Transactions of the Moscow Mathematical Society, 2011, 72, 97–142
Bibliographic databases:
UDC: 512.647+512.552+515.145.2
MSC: Primary 16W20, 17A42; Secondary 57M12
Revised: 05.01.2011
Citation: D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Tr. Mosk. Mat. Obs., 72, no. 1, MCCME, Moscow, 2011, 127–188; Trans. Moscow Math. Soc., 72 (2011), 97–142
Citation in format AMSBIB
\Bibitem{Gug11} \by D.~V.~Gugnin \paper Topological applications of graded Frobenius $n$-homomorphisms \serial Tr. Mosk. Mat. Obs. \yr 2011 \vol 72 \issue 1 \pages 127--188 \publ MCCME \publaddr Moscow \mathnet{http://mi.mathnet.ru/mmo14} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184814} \zmath{https://zbmath.org/?q=an:06026282} \elib{https://elibrary.ru/item.asp?id=21369339} \transl \jour Trans. Moscow Math. Soc. \yr 2011 \vol 72 \pages 97--142 \crossref{https://doi.org/10.1090/S0077-1554-2012-00191-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84959573947}
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This publication is cited in the following articles:
1. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms, II”, Trans. Moscow Math. Soc., 73 (2012), 167–182
2. D. V. Gugnin, “Lower Bounds for the Degree of a Branched Covering of a Manifold”, Math. Notes, 103:2 (2018), 187–195
3. D. V. Gugnin, “Razvetvlennye nakrytiya mnogoobrazii i $\boldsymbol{nH}$-prostranstva”, Funkts. analiz i ego pril., 53:2 (2019), 68–71
4. Johnson K.W., “Group Matrices, Group Determinants and Representation Theory the Mathematical Legacy of Frobenius Preface”: Johnson, KW, Group Matrices, Group Determinants and Representation Theory: the Mathematical Legacy of Frobenius, Lect. Notes Math., Lecture Notes in Mathematics, 2233, Springer International Publishing Ag, 2019, IX+
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## The power of correlation functions
During my physics studies, I got to know several mathematical tools that turned out to be extremely useful to describe the world and to analyze data, for example vector calculus, fourier analysis or differential equations. Another tool that I find particularly useful for my current work as a neuroscientist and which is, however, rarely mentioned explicitly are correlation functions. In the following, I will try to give an intuition of the power of correlation functions using a couple of examples.
.What are correlation functions?
To put it in very simple terms, a correlation coefficient measures how similar two signals ($A$ and $B$) are after being normalized. Different from correlation coefficients, correlation functions are not single values, but functions of two input signals $A$ and $B$. This can be a correlation function $C_{AB}$ of a time lag, $C_{AB}(\tau)$, or of a distance in space, $C_{AB}(\Delta x)$ . The correlation function at a time lag or distance of zero, recovers the correlation coefficient, $C_{AB}(0)$, except for a normalizing factor.
The value of a correlation function at a given value of $\tau$ or $\Delta x$ indicates how similar the two input signals $A$ and $B$ are when one of the signals is shifted in time by $\tau$ or in space by $\Delta x$.
To make the result of this operation more clear, here are two simple examples, with signals $A(t)$ (black) and $B(t)$ (gray) as noisy sine waves that are in phase (left) or out of phase (right):
While the cross-correlation function peaks at a time lag of $\tau = 0$ for the synchronous case, the peak is shifted to $\tau \neq 0$ for the out-of-phase case. The value at a time lag of 0 is proportional to the correlation coefficient: a high value for the left side, a value close to zero for the right hand side. Also note that the correlation function used averaging over the full signal duration to get rid of the noise.
Computing the correlation function $C_{AB}$ in Matlab or Python
Computing the correlation function is actually straightforward in Matlab or Python.
Matlab:
A = rand(1000,1); B = rand(1000,1); C = xcorr(A,B,'unbiased');
Python:
import numpy as np A = np.random.norm(0,1,1000) B = np.random.norm(0,1,1000) C = np.correlate(A,B,'full')
or
import scipy.signal as signal C = signal.correlate(A,B)
It is a good but a bit tedious exercise to write one’s own cross correlation function in a basic programming language. Usually the normalization at some point can cause headaches.
1. Spatial correlation functions for image registration
In microscopy, there’s often the problem to map two images onto each other. The following examples are two average images of the same brain region, recorded at different time points and therefore shifted meanwhile due to drift. I included a horizontal line for orientation:
To find out the drift, we can use correlation functions, measuring the similarity of the two images for all possible shifts, with the result that the shift in x-direction is 0 pixels, whereas the shift in y-direction is 4 pixels (here in Matlab):
movie_AVG1; % average image 1 movie_AVG2; % average image 2 result_conv = fftshift(real(ifft2(conj(fft2(movie_AVG1)).*fft2(movie_AVG2)))); [y,x] = find(result_conv == max(result_conv(:))); shift_y = y - ( size(movie_AVG1,1)/2 + 1 ) shift_x = x - ( size(movie_AVG2,2)/2 + 1 )
Here, I calculated the correlation function using fast fourier transforms, taking advantage of a simple mathematical property of correlation functions. I could also have done the same computation with the built-in function xcorr2(movie_AVG1,movieAVG2) in Matlab, which is however much slower and requires subtraction of the respective mean from the images.
Similar algorithms are used for most image registration functions in ImageJ, Python or Matlab.
2. Local spatial correlation functions for particle image velocimetry
To go one step further, one can also compute a local instead of a global shift, for example if there are any deformations of the images that result in local deformation fields.
A more interesting application of the same principle of local deformations is a method referred to as particle image velocimetry (PIV), which has been developed in the field of experimental fluid mechanics. Using a sequence of images, correlation functions are used to extract local flow fields, as well as sinks and sources of the observed transport phenomenon. Here is, from work for my Diploma thesis, an example movie of a one-cell C. elegans embryo just before the first cell division, observed using DIC microscopy. I used the granular stuff in the cytoplasm to track the cytosolic flow patterns using PIV (with the toolbox PIVlab). The overlaid yellow arrows indicate the (wildly changing) direction of the local cytosolic flow field:
3. Temporal cross-correlation functions
One of most fascinating usages of cross-correlation functions for analysis of experimental data is for fluorescence cross-correlation spectroscopy (FCCS), or its more commonly used simpler version, fluorescence correlation spectroscopy (FCS), the latter of which is based on auto-correlation instead of cross-correlation functions.
Peri-stimulus time histograms (PSTHs) are a much more basic analysis tool that is commonly used by electrophysiologists to quantify the occurrence of a quantity triggered by certain events. Sometimes, events as ill-defined as the crests of an oscillatory signal are used as a trigger for a PSTH. Using correlation functions gets rid of this mess by measuring how much a quantity is affected depending on the quantitative history of the trigger signal.
In electrophysiologal work published in 2018, I used correlation functions to measure the phase relationship between an oscillatory local field potential (LFP) signal and an oscillatory component in a simultaneous whole-cell recording (for details, check out a part of figure 7 in the paper):
4. Autocorrelation functions for time series analysis
Auto-correlation functions are not only a tool for non-intuitive experimental methods like FCS, but also perfect to quantify periodicities in a time series. For example, if there is an oscillatory behavior in a swim pattern of a fish, in the firing of a neuron or in the spatial density of clouds, autocorrelations can easily quantify this periodicity.
Here is an example, again from an LFP recording. On the left, the signal seems clearly oscillatory, but how can we properly quantify the oscillatory period? We use an auto-correlation function, and the peak at around 40 ms in the plot on the right clearly indicates the oscillatory period (black arrow):
Correlation functions in physics
If you find the above examples interesting and want to understand what correlation functions can be used for, it could be a good idea to dive into physics, where correlation functions are all over the place:
In addition, the mathematical aspects of correlation functions are quite rewarding to explore, for example the intimate relationship between auto-correlation functions and power spectra.
As another interesting use of auto-correlation functions, the fluctuation-dissipation theorem gives an idea how spontaneous fluctuations of a system close to thermodynamical equilibrium can predict the linear response of the system towards external perturbations. It’s a bit discouraging for biologists to understand that this theorem can hardly be applied to biological systems, which live far from thermodynamic equilibrium and which show responses that are rarely linear.
Still, it is amazing to see what physics can do with correlation functions and how powerful correlation functions are at extracting precise measurements from sometimes very noisy data.
## Annual report of my intuition about the brain
There are not many incentives for young neuroscientists to think aloud about big questions. Due to lack both of knowledge and authority, discussing very broad questions like how the brain works risks to be embarrassing at best. Still, I feel that not doing so is even more detrimental since it restrains the potential for internal development: Exposing one’s thoughts comes with the potential to refine them and to dissociate from them, thereby bringing down or advancing ideas that might have got stuck.
I want to make it a habit to report some of the thoughts about the brain that marked me most during the past twelve month at the end of each year, with the hope to advance and structure the progress in the part of my understanding of the brain which is not immediately reflected in journal publications.
How I got interested in dendrites
The lines of thought described in the following actually go back as far as to 2015. I was planning to switch from calcium imaging to whole-cell recordings as my main laboratory technique and started understanding the power of studies relying on this technique. In summer 2015, I came across a paper by Katie Bittner in Jeff Magee’s lab [1] (followed up by another paper [2]). Those papers showed that electrical “plateau potentials” can drive the formation of a place cell within a single trial. The authors established in vivo whole-cell recordings deep in the CA1 region of the hippocampus. Using this technique, they could generate plateau potentials by somatic current injection and thereby trigger the generation of a place field. As probably many others, I was immediately struck by this single-shot learning behavior, but, also due to lack of background knowledge, I was not yet able to see it in a larger context.
Later, when I was searching for potential postdoc positions in 2018, I first fully encountered the mystery of the apical dendrites of pyramidal neurons. Pyramidal neurons in layer 5 of the mammalian cortex grow from their soma a “basal” dendritic tree that remains rather local in layer 5, and in addition a thick “apical” trunk that goes up to layer 1, where it branches into many small apical dendritic processes (the apical “tuft”).
I was particularly intrigued by a review by Matthew Larkum from 2013 suggesting a specific function for the apical tuft of L5 neurons. This suggested function would be to detect almost coincident somatic activity and strong input to apical dendrites, resulting in a calcium spike in the apical trunk and leading to somatic bursting [3].
Problem 1: Top-down input to the apical dendrites of pyramidal neurons
Apical dendrites of L5 neurons are thought to receive mainly top-down input, whereas the basal dendrites are predominantly contacted by bottom-up input. For example, basal dendrites of the primary visual cortex would be expected to receive more sensory input from the primary thalamic region, whereas apical dendritic processes would receive rather context-related input from brain regions higher in the sensory hierarchy. I do not know how well this separation of top-down and bottom-up inputs for apical and basal dendrites holds true – in an earlier blog post I have described why I am generally not a fan of hierarchies like this top-down/bottom-up connectivity scheme, although I still find it a fascinating idea.
Since I’m currently working next door to the lab of Georg Keller, who is interested in predictive processing in visual cortex (check out his 2018 review [4]), I could not prevent myself from wondering whether this top-down contextual input to the apical dendrites could simply be predictions. This possibility is also mentioned in the Larkum review [3]. However, in the theory of predictive processing, predictions (here: apical input) should be subtracted from the sensory input (here: basal input), or the sensory input should be subtracted from the predictions. As mentioned above in the review by Matthew Larkum, however, the apical trunk seems to compute the coincidence of those inputs rather than the difference. Therefore, this somehow does not seem to make sense.
There are ideas how to implement predictive processing using L5 pyramidal cells nevertheless. For example, there is an interesting computational model that is pretty detailed (described by Sacramento et al. [5]). The idea here is that the apical compartment does not simply signal top-down input, but encodes an error signal between local inhibitory signals and top-down excitatory input. Some assumptions of this model seem to be unrealistic and many aspects of the model are simply unconstrained by experiments, but it is an interesting starting point nethertheless.
Problem 2: Coupling between apical dendrites and the soma
Overall, this leaves me with the impression that the apical compartment might be something crucial to understand. The separation of processing in apical and somatic compartments is an assumption that seems legit given the large electrotonic distance between soma and the apical dendrite. In addition, this assumption is supported by experimental data (e.g., Cichon & Gan, 2015 [6]; Seibt et al., 2017 [7]; and some other studies). However, for all of those studies, no direct evidence for the decoupling of somatic and apical activity was available. Direct evidence would mean simultaneous recording of somatic and dendritic activity, which is challenging even for an indirect method as calcium imaging due to the large spatial separation of soma and distal apical dendrite. Probing of calcium dynamics in a direct way so far has not shown strong decoupling of somatic and dendritic activity (e.g., Helmchen et al., 1997 [8]; or more recently Kerlin et al., 2018 [9]). To be more precise, these studies showed that almost all calcium events in the apical dendrites – with very few exceptions – were temporally coupled to backpropagating action potentials. This seems to be somehow at odds with the idea of separate processing in somatic and apical compartments.
Of course, this is only about dendritic calcium signals, not about the voltage. Recording of the voltages over multiple locations of a dendritic tree, for which there is currently no reliable method, could potentially result in a different picture. Plus, the brain areas and behavioral contexts are not immediately comparable between the behavioral tasks in these experiments. For example, Helmchen et al. [8] used anesthetized rats; Kerlin et al. [9] trained their mice extensively before experiments; Cichon et al. [6] recorded dendritic activity during a weird learning paradigm that might have resulted in a lot of confusion in the mice; and Seibt et al. [7] focused on dendritic activity in mice and rats during sleep.
As a result of these (seemingly) contradictory results, I’m intrigued by the unresolved question how tightly the activities of soma and apical dendrites of L5 neurons are indeed coupled. Or rather, under which circumstances both compartments become uncoupled. The answer to this question is completely unclear to me.
Problem 3: What do bursts of pyramidal neurons signal?
It is however clear that somatic action potentials to some extent invade the apical dendritic tree. This does not seem to be a random side effect, since it was reinforced by evolution by the insertion of active conductances into the dendritic tree. One possible purpose of this backpropagating action potential could be to activate the inputs of the apical dendrite, resulting in non-linear amplification in the distal dendrites or in the apical trunk (as described by the Larkum review [3]) and thereafter in somatic burst firing. What is the purpose of these bursts? I can come up with two possible explanations:
(1) As suggested by the experiments in the Magee lab ([1][2]), the bursts could be a strong intracellular signal to reinforce recently activated context synapses. – If so, in which synapses would plasticity occur, in basal or rather apical dendrites? In the studies from the Magee lab in the hippocampus plasticity in synapses of the stratum radiatum of CA1 was observed [2]; those synapses are thought to provide spatial context. How would this translate to cortex?
(2) A second possible function of bursts could be to signal not inside of a neuron, but between neurons. Regular spiking is ideally suited to drive postsynaptic neurons with depressing synapses, i.e., only the first spike of a rapid sequence of spikes would trigger substantial synaptic release of neurotransmitters. Bursting, however, is ideally suited to drive postsynaptic neurons that are connected via facilitating synapses. The bursts would therefore be a very sparse code that could signal a coincidence of somatic spiking and apical input to the downstream neuron. In a theoretical study, Naud & Sprekeler [10] investigated the potential of such multiplexing through simple spikes and bursts for separate processing of top-down and bottom-up input in a hierarchical network. And Blake Richards mentioned (in a talk that I’ve watched on youtube, start at min 22:14), while not going into the details, the possibility to use this multiplexing for helping to solve the “credit assignment problem”.
Brief digression: The credit assignment problem is about the question how a neuron somewhere in the brain network can learn to weigh the incoming information in order to become better at a given task. This problem is also addressed by the previously mentioned paper by Sacramento et al. [5], and there is a paper by Guerguiev et al. [11] that goes into a similar direction but is a bit chaotic. Lillicrap and Richards just published a review on how the credit assignment could be solved using (apical) dendrites [12]. It is important to note that both Sacramento et al. and Guerguiev et al. suggest solutions that are approximations of “backpropagation” of errors, the solution to the credit assignment problem that has been fameously found for artificial neural networks. Backpropagation, however, follows a linearized gradient in error space and therefore only allows for small synaptic changes – and thus does not really allow for the single-shot learning behavior that has been observed in animals ([1][2]). Therefore I’m not sure whether it is good idea to search for an implementation of an algorithm similar to backpropagation in the brain.
Problem 4: It’s getting ever more complex
In addition to the open questions mentioned above, some other points related to the function of apical dendrites are also not clear.
For example, what role do inhibitory neurons play that specifically target the apical dendrite? With a disinhibtory circuit motif, interneurons could specifically gate plasticity by blocking inhibition of an apical dendrite (check this review by Letzkus et al. [13]). Following this line of thought, it is, however, not clear to me whether disinhibition is (branch- or neuron-) specific or rather a broad, global gating mechanism of plasticity that allows for specific plasticity by other means.
As another example, it is to some extent clear how the membrane potential behaves in vivo in the soma – but less so in the dendrites of the very same neurons. Dendrites might integrate much fewer inputs than a soma and thereby exhibit much stronger voltage fluctuations – unless there is a precise local balance of excitatory and inhibitory inputs to a single dendrite (this question is based on work I did in zebrafish). A recent study addressed this question of balancedness partially by mapping the co-localization of excitatory and inhibitory neurons on a full tree of L2/3 pyramidal neurons [14].
In the context of balanced networks, I’m also wondering whether apical dendrites in living, unanesthetized brains operate in a high-conductance state as a result of strong excitatory and inhibitory inputs, which has been suggested for balanced networks. If so, I would be interested to know how the mass of open input channels in vivo would affect the coupling between dendritic segments compared to ex vivo slice studies. For this particular question, I’m not sure whether I’m just ignoring existing literature on the subject or whether these questions have simply not yet been addressed experimentally.
Summary
What happens in apical dendrites of L5 pyramidal neurons remains mysterious to me, in particular if the neuron is integrated in the cortex of a behaving animal or human being. I do not know which kind of events trigger activity of the apical dendrite, and I do not know under which circumstances and how the apical compartment communicates with the soma. I wonder how well the compartments are connected electrically in vivo. And it is an open question how both synaptic plasticity and non-plastic information processing are affected by activation of the apical trunk or the more distal apical tuft. Despite this lack of knowledge, I would guess that understanding apical dendrites is not sufficient, but probably necessary to understand what a cortical region does as a whole.
Maybe I’m wrong about some of my interpretations; probably I’m overlooking some import studies. If something is wrong in my guesses and interpretations, or if I am missing an important piece of experimental or theoretical evidence, please let me know. I do not have an agenda that I want to defend but instead would like to understand. Therefore, critical comments are even more welcome than positive feedback!
.
References
[1] Bittner KC, Grienberger C, Vaidya SP, Milstein AD, Macklin JJ, Suh J, Tonegawa S & Magee JC. Conjunctive input processing drives feature selectivity in hippocampal CA1 neurons. Nature Neuroscience (2015).
[2] Bittner KC, Milstein AD, Grienberger C, Romani S, & Magee JC. Behavioral time scale synaptic plasticity underlies CA1 place fields. Science (2017).
[3] Larkum ME. A cellular mechanism for cortical associations: an organizing principle for the cerebral cortex. Trends in Neurosciences (2013).
[4] Keller GB, & Mrsic-Flogel TD. Predictive Processing: A Canonical Cortical Computation. Neuron (2018).
[5] Sacramento J, Costa RP, Bengio Y, & Senn W. Dendritic cortical microcircuits approximate the backpropagation algorithm. Advances in Neural Information Processing Systems (2018).
[6] Cichon J, & Gan WB. Branch-specific dendritic Ca2+ spikes cause persistent synaptic plasticity. Nature (2015).
[7] Seibt J, Richard CJ, Sigl-Glöckner J, Takahashi N, Kaplan DI, Doron G, Limoges D, Bocklisch C, & Larkum ME. Cortical dendritic activity correlates with spindle-rich oscillations during sleep in rodents. Nature Communications (2017).
[8] Helmchen F, Svoboda K, Denk W, & Tank DW. In vivo dendritic calcium dynamics in deep-layer cortical pyramidal neurons. Nature Neuroscience (1999).
[9] Kerlin A, Mohar B, Flickinger D, MacLennan BJ, Davis C, Spruston N & Svoboda K. Functional clustering of dendritic activity during decision-making. bioRxiv (2018).
[10] Naud R, & Sprekeler H. Sparse bursts optimize information transmission in a multiplexed neural code. PNAS (2018).
[11] Guerguiev J, Lillicrap TP, & Richards, BA. Towards deep learning with segregated dendrites. eLife (2017).
[12] Richards BA, & Lillicrap TP. Dendritic solutions to the credit assignment problem. Current Opinion in Neurobiology (2019).
[13] Letzkus JJ, Wolff SB, & Lüthi A. Disinhibition, a circuit mechanism for associative learning and memory. Neuron (2015).
[14] Iascone DM, Li Y, Sumbul U, Doron M, Chen H, Andreu V, Goudy F, Segev I, Peng H, & Polleux F. Whole-neuron synaptic mapping reveals local balance between excitatory and inhibitory synapse organization. bioRxiv (2018).
## Whole-cell patch clamp, part 4: look and feel
In previous blog posts, I have been discussing some aspects of whole-cell patch clamp recordings ([1], [2], [3][4]). Today, I will show some instructive videos that I recorded during experiments. I’m hoping that they will convey the look and feel of the procedure of whole-cell patching in an intact brain using two-photon microscopy to target neurons.
Two-photon targeted patching is the core method of my recent paper on a precise synaptic balance of input currents, focusing on biological questions instead of methods (Rupprecht et al. (2018)). The underlying method has been described before as “shadow-patching”. Dye ejected from the pipette allows to visualize cell bodies as dark shadows in a sea of fluorescence (see Kitamura et al. (2008); also check out Judkewitz et al. (2009)Margrie et al. (2003) and Komai et al. (2008)). Although these papers are very useful resources, they do not allow to understand how the procedure of patching a neuron looks and feels like to the experimenter.
For camera-based whole-cell patch clamp recordings in slices or dissociated cultures, on the other hand, there are a handful of videos on the internet (for example this one). However, this looks quite different from patching targeted by two-photon imaging. Camera-based imaging only allows to patch in thin tissues like cultures or slices since camera-imaging does not provide good optical sectioning and penetration. Here, I will show (uncut) movies of the process of patching, while monitoring the applied pressure and the test pulses.
As a brief introduction to two-photon targeted shadow-patching, the pipette approaches the brain surface while blowing out dye (1). After entering the tissue and after lowering of the pressure (2), the pipette closely approaches a target neuron (3). After gigaseal formation and break-in, allowing for electrophysiological recordings, the targeted neuron fills with the dye while the surroundings turn dark again (4).
All of the videos below are patch clamp recordings in the olfactory cortex homolog of zebrafish in an ex vivo preparation where the entire brain including the nose remain intact. Neurons are labeled using pan-neuronal expression of GCaMP6f, which is barely visible compared to the dye; shadow-imaging is performed using a resonant scanning microscope (described before on this blog). Neuronal somata in this brain area are quite small (5-8 μm in diameter, which is probably half or less the size of a typical mouse principal neuron), which can render targeted patching quite difficult, especially in deeper structures, where resolution degrades due to scattering. All of the recordings below are in more or less superficial regions (<200 μm below the brain surface). Patching deeper neurons usually required much more focused attention from my side, and the pipette tip could not be localized as easily as in the movies below.
For the paper, I produced a “Methods Video”, which due to restrictions from Neuron is limited to a duration of 1 min. I wanted to record not only the fluorescence movie during patching, but also the pressure and the test pulses applied to the electrode. For screen capture, I used the software TinyTake; for video editing, KdenLive (Linux); for text to speech synthesis of the next video, Wavenet provided by Google Cloud, which I have discussed before on this blog). The video is available in the Star Methods section of the paper, and also here:
However, the short duration of the video is maybe appropriate as a short visual summary for a paper, but not ideal for somebody who wants to get an intuition on how shadow patching can be done in reality. Therefore, here’s a longer excerpt of the same recording. I sometimes use this excerpt for presentations:
Still, this is a bit too condensed. Therefore, below you will find the uncut version of this particular patching experience. I admit it is really boring to watch, but I think it is also instructive. Not shown in the video are the changing positions of the two micromanipulators that move the pipette tip and the focus of the microscope; also not shown are small modifications to the laser power, zoom settings, bidirectional scan phase or the electrophysiological recording conditions. And yes, I’m aware that this recording is far from being perfect, but I think it can still be a useful starting point for a prospective electrophysiologist.
Next comes the patching of a different neuron. Usually, I’m using a syringe to apply pressure and suction to the pipette (other people prefer to apply the suction with the mouth). Here, after establishment of the giga-seal, the syringe somehow broke down and was not useable any more. I quickly constructed a temporary mouthpiece out of some tubings and finally managed to successfully break into the cell.
And here yet another successful attempt:
In total, I made around 20 such simultaneous recordings of the two-photon video, the pressure indicator and the test pulses window. Assembling the videos, however, turned out to take quite some time, and therefore I will show only one more movie, this time of a failed attempt. Almost immediately after entering the tissue, I realized that this recording would probably be not successful (the dura covering the brain sticked to the pipette tip for too long). Usually I would have stopped the attempt as early as possible in order not to waste time. In this case, I still tried to patch a neuron in order to get a nice recording of a failed attempt. Failures are not really rare when you try patching, especially in deep and small neurons.
Of course, shadow-patching might look somewhat different in a different brain region or with a different microscope or at a different tissue depth. To give you an idea, here is a recording with lower light levels due to lower dye concentration and laser power and with some problems related to the microscope (which I was too lazy to debug back then) which did not allow to zoom in as much as for the previous recordings. For someone not familiar with the particular setup, it is probably quite difficult to accurately see the pipette tip – which is crucial to move the pipette to the right location, in particular in the z-direction.
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## Precise synaptic balance of excitation and inhibition
The main paper of my PhD just got published: Rupprecht and Friedrich, Precise Synaptic Balance in the Zebrafish Homolog of Olfactory Cortex, Neuron (2018). (PDF)
You might like it if you are also interested in
• Classical balanced networks
• Things you can do with whole-cell voltage clamp
• Olfactory cortex
• More recent ideas about balanced networks
• Coordination of excitatory and inhibitory synaptic inputs in single neurons
To summarize this work in one sentence, this is a study of the coordination of excitatory and inhibitory synaptic inputs in single neurons. If you want to know the details, you should definitely read the paper.
The main part of the study is purely experimental, but one of its strengths is that it connects the experimental findings with computational concepts about balanced networks. The concept of a balanced state has been brought up in the mid-90s by Shadlen & Newsome and van Vreeswijk & Sompolinsky (among others). More recent theoretical work has, in my opinion, contributed a lot to identifying and correcting some weaknesses of the classical balanced network, and has come up with new concepts about circuit function of balanced networks that are of general interest to those who want to understand how the brain works. If you’re interested in a discussion of these concepts, I can recommend the following review articles as starting points (which are also discussed in our paper):
But let’s for one moment think beyond the scope of this work, which focused on synaptic inputs on the single-cell level – let’s think about the subcellular level. One thing I’d be interested in would be to have a closer look at the coordination of synaptic inputs on small dendritic segments instead of entire neurons. There is already a handful of studies that go into that direction, using mechanisms of synaptic plasticity (Chiu et al., Neuron, 2018) or the anatomical distribution of synapses (Iascone et al., bioRxiv) as entry points.
I’m really looking forward to seeing more research going into this subcellular level of neuronal processing. I can understand that people find population codes as observed by calcium imaging and extracellular recordings of interest, especially with respect to behavior. But I’m also convinced that mechanistic insights into how neurons work can be better obtained by investigating cellular and sub-cellular processes. Our published study investigates a variety of details on the cellular level; but this is only a small fraction of the many things that go unnoticed if you only look at the firing of neurons and not at underlying processes, for example the synaptic inputs.
## Alvarez lenses and other strangely shaped optical elements
In typical microscopes, lenses or mirrors are moved forth and back to change the position of their focus. Tunable lenses like the electro-tunable lens or the TAG lens, on the other hand, are deformed by an external force and thereby change their focal length. One interesting concept that I had not noticed until recently is the idea of the Alvarez lens, named after its inventor (described in this 1964 patent). I came across it in a 2017 paper from the lab of Monika Ritsch-Marte from Innsbruck/Austria. The following picture adapted from their paper very nicely illustrates the effect:
By lateral displacement of the two lens elements against each other one can focus or de-focus the beam. In two papers from this lab (paper 1, paper 2), the authors used a method that sort of replaces this lateral (slow) movement with a (fast) rotation of a galvo mirror by using a creative optical configuration (check out the paper for the details, it is a pleasure to read).
There a couple of things to notice: The Alvarez lens is a bit more complex in 2D (the above schematic illustrates a (de-)focusing system for 1D only). The authors use diffractive instead of refractive Alvarez lenses. They use only visible light (no near-infrared light, which I would prefer). And they mention some other shortcomings of their approach.
Still, I find the principle very interesting and inspiring, and I hope that somebody will invest his or her time to put a system together that is not only a proof-of-principle, but an optimized system that reaches the best possible performance. This would probably also be a nice playground for a study of optical modeling and optimization: to find out which shape of the lens could perform much better than the Alvarez lens (like this study, but a bit more systematic with respect to possible lens surfaces).
Overall, this is a fascinating piece of optics, and I got interested also because I had always been intrigued by optical scanning methods where a simple movement of the beam is translated to a complex scanning scheme by an optical element (see for example this blog post on entirely passive scanning at MHz rates). For a long time, I hoped that a method similar to an Alvarez lens and based on a strangely shaped mirror (or lens) surface could be used to transform a linearly scanned pattern into something more complex (like a spiral scan, or a 2D raster scan). In theory, this is possible, but in practice the finite beam diameter would create a lot of problems. In addition, constructing an arbitrarily shaped mirror with good surface flatness and broadband reflective coatings would be quite costly.
One field where I long thought that such an approach could be applied, because it would be applicable for many microscopes, is the un-distortion of the non-linear angular scanning trajectory of resonant scanners (described in detail in a previous blog post). The idea would be that an optical element (the ‘black box lens’ in the schematic below) placed after the resonant scanner would somehow convert the angular dependency $\sim \sin(\omega t)$ into a relationship that is rather linear in time, $\sim t$. Such that at time points close to the turnaround of the sine (blue time point below), the ‘black box lens’ would increase the angular deflection angle, eventually inversing the sine function:
I have the suspicion that this problem is practically not solvable due to the finite beam diameter, but it would be interesting to know whether there is a solution for this problem at least for the assumption of infinitely small scanning beams using geometric optics. This could be done by a lens whose diffractive power increases with distance $x$ from the center of the lens.
Let’s assume a scan angle $\alpha = \sin( \omega t)$. The scanned beam hits the black box lens at a position $x(t) = \tan(\alpha) \cdot d$ with the distance $d$ between the resonant scanner and the lens. The refractive power $f(x)$ of the lens must therefore change with $x$ such that the outgoing beam is linear in time. In approximative ABCD optics:
$\left( \begin{array}{cc} 1 & 0 \\ -\frac{1}{f(x(t))} & 1 \end{array} \right) \cdot \left( \begin{array}{cc} x(t) \\ \alpha (t) \end{array} \right) \stackrel{!}{=} \left( \begin{array}{cc} x(t) \\ t \end{array} \right)$
This results in the following expression for the local radius of the lens depending on the location $x$:
$f(x) = \frac{x}{\arctan(x/d) - \arcsin(\arctan(x/d))}$
You can find the equations and some plots also in a Jupyter notebook on Github. For small absolute values of $x$, $f(x)$ diverges, indicating an infinite curvature, i.e., a lens that simply transmits light without deflection. With increasing/decreasing $x$, $f(x)$ tends to zero, indicating increasing refractory power and a stronger local curvature of the lens surface.
Such a lens would only work optimally at one single zoom setting, which is probably one of the many reasons why nobody ever has tried this out. But it’s still interesting to think about it.
## Entanglement of temporal and spatial scales in the brain, but not in the mind
In physics, many problems can be solved by a separation of scales and thereby become tractable. For example, let’s have a look at surface waves on water: they are rather easy to understand when the water wave-length is much larger or much smaller than the depth of the water, but not if both scales are similar (wikipedia).
To give another example, light scattered by small particles (like fat bubbles in milk, or water drops in a cloud) can be described more easily if the wavelength of the light is much larger (Rayleigh scattering) or much smaller than the particles, but not if it is of the same order of magnitude (Mie scattering). Separation of scales is often key to making a problem tractable by mathematics.
What physicists like even more than separation of spatial scales, is the separation of different temporal scales. For example, consider two variables $A(t)$ and $B(t)$ that are influenced by each other:
$\tau_1 \frac{d A}{d t} = f(A,B) \\ \\ \tau_2 \frac{d B}{d t} = g(A,B)$
If the timescales separate, for example $\tau_1 \gg \tau_2$, the variable $B(t)$ is basically seen as constant by the variable $A(t)$. In this case, the variables can be decoupled, and the problem is often solvable. (Sidenote: In very simple and idealized systems without separations of scales, for example during some sort of phase transitions, mathematical physics can still come to the rescue and provide some clean solutions. But in most systems, this is not the case.)
I am convinced that problems do not become easier by a separation of scales only for physics or mathematics. I think that this applies even more to our intuition and our own understanding of the world. Automatically, we try to disentangle systems by using hierarchies and separations of length- and timescales, and if we are unable to do so, our intuition fails, as does the physics analysis.
What about the brain? In my opinion, the brain is one of those system that will defy human attempts to understand it by separating temporal processes or spatial modules. The brain consists of an enormous amount of different temporal and spatial scales that, however, overlap with each other and cannot be easily segregated. For example, on the timescale of few 100 ms, many different processes are non-stationary and therefore relevant at the same time: neuromodulation of many kinds; spike frequency adaptation and presynaptic adaptation and facilitation; diffusion of proteins across spines, or ions across dendrites; calcium spikes; NMDA currents; et cetera. At a timescale of 1000 ms or 10 ms, it is a different but overlapping set of processes that are non-stationary. To put it short, it seems likely to me that the brain consists of a temporal and spatial continuum of processes, rather than a hierarchy.
Why would this be so? Because, as far as I can see, there is no incentive for nature to prevent the entanglement of temporal and spatial scales of all those processes. In contrast, those interactions may offer advantages that emerge randomly by evolution, at the cost of a higher complexity. Nature, which does not need to understand itself, probably does not care much about an increase of complexity, unlike the biologists working to disentangle the chaos.
It is perhaps misleading to personify ‘nature’ and to speak of an ‘incentive’. It is probably more acceptable to derive these processes from ‘entropic forces’, which make any ordered system, including the organic and cellular systems invented by evolution, less ordered and therefore more chaotic over time. Even if there was order once (think of a glass of water which is strictly colored green in the left and blue in the right half), random changes, which is the driving force of evolution, will undo this order (nothing can prevent that green and blue water will mix over time by random motion of its molecules, that is, diffusion).
In addition to the deficiency of our mind and of mathematical tools when it comes to entangled scales, I suspect based on personal experience that humans are to some extent unable to bring together knowledge from different hierarchies. In neuroscience, most researchers stick to one small level of observation and the related processes; and in most cases it is very difficult to bridge the gaps between levels. For example, “autism” can be addressed by a neurologist who thinks about case studies and very specific behavioral observations of her patients; by a geneticists looking for combinations of genes that make a certain autistic feature in humans more likely; or by a neurophysiologist studying neurons in animals or in vitro models of autism, trying to dissect the contribution of neuronal connectivity or ion channel expression. Many people believe (or hope) that with sufficient knowledge and understanding, these different levels of observation will fuse together, resulting in a complete understanding that pervades all levels. I would argue – and I’d like to be disproven – that a more pessimistic view seems to be more realistic and that humans will probably never achieve an understanding of neuronal circuits and the brain that is deep enough to bridge the gaps between the levels.
The limitations of both our mathematical tools and our mind when it comes to complex systems is obvious when we think of deep learning. For this field of machine learning, other than for the brain, we know all the basic principles (because we have defined them ourselves): Back-propagation of errors, gradient descent algorithms for optimization, weight-sharing in convolutional networks, rectified linear units (or maybe LSTM units), and a few more. Compared with the brain, the system is not very complex, and we can observe everything throughout the process without interfering with its operation. Still, although the process is 100% transparent, people struggle and fail to understand what is happening and why. There does not seem to be a simple answer to the question how it works. “What I cannot create, I do not understand”, Feynman famously wrote. But the act of creation does not automatically come with understanding.
Experimental neuroscience might face similar, but probably even more complex problems. The way to “understand” a neuronal process that is accepted by most researchers is a (mathematical or non-mathematical) model that can both reproduce and predict experimental results. However, if biology indeed consists of many processes and components that are entangled in space and time, also a model needs to be built that is entangled on several temporal and spatial scales. This can be done – no problem. However, this model will again resist attempts by mathematics or human intuition to understand it, similar to our current lack of understanding of the less complex deep networks. Therefore, the machine (the model, the computer program) will still be able to deal with the complexity and “understand” the brain, but I am not sure that human intuition will be able to follow.
I don’t want to deny all the pieces of progress that have been made to achieve a better understanding of the brain. I rather want to point out the limitation of the human mind when it comes to putting the pieces together.
## Blue light-induced artifacts in glass pipette-based recording electrodes
Recently, I was carrying out whole-cell voltage-clamp and LFP recordings with simultaneous optogenetic activation of a channelrhodopsin using blue light. Whole-cell voltage clamp techniques can record the input currents seen by a neuron (previously on this blog [1], [2]); an LFP records the very small synaptic currents in bulk brain tissue (nicely reviewed by Oscar Herreras); and optogentics with genetically encoded rhodopsins can make neurons fire using light pulses.
For the LFP recordings, I used the same glass pipette that I had used before for the whole-cell recording of a nearby neuron. In the LFP, I saw a light-evoked response which I first thought was a rhodopsin-evoked synaptic current. However, it turned out that I could make the same observation when positioning the pipette tip in the bath instead of in the tissue, which meant that this was clearly not a synaptic current, but an artifact. When changing the pipette resistance by gently breaking the pipette tip, the light-evoked voltage remained the same, whereas the evoked currents changed proportionally with the pipette resistance Rp, or more generally with the resistance between the two electrodes:
I found out that this sort of artifact has been described in the context of tetrode recordings several years ago by Han et al. (2009; supplementary figure 1) and has been sort of explained with the Becquerel effect (here), which is better known as the photovoltaic effect. According to Han et al., the effect is stronger for blue light and affects the recorded currents on a slow timescale, such that highpass-filtering of the recorded signal, which is used to detect spikes in tetrode recordings, gets rid of this artifact.
In addition, Han et al. state:
We have not seen the artifact with pulled glass micropipettes (such as previously used in Boyden et al., 2005 and Han and Boyden, 2007, or in the mouse recordings described below). Thus, for recordings of local field potentials and other slow signals of importance for neuroscience, hollow glass electrodes may prove useful.
Contrary to this suggestion, my above measurements indicate that using a glass electrode does not or not always get rid of the artifact. To better understand this artifact, I checked whether it was mediated by the chloride silver electrode in the glass pipette or rather by the ground electrode, and found that both contributed more or less equally to the artifact in this experiment. Protection of the electrode by some sort of cover reduced the magnitude of the artifact.
What does this mean for whole-cell or LFP recordings using a glass pipette? For whole-cell recordings, the resistance between the two electrodes is much larger than for the two traces shown in the plots above, typically between 50 and 2000 MΩ. This reduces the artifact-induced current recorded in voltage-clamp to something less than 5 pA for 50 MΩ cells, and much less for neurons with higher membrane resistance. In most cases, this is negligible.
For glass pipette-based LFP recordings, however, the light-induced voltage change (few hundred μV, as shown above) is of the same magnitude as a strong LFP signal (see for example figure 1 in Friedrich et al., 2004). Therefore, in order to measure a LFP signal in response to blue light-activated rhodopsins, one needs to take into account the artifacts induced by the photovoltaic effect. This can for example be done by measuring the light-evoked voltage change with the glass pipette both in the tissue and in the bath and subtracting the latter measurement from the previous one on a pipette-by-pipette basis.
I would also be curious about other reports (if there are any) on light-induced artifacts with recording electrodes and under which circumstances (if there are any) they might play a non-negligible role.
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http://equamount.com/how-much-ttov/do-quartermasters-see-combat-9bf63b
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In the game subsystems for a starship are split into four sections: Weapons, Engines, Shields and Auxiliary. save. Combat-related service includes training that simulates war, hazardous duty, using an instrument of war, or armed conflict. It is a quest reward. Quartermaster's Note is a quest item needed for Quartermaster's Bounty. Combat definition is - a fight or contest between individuals or groups. The Quartermaster School (QMS) is a subordinate command of the United States Army's Combined Arms Support Command. Quartermasters learned valuable lessons in supporting a large, modern Army overseas that would be carried into the next conflict. MEDAL OF HONOR. They work with patrol quartermasters as they check out equipment and return it. Ground Forces are warriors. Synonym Discussion of combat. Specifically, 35L (counter-intelligence.) 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2021-05-06 18:05:49
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https://aptitude.gateoverflow.in/4309/cat1994-73?show=4314
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75 views
A and B walk from X to Y, a distance of 27 km at 5 kmph and 7 kmph respectively. B reaches Y and immediately turns back meeting A at Z. What is the distance from X to Z?
1. 25 km
2. 22.5 km
3. 24 km
4. 20 km
According to the question , B's speed is more than A so it will take less time to cover the distance between X and Y.
Let distance between X and Z = x
So distance between Y and Z = 27 - x [ As B returns towards X and meet A at Z in that process ]
For meeting , time taken throughout must be same . Thus we have :
x / 5 = (27 + 27 - x) / 7
==> x / 5 = (54 - x) / 7
==> 7x = 270 - 5x
==> 12x = 270
==> x = 22.5
Hence B) should be the correct option.
answered by (1.4k points) 1 3 12
selected by
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2018-08-18 02:18:26
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https://www.physicsforums.com/threads/nasty-autocorrelation-integral.731436/
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# Nasty Autocorrelation Integral
1. Jan 5, 2014
### krobben92
Hi guys,
Long story short, I need to compute an autocorrelation integral. Here's the problem:
There are two arbitrary gaussian pulses, one following the other by a fixed distance. By computing the autocorrelation over space(not time) and taking the derivative of the space-shift autocorrelation and setting it equal to zero, important information hopefully could be obtained.
The mathematics of this would be as followed:
$\frac{\partial}{\partial\tau}\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(Ae^{-a(x-c-\tau)^2}+Be^{-b(x-d-\tau)^2})dx=0$
$\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(a(x-c-\tau)Ae^{-a(x-c-\tau)^2}+b(x-d-\tau)Be^{-b(x-d-\tau)^2})dx=0$
I am NOT asking anyone to do this for me - I'll do it myself but I just need some ideas or directions on how to go about it.
I have experience in fourier transforms, complex analysis and calculus of course. I've considered doing a complex contour integral but I'm not sure how reasonable that is after seeing how big of a pain the normal gaussian contour integral is. I've considered fourier transforms a little - I didn't immediately see much help due to the fourier transform of a gaussian just being another gaussian. I've thought about parametrization or even centering the integral about the center of the two gaussians but I don't know where to start I guess.
It's clearly a bound integral but is it just too impossibly hard to try?
Last edited: Jan 5, 2014
2. Jan 5, 2014
### Staff: Mentor
You can reduce the problem to several integrals of the types $\int dx e^{-(x+d)^2 - (x-d)^2}$, $\int dx x e^{-(x+d)^2 - (x-d)^2}$ and maybe something I missed and look for solution methods for those integrals.
3. Jan 5, 2014
### krobben92
Yes, that's definitely one way to do it. However, this may take a couple dozen sheets of paper and a few hours considering the factoring. I guess after seeing so many tricks in math classes I just assumed there might be a quick way around this... But real world problems versus classroom problems aren't a fair comparison I suppose.
4. Jan 5, 2014
### krobben92
Well I tried it the old fashion way and it turned out better than I thought - but I don't think $\tau$ can be solved for explicitly.
$\frac{-A^{2}}{2\sqrt{2a}}e^{\frac{-a\tau^{2}}{2}}+\frac{-B^{2}}{2\sqrt{2b}}e^{\frac{-b\tau^{2}}{2}} = \frac{2ABe^{\frac{-ab(j(j-2k)+k^{2})}{a+b}}}{(a+b)\sqrt{a+b}}e^{\frac{-ab\tau^{2}}{a+b}}(\tau cosh(\frac{2ab\tau(j-k)}{a+b})-(j-k)sinh(\frac{2ab\tau(j-k)}{a+b}))$
Any other ideas? I'm not an expert on fourier transforms but I'm beginning to think that's the only way because it should be a multi-valued answer - just not sure how to approach it.
Last edited: Jan 5, 2014
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2017-11-21 11:40:39
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http://mathhelpforum.com/pre-calculus/24921-afew-questions.html
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# Math Help - Afew Questions
1. ## Afew Questions
Any help with the following questions is appreciated
Linear/Non-Linear Systems of Equations
1. Write an equation to form a system with $y=5x-1$ so that the system has (2,9) as its only solution.
Answer is $y=2x+5$
But how do i get that answer?
2. The range of the graph $y=\frac{1}{2}(x-12)^2+15$ is: ?
1. What is the new equation when $y=(x-2)^2+9$ is translated 5 units left and 3 units down?
1. Determine the values of K so that the graph $f(x)=x^2+4x+k$ crosses the x-axis twice.
$b^2-4ac>0$
$(4)^2-4(1)(k)>0$
$16-4k>0$
$-4k>-16$
$k<4$
Im wondering how to tell if i need a greater/less or equal sign (I think it is > for 2 real roots, < 2 unreal, = roots that are equal) also why it switched on the last step from -4k>-16 to K<4
Polynomial Functions/Equations
1. What are the zeroes of $7(2x+3)^3(x-1) > 0$?
I got $x=\frac{-3}{2},1$
But i think there are others?
2. I must ask you what in the world that posting means?
It seems to be just series of meaningless statements.
3. Was in a rush sort of, i can't believe i forgot my notes at school >.<
I think its more clearer now.
4. Originally Posted by Raj
1. Write an equation to form a system with $y=5x-1$ so that the system has (2,9) as its only solution.
Answer is $y=2x+5$
But how do i get that answer?
I presume that you are looking for another linear function, else there are an infinite number of answers.
So let's assume we have a new equation of the form $y = mx + b$.
So solve the system
$y = 2x + 5$
$y = mx + b$
such that the solution is (2, 9).
I'll start you off:
Put the solution point into the second equation:
$9 = 2m + b$
So
$b = 9 - 2m$
Thus the second equation is
$y = mx + (9 - 2m)$
Now we need to solve the system
$y = 2x + 5$
$y = mx + (9 - 2m)$
for m.
See what you can do with this.
-Dan
5. Originally Posted by Raj
2. The range of the graph $y=\frac{1}{2}(x-12)^2+15$ is: ?
The range of a graph is the set of all possible y values that the function takes. So what y values can this function take?
(Hint: Graph it.)
-Dan
6. Originally Posted by Raj
1. What is the new equation when $y=(x-2)^2+9$ is translated 5 units left and 3 units down?
Consider the function $y = f(x)$.
When we translate this graph h units to the right the corresponding function is $y = f(x + h)$.
When we translate this graph h units to the left the corresponding function is $y = f(x - h)$.
When we translate this graph k units up the corresponding function is $y - k = f(x) \implies y = f(x) + k$.
When we translate this graph k units down the corresponding function is $y + k = f(x) \implies y = f(x) - k$.
-Dan
7. Originally Posted by Raj
1. Determine the values of K so that the graph $f(x)=x^2+4x+k$ crosses the x-axis twice.
$b^2-4ac>0$
$(4)^2-4(1)(k)>0$
$16-4k>0$
$-4k>-16$
$k<4$
Im wondering how to tell if i need a greater/less or equal sign (I think it is > for 2 real roots, < 2 unreal, = roots that are equal) also why it switched on the last step from -4k>-16 to K<4
Note that if the discriminant is equal to 0 then there is only one solution to the quadratic equation. Thus we can't include the equal sign on the line $b^2-4ac>0$.
Whenever you divide both sides of an inequality by a negative number the inequality changes "direction." So
$-4k>-16$
$\frac{-4k}{-4} < \frac{-16}{-4}$
$k<4$
If you have troubles seeing this, consider a simpler example. Solve
$-x > 1$
You can easily get the solution set from the given inequality, and you should be able to see that it is $x < -1$, which is what you get when you divide both sides by -1.
-Dan
8. Originally Posted by Raj
1. What are the zeroes of $7(2x+3)^3(x-1) > 0$?
I got $x=\frac{-3}{2},1$
But i think there are others?
I presume this is really $7(2x+3)^3(x-1) = 0$?
Why should there be other zeros?
The concept behind solving something like this is the following theorem:
If ab = 0 then either a = 0 or b = 0 or both.
So when we a simpler example like
$(x - 1)(x - 2) = 0$
then either
$x - 1 = 0 \implies x = 1$
or
$x - 2 = 0 \implies x = 2$
(and we can't have both terms equal to 0 at the same time.) You've likely seen this process any number of times when solving a quadratic that factors.
In this case we have:
$7(2x+3)^3(x-1) > 0$
$7(2x + 3)(2x + 3)(2x + 3)(x - 1) = 0$
So either
$7 = 0$ <-- Not true!
or
$2x + 3 = 0 \implies x = -\frac{3}{2}$
or
$2x + 3 = 0 \implies x = -\frac{3}{2}$
or
$2x + 3 = 0 \implies x = -\frac{3}{2}$
or
$x - 1 = 0 \implies x = 1$
So we have
$x = -\frac{3}{2}, -\frac{3}{2}, -\frac{3}{2}, 1$
or more simply
$x = -\frac{3}{2}, 1$
-Dan
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2014-09-17 21:16:06
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http://www.math-only-math.com/parametric-equations-of-the-hyperbola.html
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# Parametric Equation of the Hyperbola
We will learn in the simplest way how to find the parametric equations of the hyperbola.
The circle described on the transverse axis of a hyperbola as diameter is called its Auxiliary Circle.
If $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 is a hyperbola, then its auxiliary circle is x$$^{2}$$ + y$$^{2}$$ = a$$^{2}$$.
Let the equation of the hyperbola be, $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1
The transverse axis of the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 is AA’ and its length = 2a. Clearly, the equation of the circle described on AA’ as diameter is x$$^{2}$$ + y$$^{2}$$ = a$$^{2}$$ (since the centre of the circle is the centre C (0, 0) of the hyperbola).
Therefore, the equation of the auxiliary circle of the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 is, x$$^{2}$$ + y$$^{2}$$ = a$$^{2}$$
Let P (x, y) be any point on the equation of the hyperbola be $$\frac{x^{2}}{a^{2}}$$ -$$\frac{y^{2}}{b^{2}}$$ = 1
Now from P draw PM perpendicular to the transverse axis of the hyperbola. Again take a point Q on the auxiliary circle x$$^{2}$$ + y$$^{2}$$ = a$$^{2}$$ such that ∠CQM = 90°.
Join the point C and Q. The length of QC = a. Again, let ∠MCQ = θ. The angle ∠MCQ = θ is called the eccentric angle of the point P on the hyperbola.
Now from the right-angled ∆CQM we get,
CQ/MC = cos θ
or, a/MC = a/sec θ
or, MC = a sec θ
Therefore, the abscissa of P = MC = x = a sec θ
Since the point P (x, y) lies on the hyperbola $$\frac{x^{2}}{a^{2}}$$ -$$\frac{y^{2}}{b^{2}}$$ = 1 hence,
$$\frac{a^{2}sec^{2} θ }{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1, (Since, x = a sec θ)
$$\frac{y^{2}}{b^{2}}$$ = sec$$^{2}$$ θ – 1
$$\frac{y^{2}}{b^{2}}$$ = tan$$^{2}$$ θ
y$$^{2}$$ = b$$^{2}$$ tan$$^{2}$$ θ
y = b tan θ
Hence, the co-ordinates of P are (a sec θ, b tan θ).
Therefore, for all values of θ the point P (a sec θ, b tan θ) always lies on the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1
Thus, the co-ordinates of the point having eccentric angle θ can be written as (a sec θ, b tan θ). Here (a sec θ, b tan θ) are known as the parametric co-ordinates of the point P.
The equations x = a sec θ, y = b tan θ taken together are called the parametric equations of the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1; where θ is parameter (θ is called the eccentric angle of the point P).
Solved example to find the parametric equations of a hyperbola:
1. Find the parametric co-ordinates of the point (8, 3√3) on the hyperbola 9x$$^{2}$$ - 16y$$^{2}$$ = 144.
Solution:
The given equation of the hyperbola is 9x2 - 16y2 = 144
$$\frac{x^{2}}{16}$$ - $$\frac{y^{2}}{9}$$ = 1
$$\frac{x^{2}}{4^{2}}$$ - $$\frac{y^{2}}{3^{2}}$$ = 1, which is the form of $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1.
Therefore,
a$$^{2}$$ = 4$$^{2}$$
⇒ a = 4 and
b$$^{2}$$ = 3$$^{2}$$
⇒ b = 3.
Therefore, we can take the parametric co-ordinates of the point (8, 3√3) as (4 sec θ, 3 tan θ).
Thus we have, 4 sec θ = 8
⇒ sec θ = 2
⇒ θ = 60°
We know that for all values of θ the point (a sec θ, b tan θ) always lies on the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1
Therefore, (a sec θ, b tan θ) are known as the parametric co-ordinates of the point.
Therefore, the parametric co-ordinates of the point (8, 3√3) are (4 sec 60°, 3 tan 60°).
2. P (a sec θ, a tan θ) is a variable point on the hyperbola x$$^{2}$$ - y$$^{2}$$ = a$$^{2}$$, and M (2a, 0) is a fixed point. Prove that the locus of the middle point of AP is a rectangular hyperbola.
Solution:
Let (h, k) be the middle point of the line segment AM.
Therefore, h = $$\frac{a sec θ + 2a}{2}$$
⇒ a sec θ = 2(h - a)
(a sec θ)$$^{2}$$ = [2(h - a)]$$^{2}$$ …………………. (i)
and k = $$\frac{a tan θ}{2}$$
⇒ a tan θ = 2k
(a tan θ)$$^{2}$$ = (2k)$$^{2}$$ …………………. (ii)
Now form (i) - (ii), we get,
(a sec θ)$$^{2}$$ - (a tan θ)$$^{2}$$ = [2(h - a)]$$^{2}$$ - (2k)$$^{2}$$
⇒ a$$^{2}$$(sec$$^{2}$$ θ - tan$$^{2}$$ θ) = 4(h - a)$$^{2}$$ - 4k$$^{2}$$
⇒ (h - a)$$^{2}$$ - k$$^{2}$$ = $$\frac{a^{2}}{4}$$.
Therefore, the equation to the locus of (h, k) is (x - a)$$^{2}$$ - y$$^{2}$$ = $$\frac{a^{2}}{4}$$, which is the equation of a rectangular hyperbola.
The Hyperbola
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2017-02-19 16:35:59
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http://www.thestudentroom.co.uk/showthread.php?t=1960613
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# power series - radius of convergence Tweet
Maths and statistics discussion, revision, exam and homework help.
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1. power series - radius of convergence
We went through an example in class the other week but I don't quite follow.
Basically, we are given a power series from k=0 to infinity: [(k!)^3 / (3k!)!]*x^k
Using the ratio test: mod of bk+1/bk eventually leads to:
[(k+1)^3 * mod x] / [(3k+3)(3k+2)(3k+1)]
Then this is meant to converge to [mod x] / 27, thus giving a radius of convergence of 27.
But I don't see how you get that it converges to [mod x]/27.
2. Re: power series - radius of convergence
(3k+1), (3k+2), (3k+3) ~3k each as k--> infinity
k+1~k as k-->infinity
So you get approximately k^3|x|/(3k)^3
Make sense?
(I don't know how much precision you want here, so if you want slightly more rigour, I'm sure that's possible)
3. Re: power series - radius of convergence
Yeah I guess I can see how that one works. But then for the next example (which the lecturer didn't go through so I did this working out on my own, I don't think I quite get it.
I have a power series again from k=0 to infinity. [(-1)^k / 2k!] * x^2k
Using the same method again, it brings me to x^2 /(2k+1)(2k+2).
This example is supposed to have a radius of convergence of 30. So maybe I did something wrong in my calculation?
4. Re: power series - radius of convergence
I agree with the x^2/(2k+1)(2k+2) bit. I'm not sure how they're getting to 30 from there either.
5. Re: power series - radius of convergence
It's a Maclaurin series, isn't it (exponential)? I've no idea where the value 30 comes from for the radius of convergence, since it ought to converge for all real x.
6. Re: power series - radius of convergence
(Original post by james.h)
It's a Maclaurin series, isn't it (exponential)? I've no idea where the value 30 comes from for the radius of convergence, since it ought to converge for all real x.
No, it's
7. Re: power series - radius of convergence
(Original post by Mathlete29)
No, it's
It's cos, so convergence is still as he said.
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2013-06-20 12:01:12
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https://issues.gradle.org/browse/GRADLE-2123
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# WTP Plugin should add org.eclipse.jst.component.dependency to all WEB-INF/lib deps and not add them to the WTP component file
XMLWordPrintable
#### Details
• Type: Bug
• Status: Resolved
• Resolution: Fixed
• Affects Version/s: 1.0-milestone-7, 1.0-milestone-8
• Fix Version/s:
#### Description
Currently Gradle is generating classpath entries like this for JAR dependencies in the WAR:
<classpathentry kind="lib" path="C:\Documents and Settings\it403\.gradle\caches\artifacts\ch.qos.logback\logback-core\ef28c670c723aafc18a1f247a00c89bb\jars\logback-core-0.9.29.jar" exported="true" />
It should be generating:
<classpathentry kind="lib" path="C:\Documents and Settings\it403\.gradle\caches\artifacts\ch.qos.logback\logback-core\ef28c670c723aafc18a1f247a00c89bb\jars\logback-core-0.9.29.jar" exported="true">
<attributes>
<attribute name="org.eclipse.jst.component.dependency" value="WEB-INF/lib"/>
</attributes>
</classpath>
And once it starts using "org.eclipse.jst.component.dependency", the entries from the WTP component file should be removed. Without removing them the JARs may be included in the classloader twice when using a server within Eclipse.
<dependent-module deploy-path="/WEB-INF/lib" handle="module:/classpath/lib/C:\Documents and Settings\it403\.gradle\caches\artifacts\ch.qos.logback\logback-core\ef28c670c723aafc18a1f247a00c89bb\jars\logback-core-0.9.29.jar">
<dependency-type>uses</dependency-type>
</dependent-module>
This seems related to a bunch of issues that have been reported (and some of them fixed before):
This Maven issue was linked from one of them, which recommends the approach I documented above:
http://jira.codehaus.org/browse/MECLIPSE-264
Bear in mind that one difference in my use case is that we are using SelfResolvingDependencies. I don't know if these are handled differently in the WTP plugin than the module dependencies you would get from using a Maven/Ivy repo.
Even if that is the issue, they should be handled consistently.
#### People
Assignee:
Donát Csikós
Reporter:
Andrew Oberstar
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2020-08-04 08:34:25
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https://elibm.org/article/10000266
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## Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces
### Summary
Summary: We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with $K_S^2 = 4$, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with $K_S^2 = 4$, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=4$ form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=3$ form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms $G$. In fact, we show that for the minimal model $S$ of a nodal Burniat surface $(G = (\ZZ/2 \ZZ)^2)$ we have $\Def(S,G) \neq \Def(S)$, whereas for the canonical model $X$ it holds $\Def(X,G) = \Def(X)$. All deformations of $S$ have a $G$-action, but there are different deformation types for the pairs $(S,G)$ of the minimal models $S$ together with the $G$-action, while the pairs $(X,G)$ have a unique deformation type.
### Mathematics Subject Classification
14J29, 14J25, 14J10, 14D22, 14H30, 32G05
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2023-02-05 23:06:14
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https://www.qalaxia.com/questions/Equivalent-fractions
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D
Equivalent fractions
16 viewed last edited 1 year ago
Sherlyn Casco Noperi
0
I didn't try anything cause I don't know how to do it.
Equivalent fractions of 10/15=?/3
Mahesh Godavarti
1
Hi Sherlyn, we have to understand what fractions are. First, I will give you the quick answer then I will explain the meaning behind it in a comment. First, notice that 10 = 5 X 2, 15 = 5 X 3. Therefore, 10/15 = (5 X 2)/(5 X 3) = 2/3 (Removing 5 as a common factor between numerator and denominator)
Mahesh Godavarti
0
Think of what fractions are. Let's say there are 15 apples in a bag and you take out 10 of them. Then you have taken 10 / 15 fraction of apples. But, now let's say apples in the bag are bundled together in a mini-bag. Let each mini-bag contain 5 apples. Let there be 3 such mini-bags. If we pick out 2 mini-bags, then the fraction of mini-bags that we picked is 2/3 (we picked 2 mini-bags out of 3). But, note that this is the same as picking 10 apples out of 15 apples. Therefore, whether we think of it as picking 10 apples out of 15 apples or 2 mini-bags out of 3 mini-bags, it is basically the same thing! Therefore, 10/15 = 2/3
Vivekanand Vellanki
0
You seem to be asking for an equivalent fraction in simplest form. 10/15 is equivalent to 2/3; it is also equivalent to 4/6; or 20/30, etc. Mahesh's answer shows how to find the equivalent fraction in simplest form. To do this, you need to do the following: 1) find the common factors between the numerator and the denominator. In this case, 5 is the common factor 2) Write the numerator and denominator as multiples of the common factor. 10/15 = 2*5/3*5 3) Now, eliminate the common factor to get 10/15 = 2/3 Continue this process until there are no more common factors
Sangeetha Pulapaka
0
The question you have is how to find the missing number when the two fractions are equivalent. Equivalent fractions are fractions which look different but have the same value. All we need to know to find the missing number in two fractions which are are equivalent is the multiplication table or the division table. \frac{10}{15}= \frac{?}{3} Look at the denominators of both the fractions given. When you divide 15 by which number do you get a 3? You divide 15 by 5 you get 3. To make sure that the value of the fraction does not change when the denominator is divided by 5, the numerator should also be divided by 5. So dividing the numerator 10 by 5 we get the missing number as 2. How to check if this answer is right or wrong? Do the multiplication table of 2 which is 2\times 5 = 10 Do the multiplication table of 3 which is 3\times 5 = 15. Since we get the same fraction this means that the missing number is 2.
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2019-02-19 14:06:12
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https://gamedev.stackexchange.com/questions/165435/why-are-committed-resources-better-than-placed-resources/184428
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# Why are committed resources better than placed resources?
This NVIDIA document regarding dos and don'ts in DirectX 12 states the following:
Use committed resources where possible to give the driver more knowledge
• This allows the driver to better manage GPU memory
• A good use case for placed resources are resource heaps that are e.g. used during streaming and do hold different sets of read-only textures over their life time
But this raises more questions than it answers:
• What additional knowledge is given to the driver if I use a committed resource? The information on the resource that I pass to both CreateCommittedResource and CreatePlacedResource is pretty much the same...
• What does it mean that the driver will better manage GPU memory?
• If I know what the allocation/deallocation patterns look like in my game, isn't it better if I manage GPU memory manually with placed resources, instead of letting the GPU decide where to place each resource, and potentially run into VRAM fragmentation in the long run? Or will the driver perform heap defragmentation?
• CreateCommittedResource creates an implicit heap each time it is called, while CreatePlacedResource simply creates the resource over the memory I choose on an existing heap, and is therefore much faster. Wouldn't it be preferable in most cases to use placed resources instead? Especially if I am interested in creating loading screens that don't miss a single frame deadline?
I am quite surprised to read from NVIDIA that committed resources are preferable to placed resources. Could anybody explain in more detail what else is happening under the hood, so committed resources are preferable to placed resources?
• The general answer is that the more flexibility you provide the driver, the better it can choose the right kind of memory based on the specific device. CreatePlacedResources is more proscriptive than CreateCommittedResource for technologies like nVidia TurboCache as well as dealing with video memory overcommittment. What actually happens is up to the driver writer here, which is basically what nVIDIA would prefer as noted in their article. In the end it's general advice, so you need to test on a variety of cards to see what actually works. – Chuck Walbourn Dec 4 '18 at 18:17
• @ChuckWalbourn: Thanks for the insight. I thought the entire point of Direct3D 12 was to leave less decisions to the driver so you can optimize for your use case, so I'm surprised that they would prefer to defer those decisions to the driver. – Panda Pajama Dec 7 '18 at 10:53
Page 14: "Sub-allocation from a larger heap (as intended) improved performance by ~15%"
Page 46: "Creating committed resources for everything will use tons of memory"
Pages 25 ~ 26
The philosophy of the new-generation APIs is to offload burden from drivers (implicit, extra guess work) to applications who know the needs better (explicit, do only what you need), to boost CPU performance and save GPU memory (less fragmentation, better reuse). D3D12 Committed resources are just like D3D11 resources, if you use Committed resources then you go back to D3D11 with driver and possibly paging (on over-subscription) costs.
https://software.intel.com/content/www/us/en/develop/blogs/direct3d-12-overview-part-3-resource-binding.html
"Resource residency management:
GPUs like to use a lot a video memory, more in fact that is often available. This can be more of problem with discrete video cards with a set amount of memory. So we have resource residency management that pages things in and out of memory as the commands flow. To the game it looks like there is unlimited memory, when really it is just memory management. Once again this comes at a cost of reference counting and tracking.
Just like resource lifetime, the game gets explicit control over resource residency. D3D 11 tracks the residency counts and control flow in the operating system. Typically though the game already knows that a sequence of rendering commands refers to a set of resources. In D3D 12 the game can now explicitly tell the operating system to move them into memory. Later on when the commands have been executed the game can have the resources removed from memory."
Allocations call into the kernel-model driver, which is costly.
Vulkan even has a limit in # of allocations.
https://docs.microsoft.com/en-us/windows/win32/direct3d12/memory-management-strategies
Search for "sub-allocation" + "Direct3D 12"/"Vulkan"/"Metal", you will basically get the same answer except in this NVIDIA DX12 Do's And Don'ts.
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2021-01-27 00:51:12
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https://maker.pro/forums/threads/pwm-dc-motor-circuit.4959/
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PWM DC motor circuit?
T
Terry Pinnell
Jan 1, 1970
0
Could someone kindly save me a bit of time in finding a suitable
circuit for driving a DC motor with PWM please? My first hour of
googling has found many threads, but so far no specific relevant
schematics.
The DC motor is from a cordless screwdriver, used originally with two
NiCad C cells, and duly marked as '2.4V'. I ran it briefly at 5V and
assume it would tolerate that, giving me higher torque potential. It's
geared, and (unloaded with 2.4V applied) gives 250 rpm. That's too
fast for my curtain control application so I want to be able to reduce
it. Hence PWM.
I'll probably use limit switches and a C/O relay for the
forward/reverse/off control, at least while getting the challenging
mechanicals sorted (as per separate thread). So I don't need to
consider the complications of combining PWM with an H-Bridge.
My first thoughts are to use a 5V supply (derived from a convenient
fairly heavy duty 12V supply I have nearby) with a variable duty cycle
555 (or maybe a 4001/4011 equivalent), directly driving a medium power
MOSFET, with the motor between its drain and 5V. I'll breadboard that
shortly, but meanwhile:
- will that simple approach be reliable?
- any snags or protective measures to watch out for?
- is there an 'optimum' frequency range?
- if I designed a *high* duty cycle from 555, which I recall makes it
easier to achieve a wide d/c range, what is most efficient way to
configure the output MOSFET stage?
- I happen to have a 2N3055 (or similar) already neatly mounted on a
h/sink, from a previous project; any major reason why I shouldn't use
As mentioned in the curtain thread, I'd like to build this asap, as
the full-speed motor action is adding to my design and testing
problems.
TIA.
P
Paul Burridge
Jan 1, 1970
0
Could someone kindly save me a bit of time in finding a suitable
circuit for driving a DC motor with PWM please? My first hour of
googling has found many threads, but so far no specific relevant
schematics.
This is a really clever design that works very well indeed in
practice:
http://www.solorb.com/elect/solarcirc/pwm1/
P
Paul Burridge
Jan 1, 1970
0
finished printing out that very circuit! One of two I'd found at
following further googling this afternoon.
I'm also going to try my own 'simple 555' design mentioned in my OP.
The main problem with using a 555 based design is going to be the
1/3-2/3 range of available supply volts. That may not be a problem in
your particular case but it's nice to have the full rail-rail o/put
this quad op-amp gives you, IMHO.
J
John Jardine
Jan 1, 1970
0
[clp]
Terry Pinnell
Hobbyist, West Sussex, UK
My cooker extracter uses a pot', setting a 555 pwm, driving a TIP110
darlington, directly feeding an ex Alfa-Romeo radiator fan. Frequency is
arbitrary at a couple of kHz. No heatsinking, no protection or
jiggery-pokery parts, other than a diode across the motor.
There must be all of 50p's worth of bits and it's run fine for years.
regards
john
T
Terry Pinnell
Jan 1, 1970
0
John Jardine said:
[clp]
Terry Pinnell
Hobbyist, West Sussex, UK
My cooker extracter uses a pot', setting a 555 pwm, driving a TIP110
darlington, directly feeding an ex Alfa-Romeo radiator fan. Frequency is
arbitrary at a couple of kHz. No heatsinking, no protection or
jiggery-pokery parts, other than a diode across the motor.
There must be all of 50p's worth of bits and it's run fine for years.
regards
john
Thanks. I'm probably going to have to get a bit more complex in this
case, due to a combination of the reversal requirement plus automatic
stopping at each extreme. In considering that very simple '555 driving
MOSFET' appoach, (or the epanorama site version), I'd vaguely thought
I could implement these additional facilities with a relay and
microswitches. But of course that's not possible; the motor will be
permanently connected with one specific polarity, one end to Vcc and
the other via the MOSFET (or NPN BJT) to ground.
However, I might *still* build such a simple PWM circuit, purely to
get the speed down while I experiment. At present, I barely get time
to apply power before the cord flies off the pulley or fouls up in
some way!
But for my final circuit, it now looks as if I'll need an H-Bridge
approach. That looks more easily done with two NPN and two PNP BJTs,
but I'd welcome others' views please.
How's the Alfa getting on without its fan?
P
Paul Burridge
Jan 1, 1970
0
How's the Alfa getting on without its fan?
Alfas don't have fans anymore. They've all gone over to Ferrari.
J
John Jardine
Jan 1, 1970
0
Terry Pinnell said:
John Jardine said:
[clp]
Terry Pinnell
Hobbyist, West Sussex, UK
My cooker extracter uses a pot', setting a 555 pwm, driving a TIP110
darlington, directly feeding an ex Alfa-Romeo radiator fan. Frequency is
arbitrary at a couple of kHz. No heatsinking, no protection or
jiggery-pokery parts, other than a diode across the motor.
There must be all of 50p's worth of bits and it's run fine for years.
regards
john
Thanks. I'm probably going to have to get a bit more complex in this
case, due to a combination of the reversal requirement plus automatic
stopping at each extreme. In considering that very simple '555 driving
MOSFET' appoach, (or the epanorama site version), I'd vaguely thought
I could implement these additional facilities with a relay and
microswitches. But of course that's not possible; the motor will be
permanently connected with one specific polarity, one end to Vcc and
the other via the MOSFET (or NPN BJT) to ground.
However, I might *still* build such a simple PWM circuit, purely to
get the speed down while I experiment. At present, I barely get time
to apply power before the cord flies off the pulley or fouls up in
some way!
But for my final circuit, it now looks as if I'll need an H-Bridge
approach. That looks more easily done with two NPN and two PNP BJTs,
but I'd welcome others' views please.
How's the Alfa getting on without its fan?
I really missed that Alpha after they sent it to the
great-scrapyard-in-the-sky. At the time it had been a useful source of 12V
project motors and that nice multi-strand cabling that only seems to turn up
in car wiring looms.
For the project I wouldn't dream of using H bridges. Damned things are
always messy, ugly and require effort to make 'em work reliably .
Why the motor can't be reversed?. Surely just regard the Vcc and fet wires
like what they basically are, i.e. A pos and neg supply for the motor. Run
'em through the reversing relays and limit switches as normal. The wiring
runs will add a bit of resistance and inductance but at (say) audio chopping
frequencies, the motor won't remotely notice any difference.
regards
john
B
Bob Wilson
Jan 1, 1970
0
that, or a cheap single-chip microcontroller + MOSFET + capacitor/diode
snubber? =)
Al
Using a microcontroller to replace a simple quad-comparator-based PWM circuit
(such as the one mentioned) would seem to be a little silly. Not only is the uC
more expensive than the comparator, it also requires time spent on programming
(and debugging). The comparator-based PWM is so simple, that it would be up and
running before the code for a uP version wer even half finished.
Bob.
T
Terry Pinnell
Jan 1, 1970
0
Another potentially interesting link I found on the epanorama site was
Could someone give it a try and confirm that it's inaccessible please?
Googling on some of the key words from the epanorama outline also so
far unsuccessful.
"Motor controller uses fleapower - A simple, permanent-magnet dc motor
is an essential element in a variety of products, such as toys, servo
mechanisms, valve actuators, robots, and automotive electronics. In
many of these applications, the motor must rotate in a given direction
until the mechanism reaches the end of travel, at which point the
motor must automatically stop. This circuit implements The design is
optimized for a supply voltage of 3 to 9V, making it well-suited to
battery-powered applications."
J
John Woodgate
Jan 1, 1970
0
I read in sci.electronics.design that Terry Pinnell <[email protected]
Edial.pipex.com> wrote (in said:
Another potentially interesting link I found on the epanorama site was
Could someone give it a try and confirm that it's inaccessible please?
Remove '#dc' from the URL and instead scroll down the page you then get
'Circuits'.
A
Al Borowski
Jan 1, 1970
0
Using a microcontroller to replace a simple quad-comparator-based PWM circuit
(such as the one mentioned) would seem to be a little silly. Not only is the uC
more expensive than the comparator, it also requires time spent on programming
(and debugging). The comparator-based PWM is so simple, that it would be up and
running before the code for a uP version wer even half finished.
Want to bet on that?
I already have code for simple things like PWM - I'd use a cheap 8 pin
PIC with ADC ($2 AUD in single quantites). The only parts you'd need would be an 8 pin uC, a pot for PWM adjustment, and the FET and snubber. I reakon, if I had the parts on hand, I could get it going in 10 minutes on breadboard. No having to hunt down a quad comparitor or the needed resistor values - I just have a heap of cheap micro's for jellybeans when I do things like this. cheers, Al S Steve Taylor Jan 1, 1970 0 Al said: Want to bet on that? I already have code for simple things like PWM - I'd use a cheap 8 pin PIC with ADC ($2 AUD in single quantites). The only parts you'd need
would be an 8 pin uC, a pot for PWM adjustment, and the FET and snubber.
I reakon, if I had the parts on hand,
....and all the software written and debugged.....
;-)
Steve
A
Al Borowski
Jan 1, 1970
0
Steve said:
...and all the software written and debugged.....
;-)
/me points to where I wrote "I already have code for simple things like
PWM"
cheers,
Al
T
Terry Pinnell
Jan 1, 1970
0
John Woodgate said:
I read in sci.electronics.design that Terry Pinnell <[email protected]
Remove '#dc' from the URL and instead scroll down the page you then get
'Circuits'.
Thanks John (where've you been?), but that gives same result I got.
IOW, I end up on page
http://www.reed-electronics.com/ednmag/index.asp?layout=siteInfo&doc_id=30982
which says: "We are sorry, but the article you've requested cannot be
found. You may try searching for the title in our Article Search
above."
T
Terry Pinnell
Jan 1, 1970
0
John Jardine said:
I really missed that Alpha after they sent it to the
great-scrapyard-in-the-sky. At the time it had been a useful source of 12V
project motors and that nice multi-strand cabling that only seems to turn up
in car wiring looms.
For the project I wouldn't dream of using H bridges. Damned things are
always messy, ugly and require effort to make 'em work reliably .
Why the motor can't be reversed?. Surely just regard the Vcc and fet wires
like what they basically are, i.e. A pos and neg supply for the motor. Run
'em through the reversing relays and limit switches as normal. The wiring
runs will add a bit of resistance and inductance but at (say) audio chopping
frequencies, the motor won't remotely notice any difference.
regards
john
I used to *add* stuff to my cars, rather than take bits away <g>.
Started more than a couple of decades ago on my own cars, when even
basics like windscreen wiper speed control were not always provided.
Later, whenever I got a new company car, I'd have to strip out any
increasingly generous, there was less need to add my own projects
anyway. What car now doesn't have a burglar alarm, ice alert,
sidelights-on warning, etc? In fact, it was only as recently as last
week that I got around to adding my very first gadget to my '96 BMW
328SE. It's one of the few things apparently not covered in all modern
cars: what I call a 'Pedestrian Horn'. A footswitch just above my left
foot (the car's auto) initiates a rapid and *relatively* quiet
bip-bip-bip of the horn/siren, to warn little old ladies or horse
riders etc just around the next bend, or that apparently oblivious
driver or passenger emerging from a car ahead. Also has advantage that
I don't need to take hands from steering wheel. The challenge was
access to the horn wiring and getting it inside the car. (Just a low
duty cycle astable driving the existing horn relay.)
Back to curtain motor control. (Can't postpone the admission any
longer!) I'm embarrassed to say that you're quite right, of course.
Don't know what I was thinking. Clearly I should be able to use a
simple single MOSFET or BJT circuit for the PWM speed control, and my
relays/microswitches for F/Off/R control, as you say.
Your post couldn't have been more timely, as I was going to get stuck
in today and build the H-Bridge design I found yesterday in this
month's edition of EPE mag. That's actually a neat design (NPN & PNP
pairs), and incorporates all the facilities I want. But it would have
been time-consuming to biild and test. Thanks for the heads up.
C
cpemma
Jan 1, 1970
0
Paul said:
The main problem with using a 555 based design is going to be the
1/3-2/3 range of available supply volts. That may not be a problem in
your particular case but it's nice to have the full rail-rail o/put
this quad op-amp gives you, IMHO.
At lowish PWM frequencies (50-150Hz) a simple 555 + switcher transistor
circuit can give about 5%-95% duty cycle at almost full supply volts, but a
2-3 opamps will give 100% for little more in parts count or cost.
Some more circuits at http://www.cpemma.co.uk/pwm.html
J
John Woodgate
Jan 1, 1970
0
I read in sci.electronics.design that Terry Pinnell <[email protected]
Thanks John (where've you been?),
Very busy, and travelling abroad a lot.
but that gives same result I got.
IOW, I end up on page
http://www.reed-electronics.com/ednmag/index.asp?layout=siteInfo&doc_id=30982
which says: "We are sorry, but the article you've requested cannot be
found. You may try searching for the title in our Article Search
above."
I think you may be picking up a bad proxy. Even the URL with the #dc in-
page reference works for me. I've copied the text on DC motors and I can
send it to you as a Word document if you want it.
T
Terry Pinnell
Jan 1, 1970
0
John Woodgate said:
I read in sci.electronics.design that Terry Pinnell <[email protected]
Very busy, and travelling abroad a lot.
I think you may be picking up a bad proxy. Even the URL with the #dc in-
page reference works for me. I've copied the text on DC motors and I can
send it to you as a Word document if you want it.
http://www.reed-electronics.com/ednmag/index.asp?layout=siteInfo&doc_id=30982
yet you can. Must admit I don't really understand what 'picking up a
bad proxy' means. My personal hygiene is pretty good ...
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2022-09-30 03:09:52
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http://soft-matter.seas.harvard.edu/index.php?title=Geometrically_Mediated_Breakup_of_Drops_in_Microfluidic_Devices&oldid=5025
|
# Geometrically Mediated Breakup of Drops in Microfluidic Devices
Zach Wissner-Gross (February 9, 2009)
## Overview
Howard Stone and coworkers describe two techniques for breaking up droplets in an emulsion using microfluidic devices. The first method makes use of a "T-junction," in which a droplet flows down a channel that abruptly bifurcates into two channels in an orthogonal direction. The second method involves placing an obstacle in the droplet's path. Also notable is the article's analysis of the conditions under which a droplet encountering a T-junction will actually break up.
## Using T-junctions
The process of creating droplets in the first place was previously described by Steve Quake and coworkers [1], who mixed two immiscible liquids in a similar T-junction microfluidic circuit, made from PDMS using soft lithography [2]. Stone and coworkers used oil (hexadecane) and water to make the emulsion, including a surfactant to lower the surface tension between the two liquids in order to stabilize droplets.
Water droplets in the oil emulsion were flowed toward the T-junction, where they encountered a stagnation point at the intersection (Figure 1). Under certain conditions, the droplets would simply flow into one of the two side channels -- the authors mention that unbroken droplets in fact precisely alternate which channel they choose, although this phenomenon appears non-trivial to me. Perhaps when a given droplet selects one channel, it increases the fluidic resistance of that channel so that the next droplet favors the other channel. But more interestingly, under different conditions, oil flowing behind the droplet will pinch closed the surface, thus breaking the drop into two smaller droplets.
Figure 1: The top set of images (a-e) shows a droplet that will remain intact, while the bottom set (f-j) shows a droplet that will ultimately break up.
## When breakup occurs
The real physics of this paper occurs when the authors attempt to discern the conditions under which water droplets break up at the T-junction. Breakup takes place due to Rayleigh-Plateau instability [3], which predicts the breakup of a cylindrical column of liquid when its length exceeds its circumference. The authors define this geometric ratio as the extension $\epsilon_0$ of a droplet: $\epsilon_0=\frac{l_0}{\Pi w_0$, where $l_0$ and $w_0$ are the length of the drop and width of the channel that constrains it, respectively.
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2022-11-30 07:49:00
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https://documen.tv/question/write-a-system-of-equations-to-describe-the-situation-below-solve-using-substitution-and-fill-in-20692942-47/
|
Question
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. Craig’s Bakery recently spent a total of $311 on new equipment, and their average hourly operating costs are$9. Their average hourly receipts are $10. The bakery will soon make back the amount it invested in equipment. How many hours will that take? What would the total expenses and receipts both equal? in progress 0 5 months 2021-08-24T00:51:20+00:00 1 Answers 3 views 0 ## Answers ( ) 1. Answer: a. 311 hours b. i.$3,100 in receipts
ii. $2,799 in expenses. Step-by-step explanation: a. The company is making$10 per hour and spending $9 an hour. Their profit is therefore: = 10 – 9 =$1 per hour
Since they spent $311 on the new equipment, the amount of time it would take for them to make this back is: = 311 / 1 = 311 hours b. In 311 hours, the total receipts would be: = 311 * 10 =$3,110
The expenses would be:
= 311 * 9
= \$2,799
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2022-12-04 02:02:46
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https://www.rdocumentation.org/packages/raster/versions/3.0-12
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# raster v3.0-12
0
0th
Percentile
## Geographic Data Analysis and Modeling
Reading, writing, manipulating, analyzing and modeling of gridded spatial data. The package implements basic and high-level functions. Processing of very large files is supported. There is a also support for vector data operations such as intersections. See the manual and tutorials on <https://rspatial.org/> to get started.
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2020-08-14 19:24:52
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https://www.math.uni-potsdam.de/professuren/geometrie/publikationen/ansicht/manifolds-with-many-rarita-schwinger-fields
|
# Manifolds with many Rarita-Schwinger fields
#### Autoren: Christian Bär, Rafe Mazzeo (2021)
The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions.
In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kähler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.
Zeitschrift:
Commun. Math. Phys.
Verlag:
Springer-Verlag
zur Übersicht der Publikationen
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2023-03-30 12:19:42
|
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http://repository.bilkent.edu.tr/browse?type=subject&value=Dragilev%20space
|
Now showing items 1-2 of 2
• #### An application of linear topological invariants
(1997)
We consider a possible isomorphism of cartesian product of two Dragilev spaces of infinite type, and by making use of Zahariuta invariants and some structural properties, we show that if there is such an isomorphism, then ...
• #### Isomorphic classification problem and linear topological invariants
(Bilkent University, 1995)
We consider all possible isomorphisms of cartesian products of Dragilev spaces, and thanks to relations between the Dragilev functions of each factor try to show that if there exists such an isomorphism, then any factor ...
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2020-08-14 02:29:39
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https://www.hepdata.net/record/ins915980
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• Browse all
Measurement of the Cross Section for Prompt Isolated Diphoton Production in $p\bar{p}$ Collisions at $\sqrt{s} = 1.96$ TeV
The collaboration
Phys.Rev.Lett. 107 (2011) 102003, 2011
Abstract (data abstract)
Data tables can be found in the article PR D84, 052006 (2011); see https://hepdata.net/record/ins915978 .
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2019-08-25 20:23:22
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https://economics.stackexchange.com/questions/43548/why-in-the-quantitative-equation-mv-py-v-and-y-can-be-taken-as-fixed
|
# Why in the quantitative equation: $MV=PY$, $V$ and $Y$ can be taken as fixed?
To equation is
\begin{align} MV=PY \end{align}
where $$V=\frac{1}{k}$$.
Why $$V$$ and $$Y$$ can be taken as fixed or constant?
Why can $$V=\frac{1}{k}$$ too?
• Assuming that in some future period $Y$ and $V$ are fixed (or more precisely that they are independent of $M$ and $P$, as they are unlikely to be the same as earlier $Y$ and $V$) is an empirical assumption that should be tested (and would typically fail such a empirical test) – Henry Apr 23 at 8:10
There are multiple answers to this question.
1. In any model you can always make a thought experiment where you hold certain variables fixed. So one answer, although not very satisfying one, is that you can view it as a thought experiment. For example, in physics distance traveled equals velocity times time or $$D=tv$$ and you can always make a thought experiment where you choose 1 or 2 variables that will be fixed and see how others behave.
2. If you are asking why it would make an economic/intuitive sense to take $$Y$$, $$V$$ fixed then:
• Regarding $$Y$$: From economic theory we know that $$Y$$ in a long run depends on productive capacity of an economy that is independent of $$M$$, $$P$$ or $$V$$ (you can learn more about that in any 101 Macro/econ book such as Mankiw Macroeconomics or Mankiw Principles of Economics). So you can consider it to be exogenously given and holding it constant is just performing a thought experiment where you assume you have economy that is not growing so you can see how other variables behave.
• Regarding $$V$$, in the past (before 2008), empirically speaking velocity of money was very stable (see the Fed graph below). As you can see between 60s and late 2000s it was always hovering around 1.9 so on empirical grounds many economists considered $$V$$ stable, although it is worth while pointing that recently it significantly declined. Nonetheless, given that it was historically very stable this might be just transitional, or it is possible there was a structural break and now velocity will remain approximately constant at some lower level. It is also possible that it was wrong to assume it is constant in the long run based on previous empirical observations.
• I would suggest that, from your FRED chart, fairly stable $V$ stopped being a reasonable assumption after 1992 – Henry Apr 23 at 8:13
• @Henry $V$ certainly was certainly less stable after 1992, but if I would have to put one point after which this would stop being reasonable assumption I would put it in late 2000s where $V$ actually started having a negative trend – 1muflon1 Apr 23 at 8:30
• Thank you for your answer. – Verónica Rmz. Apr 23 at 17:44
• @VerónicaRmz. Well why not? 1. It is perfectly possible to have a country with zero economic growth- in fact in recent years you have some examples of countries with economic growth approximately 0. 2. Holding things constant is the most elementary science 101. For example, in physics if you want to understand how Newtonian gravitational force depends on distance, where gravitational force is given by F=M_1*M_2*G/r^2, you will hold masses M_2 and M_1 constant together with gravitational constant G and see how change in r affects force of gravity – 1muflon1 Apr 23 at 18:47
• It does not matter that mass of objects changes over time (for example our sun loses about 4.2*10^12 grams of mass every second), it’s one of the most fundamental ‘tricks’ of a scientist to preform thought experiments where some variables are held constant - in fact that principle also applies to not just thought but real experiments where precautions are made to hold as many things constant as possible – 1muflon1 Apr 23 at 18:51
To clarify the use of notation this reference shows the definition of slope change where delta notation is used to indicate a finite difference calculation:
https://www.mathsisfun.com/calculus/derivatives-introduction.html
As a thought experiment if one holds $$V = 1/k$$ and $$Y$$ constant then:
$$\frac{\Delta M}{\Delta t} = kY\frac{\Delta P}{\Delta t}$$
which means the finite change in the money supply $${\Delta}M$$ over a finite period $${\Delta}t$$ equals a constant $$kY$$ times the finite change in the price level $${\Delta}P$$ over the same finite period $${\Delta}t$$. If one could perform this experiment in reality it would tend to validate or invalidate the monetarist theory of inflation.
Edit: Below I write out the terms for finite difference calculations to check my logic.
$$M_{t} = k_{t}Y_{t}P_{t}$$
$$M_{t-1} = k_{t-1}Y_{t-1}P_{t-1}$$
$$k = k_{t} = k_{t-1}$$
$$Y = Y_{t} = Y_{t-1}$$
$$M_{t} - M_{t-1} = kYP_{t} - kYP_{t-1} = kY(P_{t} - P_{t-1})$$
$${\Delta}M = M_{t} - M_{t-1}$$
$${\Delta}P = P_{t} - P_{t-1}$$
$${\Delta}M = kY{\Delta}P$$
• Thanks for your answer. I'm not sure how it answers my question though. – Verónica Rmz. Apr 23 at 18:02
• There is no way to hold k and Y constant in the actual economy and therefore no definitive answer to your question. This is because real output Y (price deflated GDP) tends to grow historically over time but it also declines during so-called economic recessions. In practice economists measure YP or nominal output. Then they apply a price deflator to estimate real output Y. They measure the money supply as a statistical aggregate such as M1, M2, which are financial instruments in the banking sector. Then they find velocity as ratio V = YP/M. Monetarists argue change in M causes change in P. – SystemTheory Apr 23 at 18:16
• @1muflon1 The identity M = kYP is easily derived from the identity MV = YP and the identity V = 1/k which are given in the literature and not referenced in the question or in your answer. Substitute 1/k for V in equation MV = YP to write the equivalent identity M/k = YP. Multiply both sides by k to write an equivalent identity M = kYP. Then if we let M and P vary but hold kY constant from one accounting period to another the delta notation only applies for the change in M and the change in P but not to the constant term kY. – SystemTheory Apr 23 at 21:40
• @1muflon1 In the context of my answer ${\Delta}M$ is the change in the money supply stated in terms of money units {mu} and ${\Delta}P$ is the change in price level stated in terms of money units {mu}. This means the dimensions are the same for $M = kYP$ and for ${\Delta}M = kY{\Delta}P$. This is the delta notation one learns in Calculus. Perhaps for accuracy both sides of the equation should have ${\Delta}t$ in the denominator to recognize the change occurs over time from one period to another. Monetarists make a weak argument that changes in money cause changes in price level. – SystemTheory Apr 23 at 23:32
• If one comprehends the math of finite differences then this is exactly the model one gets when holding k and Y constant during a thought experiment. This model is the rational or reason for holding k and Y constant as a thought experiment. The fact that k and Y are only constant as a thought experiment is given in my answer, my comments, and in the other answer. – SystemTheory Apr 26 at 0:24
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2021-06-17 20:06:49
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https://hit.lbl.gov/previous-seminars
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# Previous HIT Seminars
### "Design Concept of Imaging Barrel Electromagnetic Calorimeter for the Electron-Ion Collider"
The Electron-Ion Collider (EIC) will be an experimental facility to explore the gluons in nucleons and nuclei, shedding light on their structure and the interactions within. Physics goals, detector requirements, and technologies at the EIC are outlined and discussed in the EIC community White Paper and Yellow Report. In particular, for the barrel electromagnetic calorimetry, the electron energy and shower profile measurements play a crucial role in the separation of electrons from background pions in deep inelastic scattering processes. Moreover, the calorimeter must measure the energy and position of photons, identify single photons originating from deeply virtual compton scattering process, and photon pairs from pi^0 decays. Based on detector requirements, we propose a design of the imaging barrel electromagnetic calorimeter. It is a hybrid design utilizing imaging calorimetry based on monolithic silicon sensors (AstroPix) and scintillating fibers embedded in Pb. We have studied the proposed calorimeter in detail through realistic simulations to test it against the requirements for the physics case described in the EIC community Yellow Report. In this talk, I will present the expected calorimeter performance based on simulations with 3T magnetic field and the outlook of the upcoming R&D program related to the imaging calorimetry will be also presented.
### "Conformal Colliders Meet the LHC"
Jets of hadrons produced at high-energy colliders provide experimental access to the dynamics of asymptotically free quarks and gluons and their confinement into hadrons. Motivated by recent developments in conformal field theory, we propose a reformulation of jet substructure as the study of correlation functions of a specific class of light-ray operators and their associated operator product expansion (OPE). We show that multi-point correlation functions of these operators can be measured in real LHC data, allowing us to experimentally verify properties of the light-ray OPE. We then discuss how this reformulation provides new ways of experimentally studying QCD at colliders, as well as new theoretical techniques for performing previously intractable calculations.
### "Exclusive hard processes for extracting Generalized Parton Distributions"
The Generalized Parton Distributions (GPDs) describe the distribution of quarks/gluons inside a colliding hadron in both longitudinal partonic momentum fraction $x$ and transverse space (“tomographic images”). Their moments of the momentum fraction $x$ provide critical information on partonic angular momentum contribution to the hadron spin and gravitational form factors of the hadron. Extracting GPDs reliably from experimental measurements has been a major science goal for studying QCD and hadron physics. However, conventional exclusive processes such as deeply-virtual Compton scattering (DVCS) cannot fully disentangle the longitudinal dependence and provide the tomographic images at fixed x. In this talk, I will introduce a class of exclusive hard processes, to be referred as single diffractive hard exclusive processes (SDHEP), for the extraction of GPDs. I will discuss the necessary and sufficient conditions for SDHEP to be factorized in terms of GPDs. I will also demonstrate that SDHEP is not only sufficiently generic to cover all known processes for extracting GPDs, but also well-motivated for the search of new processes for the study of GPDs. Finally, I will carefully examine the sensitivity of SDHEP to the parton momentum fraction $x$ dependence of GPDs.
### "Physics Program with SoLID"
The Solenoidal Large Intensity Device (SoLID) is a new experimental apparatus proposed for Hall A at the Thomas Jefferson National Accelerator Facility (JLab). SoLID will combine large angular and momentum acceptance with the capability to handle high data rates at large luminosity. As such SoLID will push JLab to a new limit at the QCD intensity frontier to exploit the full scientific potential of the 12-GeV CEBAF. The slate of approved high-impact experiments consists of the tomography of the nucleon in 3-D momentum space from Semi-Inclusive Deep Inelastic Scattering, expanding the phase space in the search for new physics and novel hadronic effects from parity-violating Deep Inelastic Scattering, a precision measurement of near-threshold J/psi production to probe the gluon field and its contribution to the proton mass, and more. In this talk, I will discuss the rich SoLID physics program and the proposed apparatus.
### "The equation of state of dense nuclear matter from heavy-ion collisions"
The equation of state (EOS) of dense nuclear matter can give insights into fundamental properties of QCD: among others, it can elucidate phases of matter in different regions of the QCD phase diagram, it can help explain the structure of both neutron stars and neutron-rich nuclei, and it can inform us about the nature of the underlying fundamental interactions. In experiments colliding heavy nuclei at relativistic velocities, the EOS is probed through observables describing the collective behavior of the produced matter, such as flow observables, the dependence of which on the EOS can be studied in simulations. In this talk, I will discuss recent progress on constraining the EOS with hadronic transport simulations, used to model heavy-ion collision experiments at intermediate energies. I will highlight results from a recent Bayesian analysis study using new data from RHIC, and I will discuss the findings in the context of other known constraints from heavy-ion collisions and neutron star studies. I will also outline developments in state-of-the-art hadronic transport codes necessary to fully utilize the potential of the forthcoming wealth of data from the BES-II FXT program at RHIC, from GSI and FAIR, as well as from FRIB and future FRIB400.
### "Overcoming exponential volume scaling in quantum simulations of lattice gauge theories"
Quantum computers hold the promise of overcoming the numerical sign problem and allowing for study of the dynamics of quantum field theories from first principles. Before performing such calculations, it is important to ensure that the quantum algorithms used do not have a cost that scales exponentially with the volume. In this talk, I will discuss an interesting test case: a formulation of compact U(1) gauge theory in 2+1 dimensions free of gauge redundancies. A naive implementation onto a quantum circuit has a gate count that scales exponentially with the volume. I will discuss how to break this exponential scaling by performing an operator redefinition that reduces the non-locality of the Hamiltonian. With the exponential volume scaling broken, I will then show how the gate count scaling with the volume can be further reduced by approximating the quantum circuit without introducing large errors.
### New Special time: Tuesday, February 14, 2023, 9:30 AM PST (Zoom)
Dr. Yousen Zhang (Rice University)
Host: Peter Jacobs and Farid Salazar
Slides and Recording available upon request to organizers
"Probing QCD matters using heavy flavor quarks at the LHC"
Quark gluon plasma created in relativistic heavy ion collisions is a novel state in which partons are deconfined from normal matter in the universe. It is characterized by the shocking collectivity of QGP and energy loss of high energy particles traversing through QGP. Recent measurements show that the collectivity can also emerge in high-multiplicity proton-proton and proton-nucleus collisions, which was originally believed to only exist in large ion collisions. However, the origin of these collective motions is still a puzzle in theoretical studies mainly debating on contributions from initial correlations and in-medium effects. Heavy flavor quarks are sensitive to both the initial stage conditions and the later-on in-medium effects of collisions, thus can provide important information for understanding the inner workings of QGP in large ion collisions and the origin of the collectivity in small systems. In this talk, I will present the recent progress of heavy flavor collectivity at the LHC from large to small colliding systems, and discuss the future opportunities with the high-luminosity LHC together with the CMS detector upgrades.
### Special time: Thursday, February 9, 2023, 2:00 PM PST (Zoom)
Dr. Nicole Lewis (Brookhaven)
Host: Peter Jacobs and Shujie Li
Slides
"Baryon Stopping in Photonuclear Collisions"
Photonuclear collisions are one of the simplest processes that can happen in a heavy-ion collision. They occur when one nucleus emits a quasi-real photon which interacts with the other colliding nucleus, similar to an electron-ion collision except that the photon tends to have a much smaller virtuality. Photonuclear collisions can be used to study bulk properties of the medium such as collectivity due to initial-state effects and hadron chemistry. In these photonuclear collisions we observed baryon stopping: more baryons that antibaryons even at midrapidity. This phenomenon is well documented in proton-proton and heavy-ion collisions, but it is not well understood and had never before been seen in photonuclear collisions. This could indicate the existence of a baryon junction within the nucleon, a nonperturbative Y-shaped configuration of gluons which carries the baryon number and is attached to all three valence quarks. These measurements will also inform future measurements using particle identification at the upcoming Electron Ion Collider.
### Tuesday, February 7th 2023, 3:30 PM PST (Zoom)
Dr. Xiaoxuan Chu (Brookhaven National Lab)
Host: Farid Salazar
"Probing nonlinear gluon dynamics at RHIC and the EIC"
The gluon distribution function grows with lower and lower momentum fraction very fast. As the total scattering cross section is bound by quantum mechanics, the raise of the gluon density has to be tamed, which is explained by gluon recombination under the color glass condensate (CGC) framework. A definitive discovery of nonlinear effects in QCD and as such the saturation regime would significantly improve our understanding of the nucleon structure and of nuclear interactions at high energy. Two particle azimuthal correlation is one of the most direct and sensitive channels to access the underlying nonlinear gluon dynamics. In this talk, we will present the recent results of forward di-hadron correlations measured at RHIC, together with the signatures of gluon saturation predicted by CGC. New opportunities for measurements with the STAR forward upgrade and future EIC to study the nonlinear effects in QCD will also be discussed.
### Special date & time: Thursday, February 2, 2023, 10:30 AM PST (Zoom)
Florian Jonas (University of Münster/ORNL)
Host: Peter Jacobs and Farid Salazar
Slides and Recording available upon request to organizers
"Prompt photon production: probing gluons in protons and nuclei at hadron colliders"
Measurements of prompt photons at hadron colliders offer unique insights into the substructure of the colliding projectiles, enabling constraints of so called Parton Distribution Functions (PDFs), which encode the inner structure of the hadron in a universal parametrization.
In particular, prompt photons are sensitive to the gluon distributions, which are one of the least constrained PDFs and not directly accessible via deep inelastic scattering.
Furthermore, studies of the gluon bound inside protons (and nuclei) are of key importance for the exploration of non-linear QCD, where in particular their low-x growth is expected to be tamed by gluon saturation.
Photons do not interact via the strong interaction, which makes them a particularly robust probe, carrying information from the initial hard scattering to the detector, unaffected by any final state effects commonly encountered in large collision system.
This talk outlines the key experimental techniques and challenges arising in isolated prompt photon measurements. An overview of the currently available experimental results at the LHC and their impact on the gluon PDF is followed by a look into the future, outlining how prompt photon measurements at hadron colliders with the ALICE Forward Calorimeter can play an important role in the search for gluon saturation and complement DIS measurements at the EIC.
### Wednesday, January 18th, 2023, 3:30 PM PST
Dr. Xiaohui Liu (Beijing)
Host: Jennifer Rittenhouse West
"Nucleon Energy Correlators for the Color Glass Condensate"
An invited talk and discussion on very recent work by Xiaohui Liu, Feng Yuan and collaborators on low-x physics, specifically a new method to access gluon saturation which is highly relevant for the future Electron-Ion Collider. All welcome, and please note this will be a pedagogical seminar with a combination of slides and whiteboard, particularly aimed toward early career people who might not be familiar with the physics.
### Tuesday, January 17th, 2023, 3:30 PM PST
Host: Jennifer Rittenhouse West
"Spatial distributions of energy and stresses in hadrons"
The Lorentz group of special relativity contains a Galilean subgroup, which manifests in the light front picture of spacetime. Light front coordinates thus allow a fully relativistic spatial description of the internal structure and dynamics of hadrons, specifically in the form of two-dimensional densities on the plane transverse to the observer's line of sight. The energy-momentum tensor allows distributions of energy, momentum, spin and stresses to be obtained. In this talk, I elaborate on the physical meaning of light front coordinates, and explain the formalism for determining internal distributions of hadrons. Concrete model examples for the internal structure of mesons are provided.
### Tuesday, January 10th, 2023, 3:30 PM PST (in-person & zoom)
Dr. Joao Barata (BNL)
Host: Farid Salazar/Xin-Nian Wang
Jet evolution in anisotropic plasmas
Over the last decades, the heavy-ion collisions at RHIC and the LHC allowed the exploration of QCD at high energies and densities. In these experiments, the nuclear matter is produced far from equilibrium, and undergoes a multiphase evolution until it thermalizes into a nearly ideal liquid: the quark-gluon plasma (QGP). The main effort of the heavy-ions community is focused in extracting the detailed properties of this phase of matter. Among the many probes used for this effect, QCD jets have proven to be able to explore the quark gluon plasma at different time and length scales, providing a differential and dynamical picture for the underlying matter. Nonetheless, a theoretical treatment of jet structure which is sensitive to the details of the QGP is still far from complete. In this talk, I will present some recent work on how to describe jet evolution in the presence of anisotropic matter. First, I will detail the basic modifications necessary to make to the standard jet quenching formalism. Afterwards, I will show that the QGP anisotropies can be taken into account to arbitrary order in a gradient expansion using an effective kinetic theory approach. Remarkably, at second order in the matter gradients, we find a novel master equation which generalizes the usual Boltzmann transport. Finally, I will discuss how such effects can be manifest at the level of jet observables.
### "Better parton showers for HL-LHC and beyond"
The Large Hadron Collider is currently running at its highest energy and luminosity. In order to maximally exploit the potential of this precise dataset to uncover physics beyond the Standard Model, it is of crucial importance to develop tools that faithfully characterise the QCD background. Parton showers lie at the core of general purpose Monte Carlo event generators. They aim at correctly describing the phase-space for QCD branchings across disparate energy scales. A natural question, largely overlooked in the literature, is up to which degree of logarithmic accuracy do parton showers meet this goal. In this talk, I will introduce recent efforts by the PanScales collaboration to establish the criteria that a parton shower should satisfy in order to reach a given logarithmic accuracy. Then, I will present the design and implementation of next-to-leading logarithmic accurate parton showers both in electron-positron annihilation and hadron collisions. I will conclude with some exploratory studies about the phenomenological impact of these highly-precise showers.
### "Achieving High Deuteron Tensor Polarization To Probe Nuclear Structure"
Advances in experimental technology have allowed for new high-luminosity scattering probes on highly spin polarized nuclear targets. These developments have made it possible to probe the structure of tensor-polarized deuterium with greater precision than ever before. Two experiments, approved to run in Jefferson Lab's experimental hall C will measure two deuteron tensor observables. The first experiment will probe the deuteron tensor structure function b1 in the x<1 deep inelastic region, and the second experiment will measure the deuteron tensor asymmetry Azz in the quasielastic region. In order to achieve a high tensor polarized deuteron target, developments are being made in nuclear polarization technology using the dynamic nuclear polarization (DNP) technique. DNP is used to enhance the nuclear spin polarization of materials. DNP works by using microwaves to continuously drive spin transitions in a material that is doped with free radicals and placed inside a 1 K environment in a high magnetic field. Further tensor polarization enhancement is achieved using additional RF saturation on the target magterial. Once enhanced, the nuclear polarization can be determined by analyzing the lineshape of the NMR absorption spectrum. This talk will describe the DNP system used at the University of New Hampshire, and explain novel techniques in inducing high tensor polarization in deuterium.
### "Physics Program and Detector Development with ALICE3"
ALICE 3 constitutes the next-generation upgrade for heavy-ion physics in LHC Runs 5 and 6. It addresses the questions about the quark-gluon plasma which remain inaccessible to other existing or planned experiments, e.g. on the transport properties and thermalisation in the QGP, the formation of hadrons, and the early stages of the plasma evolution. To this end, precise measurements of heavy-flavour probes as well as of electromagnetic radiation are key. The required pointing and tracking performance are achieved with a high-resolution vertex tracker, installed within the beam pipe, and a large-acceptance tracker, both based on monolithic silicon-pixel sensors. The particle identification relies on a combination of a silicon-based time-of-flight detector and a Ring-Imaging Cherenkov detector. In addition, an electromagnetic calorimeter, a muon identifier, and a dedicated forward detector for ultra-soft photons are foreseen. In this presentation, we will explain the detector concept and its physics reach as well as discuss the innovative R&D activities in areas of interest for HEP experiments in general.
### "Evidence for Intrinsic Charm Quarks in the Proton"
It has been argued for a long time that the proton could have a sizable intrinsic component of the charm quark.
I will discuss how to disentangle the intrinsic charm component from charm-anticharm pairs arising from high-energy radiation and present results providing evidence for intrinsic charm, by exploiting the NNPDF4.0 high-precision determination of proton Parton Distribution Functions. The existence of intrinsic charm is estabilished at the 3σ level, with a momentum distribution in remarkable agreement with model predictions. Finally I will discuss how these findings are supported by the very recent data on Z-boson production with charm jets from the LHCb experiment.
### Host: Nu Xu
"Predicting randomness in relativistic hydrodynamics"
We usually think of hydrodynamics as a deterministic description of fluid motion. The focus of this talk is on random fluctuations in fluids caused by thermal noise, inherent in systems with dissipation. The interest in this subject is driven by the progress of heavy-ion collision experiments towards the mapping of the QCD phase diagram. In particular, the search for the QCD critical point at RHIC requires quantitative understanding of fluctuations and their dynamics. I will discuss the role of the fluctuations in hydrodynamics and how we can predict their evolution using deterministic equations.
Photo credit: Jessica Rotkiewicz
### Tuesday, November 8th, 2022, 3:30 PM PDT (Zoom only)
Dr. Björn Schenke (BNL)
Host: Farid Salazar
"QCD at the Crossroads: Hot QCD Town Hall meeting summary"
In this talk, I will try to summarize the discussions/presentations at the QCD town meeting, Boston, September 23-25: https://indico.mit.edu/event/538/
### In-person (Theory Lounge 70-228) & Zoom
Dr. Feng Yuan (LBNL)
Host: Nu Xu/Farid Salazar
"QCD at the Crossroads: Cold QCD Town Hall meeting summary"
In this talk, I will try to summarize the discussions/presentations at the QCD town meeting, Boston, September 23-25: https://indico.mit.edu/event/538/
### In-person (Theory Lounge 70-228) & Zoom
Dr. Tyler Hague (LBNL)
Host: Shujie Li/Jennifer Rittenhouse West
"The EMC Effect: Light Nuclei are Weird"
In 1983, the European Muon Collaboration (EMC) recorded Deep-Inelastic Scattering data on Iron, Deuterium, and Hydrogen. Comparisons of the targets noted that the quark structure of nucleons is modified by the surrounding nuclear environment, dubbed the EMC Effect. Nearly 40 years later, the community is still working to understand the mechanism behind this. Initial studies of the EMC Effect on heavy nuclei were able to paint broad strokes pictures of how the effect scales across nuclei. However, later precision tests of light nuclei were found to be in tension with these previous understandings. Key missing pieces to understanding this are the free neutron structure function and nuclear effects. The MARATHON experiment used a novel technique to extract F2n/F2p by largely canceling nuclear effects in the A=3 mirror nuclei in the yield ratio. In this talk, I will give an overview of our current knowledge of the EMC Effect followed by a discussion of the MARATHON experimental results and recent work applying these to light nuclei to understand nuclear effects.
### Mr. Henry Klest (Stony Brook University) Host: Nu Xu
Teaching an Old Experiment New Tricks: Measuring Modern Jet Observables Using Archived H1 e+p DIS Data
A renaissance in jet physics is now underway, with novel, theoretically rigorous approaches to jet measurements being used to probe QCD in new ways using data from hadronic collisions at RHIC and the LHC. It is timely to explore these new techniques using data from high-energy collider experiments, which were recorded before these techniques were developed. In particular, colliders with leptons in the initial state provide a clean and complementary environment to study jets. The H1 experiment was a hermetic, general-purpose detector at the HERA electron-proton collider complex at DESY, which terminated data-taking in 2007. However, the H1 Collaboration continues to be active scientifically, preserving the data and updating analysis software. The H1 dataset therefore remains as accessible for high-quality analyses as that for currently running experiments, making it the ideal playground for testing modern QCD analysis approaches on DIS data before the EIC turns on. This talk will focus on the measurement of groomed event shapes in ep DIS collisions, using a set of novel inclusive observables which leverage recent advances such as the Centauro jet algorithm and precision jet grooming phenomenology. The advantages and challenges of the event-wide grooming technique will be discussed, as well as some lessons learned that are applicable to the EIC physics program.
### Host: Nu Xu
Quarkonium Production in Heavy-ion Collisions at the LHC
Measurements of J/ψ production have been a valuable probe to study the properties of the hot and dense medium created in heavy-ion collisions, also known as the quark-gluon plasma (QGP). Experimentally, the first glimpse of the QGP was provided by collisions of lead beams at the Super Proton Synchrotron (SPS) in the late 90s via the measurements of J/ψ production. The suppression of J/ψ production in Pb–Pb collisions with respect to p–A collisions was observed, which was proposed as a smoking gun of the QGP as a consequence of colour screening in the QGP medium keeping charmed quark-antiquark pairs from binding to each other.
The higher collision energy at the Large Hadron Collider (LHC) with respect to previous collision programs at SPS and RHIC opened new perspectives on quarkonium measurements due to the increased heavy quark production cross section. The large data samples collected at the LHC based on the excellent performance of the LHC and the four experiments provided precise measurements of charmonium and bottomonium production in a wide kinematic range. Moreover, a multitude of new observables including multi-differential measurements of various quarkonium states becomes accessible, which provided crucial inputs to disentangle various quarkonium in-medium effects in the presence of the QGP.
In this talk, a selection of recent quarkonium measurements at the LHC will be discussed.
### Tuesday, October 11th, 2022, 3:30 PM PDT (Zoom only)
Dr. Niklas Mueller (U. Washington)
Host: Farid Salazar
What can QIS do for high energy and nuclear physics?
Slides
The possibility to simulate quantum many-body systems with digital quantum computers and analog devices is an exciting opportunity for high energy and nuclear physics. I will present an overview over new directions in two old examples: understanding systems far-from-equilibrium, such as QCD in ultra-relativistic heavy ion collisions, and their approach to thermal equilibrium, and addressing thermal systems in regimes where Monte-Carlo importance sampling techniques face a sign problem. For those of you impatiently waiting for quantum computers to outrun classical computers, I will emphasize that QIS not only allows to make progress computationally, but more importantly conceptually, including previously unexplored aspects such as, e.g. entanglement. I will present a few relevant examples, still far from the ultimate goal QCD, but interesting because of their interdisciplinary relevance for high energy and nuclear physics, condensed matter theory and quantum information science.
### Tuesday, October 4th, 2022, 3:30 PM PDT (Zoom only)
Dr. Yoshitaka Hatta (BNL)
Host: Farid Salazar
Spin Physics Opportunities at the EIC
I give an overview of high energy QCD spin physics with particular emphasis on physics opportunities at the future Electron-Ion Collider. Topics will include longitudinal spin decomposition, orbital angular momentum, spin at small-x, Ji sum rule and the generalized parton distributions, and transverse single spin asymmetries. I review the recent developments in the field and identify what are the outstanding challenges at the EIC.
### Thursday, 08.11.2022, 2:30 PM PDT Special date and time! in the Theory Lounge (hybrid)
Dr. Raymond Ehlers (LBNL)
What can we learn about the quark-gluon plasma from reconstructed jets with Bayesian inference?
Bayesian inference provides an approach for constraining the parameters of complex models according to the available information. In particular, the wealth of information contained within multi-messenger experimental measurements provides the opportunity for detailed data-model comparison. The need for such rigorous comparison is pervasive throughout science. The JETSCAPE collaboration has now applied this Bayesian approach to the familiar problem of jet quenching. In this talk, I will discuss our recent results, in-progress analysis, and implications for the future of the field.
### Wednesday, 06.29.2022, 3:30 PM PDT
Joint Nuclear Theory/HIT seminar! Special date on Wednesday!
Dr. Jennifer Rittenhouse West (LBNL)
Host: Xin-Nian Wang
Diquark Formation as a Breakthrough of Fundamental QCD into Nuclear Physics
A diquark bond formed from valence quarks across a nucleon-nucleon pair has been proposed as the fundamental quantum chromodynamics (QCD) physics causing short-range correlations (SRC) in nuclei. Short-range correlated nucleon-nucleon pairs and the nucleon shell model are the basis for nuclear physics, with SRC accounting for 20% of the nucleons in a nucleus. While SRC have been extensively studied both experimentally and theoretically, notably by the CLAS collaboration in recent years, their underlying cause at the QCD level has remained a mystery. The diquark formation model, if shown to be the cause of SRC in nuclei, represents a breakdown of the assumption of scale separation in effective field theories. Rather than a boundary between scales, however fuzzy and broad, a case is made in this work for diquark formation as a direct breakthrough of the underlying theory, capturing 20% of the physics of nuclear structure.
### Tuesday, 06.21.2022, 3:30 PM PDT
Dr. Aihong Tang (BNL)
Host: Xin-Nian Wang
Global Spin Alignment in Relativistic Heavy Ion Collisions : A Progress Review
In relativistic heavy ion collisions, quarks can possess global spin polarization in a globally vortical system. Such process is initially induced by the spin-orbital coupling, and the evolution of polarized quarks and the subsequent formation of hadrons involves various interesting physics mechanisms. This phenomena can be studied either by global spin polarization of hyperons or global spin alignment of vector mesons. Recently the STAR collaboration released interesting results of global spin alignment for phi- and K*-mesons. It is found that the surprisingly large value of phi-meson global spin alignment cannot be explained by conventional mechanisms, but can be accommodated by a model invoking the strong force field. In this talk we will review the recent progress in the understanding of global spin alignment, and in particular we will discuss STAR's result and its implications.
### Monday, 06.13.2022, 3:00 PM PDT (Hybrid at Persevervance Hall, note special time and date!)
Prof. Ulrich Mosel (Giessen University)
Host: Jennifer Rittenhouse West
Neutrino-nucleus interactions in quantum-kinetic transport theory
The analysis of results from long-baseline experiments such as T2K, NOvA and DUNE requires knowledge of the incoming neutrino energy. The latter has to be reconstructed from only partially measured final states of the reaction. Any theory thus has to go beyond the calculation of inclusive cross sections and has to deliver the full final state of the reaction. Since all present experiments work with nuclear targets not only the initial, first interaction of the incoming neutrino plays a role. In addition, the final state interactions of the initially produced hadrons with the nuclear environment are important and determine the final state. The state-of-the-art method to treat such processes is quantum-kinetic transport theory which has been around since 50 years, but only over the last 20 years major numerical implementations to describe nuclear reactions have become available. In my talk I will illustrate some of the results obtained with such a theory, applied both to nuclear reactions and neutrino-nucleus reactions.
### Tuesday, 06.07.2022, 12 PM PDT - Special time!
Dr. Sergey Kulagin (Institute of Nuclear Research, RAS Moscow)
Host: Jennifer Rittenhouse West
What can we say about modification of the bound nucleon at the parton level from global QCD fits?
We briefly review available experimental observations on nuclear effects in deep-inelastic scattering (DIS). We then briefly discuss a few basic mechanisms responsible for nuclear corrections and review progress in understanding the observed phenomenon focusing on the valence quark region. We report the results of our global QCD analysis which includes a "standard" set of high-energy data for the proton target (DIS, DY production of lepton pair as well as W+- / Z boson production) and also nuclear 2H, 3H, and 3He DIS data. In this analysis we treat nuclear corrections in DIS in terms of a nuclear convolution approach with off-shell bound nucleon. The off-shell correction describes the modification of parton distributions in bound nucleons, which is determined along with the parton distribution functions (PDFs). A number of systematic studies have been performed aiming to estimate the uncertainties arising from the use of various deuterium data sets, from the model of high twist contributions to the structure functions, from the treatment of target mass corrections. We compare our predictions for the ratio F2n/F2p and the d/u ratio of the quark distributions with the results of other analyses as well as with the recent data from the MARATHON experiment.
### Tuesday, 05.31.2022, 3:30 PM PDT
Dr. Christopher McGinn (Univ of Colorado Boulder)
Inclusive and Electroweak Boson-tagged Jets as Probes of the Quark-Gluon Plasma and Medium Response
Jets, as proxy for hard scattered partons in initial collision of heavy and ultrarelativistic nuclei, are modified significantly relative their vacuum reference counterparts when traversing the subsequently formed hot-and-dense medium of deconfined quarks and gluons known as the Quark-Gluon Plasma (QGP). Specifically, a suppression in the overall production of jets is observed compared to vacuum expectation, as well as a modification to jet fragmentation patterns towards softer fragments. These phenomena are known collectively as 'jet-quenching'. Recently, much attention has been paid to the impact of jet-medium interactions on the medium itself, in searches for medium response and in searches for quenching in small systems (such as proton-nucleus collisions) in the context of observed high-pT v2. Using data taken with the ATLAS detector at the LHC, sqrt(sNN) = 5.02 TeV, jets produced with electroweak boson partners (unmodified by strong-force interactions in QGP) are studied to characterize both the jet production and fragmentation, with the latter providing insight into the necessity of incorporating medium response in theoretical model comparisons. Additionally, simultaneous studies of jet v2 and quenching in big-and-small systems reveals there may be more questions on the exact nature of the jet-medium interactions in both systems and how they lead to the physical final state observed.
### Tuesday, 05/24/2022,3:30 PM PDT
Prof. Mike Lisa (Ohio State University)
Host: Xin-Nian Wang
Subatomic Smoke Rings: Polarization and Toroidal Vorticity in the QGP
Since the discovery of global hyperon polarization in Au+Au collisions at RHIC about five years ago, there has been intense theoretical and experimental focus on the topic. After a brief review, I will discuss novel vortex structures that may be generated in two situations. The first is a p+A collision, which may produce droplets of QGP that develop toroidal vortex ("smoke ring") structure. Experimental observation of such a structure would provide compelling evidence supporting the hydrodynamic nature of this tiny system, a much-debated topic today. The other is an idealized "hot moving spot" that may result from thermalization of a jet in an expanding QGP; in this case, the "smoke ring" centers on the jet direction. In both cases, we suggest an experimental observable to measure the toroidal vortex structure, and present full hydrodynamical simulations to make quantitative predictions, including initial state fluctuation effects. I will mention prospects and challenges for observing this novel phenomenon in experiment.
### Tuesday, 05.17.2022, 3:30 PM PDT
Prof. Jorge Noronha (UIUC)
Host: Xin-Nian Wang
Hydrodynamic Frames: the Good, the Bad, and the Ugly
Three of the most cutting-edge experiments in modern science, RHIC, LHC, and LIGO are now producing data whose description requires a major overhaul of our current understanding of what constitutes fluid dynamic behavior in the relativistic regime. In this talk I will explain how the choice of hydrodynamic variables in a system out of equilibrium, i.e., our definition of the so-called hydrodynamic frame, affects the domain of applicability of relativistic viscous fluid dynamics formulations. I will also show how developments in relativistic viscous hydrodynamics obtained in heavy-ion collisions could be instrumental in determining the viscous properties of ultradense matter formed in neutron star mergers.
### Tuesday, 05.10.2022, 3:30 PM PDT
Prof. Kenji Fukushima (Univ. of Tokyo)
Host: Feng Yuan
Continuity or Discontinuity between Nuclear and Quark Matter and the Astrophysical Implications
In this talk I will review theoretical scenarios of quark matter at high density. Although a phase transition is not logically excluded, a crossover or weak first-order transition is likely to occur. I will explain how the equation of state is constrained by the neutron star observation and demonstrate that the future gravitational wave detection can identify the nature of the quatk matter onset.
### Wednesday, 05.04.2022, 3:30 PM PDT (Special date)
(Hybrid talk in the INPA room)
Prof. Carl Schroeder (LBNL/UCB)
Host: Peter Jacobs
Laser-Plasma Accelerators
Slides
Laser-driven plasma-based accelerators (LPAs) are able to sustain extremely large accelerating gradients, orders of magnitude larger than those achievable using conventional metallic accelerating structures. LPA experiments have demonstrated the generation of multi-GeV electron beams in cm-scale plasmas. These high gradients have attracted interest in LPA technology for high-energy collider applications. In this talk, I will review the basic physics of LPAs, LPA research in the BELLA Center at LBNL, and how LPAs can be developed for various applications. Employing LPAs as a linac for an e+e- collider is the most challenging application, and I will discuss the design considerations and expected performance for an LPA-based linear collider. .
### (This will be a hybrid seminar in theTheory Lounge)
Dr. Jan Steinheimer (FIAS)
Hyper-nuclei production in heavy-on collisions
Hypernuclei have been an interesting topic of study for more than half a century. Yet their properties are still not fully understood. In this talk, I will give a short introduction to hypernuclei and how they can be produced. Here, the production of nuclei with one or more units of strangeness from relativistic heavy ion collisions is of particular interest, due to the abundance of strange baryons created. While the absorption of hyperons offers a chance to create large hypercluster in the spectator fragments of nuclear collisions, also the observation of light hypernuclei from the central hot region is of interest. I will discuss how the production probability can be well described by a coalescence mechanism using the UrQMD model over a beam energy range of three orders of magnitude. In addition I will show how the measurement of different nuclei can be used to determine the size of the region of homogeneity from which baryons, that will eventually form (hyper-)nuclei, are emitted . Within this context I will discuss recent interpretations of the centrality dependence of (hyper-)nuclei production.
### Tuesday, 04.19.2022, 3:30 PM PDT
Dr. Brandon Kriesten (Center for Nuclear Femtography)
Host: Jennifer Rittenhouse West
Quark and Gluon Spatial Distributions in the Nucleon
Studying the role of gluonic observables in exclusive scattering processes is essential as new physics programs, such as an electron ion collider, are planned in unprecedented kinematic regimes. I will present a parameterization of quark and gluon generalized parton distributions (GPDs) calculated using a reggeized spectator model. This parameterization is constrained using a combination of lattice QCD form factor calculations and extracted deep inelastic parton distributions. We evolve our parameterization at leading order in Q2 to the scale of experimental data using a perturbative QCD evolution framework. We demonstrate expected spatial distributions under Fourier transformation using our parametrization. Understanding the behavior of gluon GPDs is a first step towards extracting the gluon contribution to deeply virtual Compton scattering (DVCS) and timelike Compton scattering (TCS) observables at the EIC.
### Tuesday, 04.12.2022, 3:30 PM PDT
Dr. Gunther M Roland (MIT)
Hot QCD with sPHENIX at RHIC
Over the last decades heavy-ion experiments at RHIC and LHC have demonstrated a range of novel QCD phenomena that emerge under conditions of extreme pressure and temperature. New efforts, sPHENIX at RHIC and the upgraded LHC experiments, will begin collecting high precision new data in the next year. These data will allow us to investigate the microscopic origins of observed phenomena in the produced Quark-Gluon Plasma (QGP). Of particular importance will be the complementarity of experiments in the two energy regimes, elucidating the temperature dependence of QGP properties. In this talk I will present the sPHENIX design, construction status and expected performance, and discuss case studies that illustrate key aspects of the sPHENIX physics program.
### Tuesday, 03.29.2022, 3:30 PM PST
Prof. MaElena Tejeda Yeomans (Universidad de Colima)
Host: Jennifer Rittenhouse West
Rise and Fall of Lambda and anti-Lambda Polarization from the Core-Corona Model
The polarization of particles produced in heavy-ion collisions provides the perfect arena to study amazing phenomena such as the collective rotation of the nuclear medium, the transference of global angular momenta to spin properties of certain particles, and the evolution of these properties when there are drastic changes in the properties of the hot and dense medium. Recently, measurements by STAR collaboration at RHIC and HADES collaboration at GSI show the rising of Lambda and anti-Lambda global polarization with decreasing collision energy. Many models predict the vanishing of global polarization due to the lack of system angular momentum, but the height and location of the expected peaks for both Lambda and anti-Lambda are still not well understood. In this talk I will report on recent work to study Lambda and anti-Lambda global polarization in heavy-ion collisions using the core-corona model, where the source of these hyperons is a high-density core and a less dense corona. I will show that the overall properties of the polarization excitation functions can be linked to the relative abundance of Lambdas and anti-Lambdas coming from the core versus those coming from the corona. We will see how the global polarization peak at the expected ranges of collision energies but, the exact positions and heights of these peaks depend on the centrality class, which is directly related to the QGP volume and lifetime, as well as on the relative abundances of Lambdas and anti-Lambdas in the core and corona regions. Finally, I will talk about the improvements we are making to this model to study polarization properties of other species, and as a probe of critical phenomena in the nuclear medium.
### Tuesday, 03.22.2022, 3:30 PM PST
Dr. Xiaojun Yao (MIT)
Host: Xin-Nian Wang
Pure quark and gluon jet observables
Disentangling quark- and gluon-initiated jets can help us to better understand fundamental interactions in QCD and use jets as probes of the quark-gluon plasma in heavy ion collisions. Many previous studies relied on the Sudakov factors of some jet substructure observables such as the soft drop jet mass in the tail region and failed to reach a 100% efficiency in the disentangling. In this talk, I will introduce novel jet observables that are made pure quark or gluon in a wide kinematic region. The construction is based on the collinear drop grooming technique and nonperturbative effects are taken into account. I will show both analytic and Monte Carlo results for these observables in proton-proton collisions and discuss the impact of initial-state radiation and multi-parton interaction. Finally, I will discuss the potential obstacles in applying these observables in heavy ion collisions.
### Tuesday, 03.15.2022, 2:00 PM PST (Special time)
Dr. You Zhou (NBI)
Host: Xin-Nian Wang
Multi-particle correlations for the new decade of QGP studies
Multi-particle correlations have been compelling tools to probe the properties of the Quark-Gluon Plasma (QGP) created in the ultra-relativistic heavy-ion collisions. In this seminar, I will present a generic recursive algorithm for multi-particle cumulants, which enables the calculation of arbitrary order multi-particle cumulants. Among them, I will emphasize a particular series of mixed harmonic multi-particle cumulants, which measures the general correlations between any moments of different flow coefficients. The study of these new multi-particle cumulants in heavy-ion collisions will significantly improve the understanding of the initial event-by-event geometry fluctuations and the hydrodynamic response in the final state. This will pave the way for more stringent constraints on the initial state and help extract more precise information on how the created hot and dense matter evolves. Last but not least, I will show the most recent study of correlations between anisotropic flow and mean transverse momentum in terms of multi-particle correlations/cumulants. I will show how we can directly access the initial conditions of heavy-ion collisions using the latest experimental measurements at the LHC and discuss the critical challenge of the state-of-the-art QGP studies via Bayesian analyses.
### Tuesday, 03.08.2022, 2:00 PM PST
Dr. Giuliano Giacalone (Heidelberg University)
Host: Xin-Nian Wang
The initial state of the quark-gluon plasma: status, prospects, interdisciplinary connections
The hydrodynamic model of the quark-gluon plasma (QGP) formed in relativistic nuclear collisions has permitted us over the years not only to obtain quantitative estimates of the transport properties of this medium from data, but also to establish a phenomenologically viable picture of its initial condition and how it emerges from the interaction of two ions at high energy. I review the current understanding of the initial condition of the QGP, emphasizing the outcome of state-of-the-art models and the overall picture that they yield. I discuss the progress made in the definition of observable quantities that offer a specific sensitivity to the physics of the initial state, with a focus on recent results (both theoretical and experimental) on mean transverse momentum-anisotropic flow correlations. Such observables pose unprecedented constraints on the parameters of state-of-the-art Monte Carlo generators for nuclear collisions, demonstrating, in particular, the importance of having an accurate implementation of the structure of the colliding ions, and the nucleons therein, in such frameworks. Consequently, the initial state of heavy-ion collisions provides fertile ground for new interdisciplinary connections involving different aspects of hadronic and nuclear physics across energy scales.
### Spin-momentum correlation in hot and dense QCD matter
The transport phenomena involving spin are instrumental in investigating quantum effects in many-body systems. In heavy-ion collisions, the recent measurement of spin polarization and spin alignment opens a new avenue to explore the properties of hot and dense QCD matter. Based on linear response theory and quantum kinetic equation, we have systematically studied spin-momentum correlation induced by hydrodynamic gradients [1]. In addition to the widely studied thermal vorticity effects, we identify an undiscovered contribution from the fluid shear [2]. This shear-induced polarization (SIP) can be viewed as the fluid analog of strain-induced polarization observed in elastic and nematic materials. The possible signature of SIP at RHIC and LHC will be elaborated. Then, I will present our prediction for the signature Spin Hall effect induced by baryon chemical gradient at RHIC beam scan energies [3,4]. If time permitted, I will briefly discuss the effect of SIP on vector mesons. .
### The End of the Proton Radius Puzzle?
For many years scientists believed that the proton radius was 0.877(6) fm based on a series of atomic Lamb shift and electron scattering measurements. In 2010, a new type of measurement, making use of muonic hydrogen, determined the radius to be 0.842(1) fm. The large systematic difference between muonic hydrogen measurements and the previous results became known as the proton radius puzzle. To solve this puzzle, a world-wide theoretical and experimental investigation has been undertaken. I will review the status of the puzzle with an emphasis on electron scattering results and the systematic differences between the different analysis techniques that led to disparate conclusions. I will leave it to the audience to decide if the puzzle is solved.
### Joint NT and HIT seminar (note special day and time)
Wednesday, 2022.02.02, 1:00 PM PST
### Host: Aaron Meyer
Lattice QCD Determination of the Bjorken-$x$ Dependence of PDFs at Next-to-next-to-leading Order
The large-momentum effective theory (LaMET) is a systematic approach to calculate parton physics from Euclidean approaches such as lattice QCD. With major progress in the lattice renormalization and perturbative matching, the lattice calculation of PDFs with LaMET is now entering the stage of precision control. In this talk, I will present a state-of-the-art lattice QCD calculation of pion valence quark distribution with next-to-next-to-leading order matching correction, which is done using two fine lattices with spacings $a=0.04$ fm and $0.06$ fm and valence pion mass $m_\pi=300$ MeV at boost momentum as large as 2.42 GeV. I will demonstrate that perturbative matching in Bjorken-$x$ space yields a reliable determination of the valence quark distribution for moderate x with a target precision, which shows considerably improved systematic uncertainty compared to a previous analysis of the same lattice data with a short-distance factorization approach in the coordinate space, and is in excellent agreement with the most recent global analyses.
### Tuesday, 2022.01.25, 3:30 PM PST
Prof. Rene Bellwied (University of Houston)
Matter under extreme conditions - Particle Collisions along the QCD Phase Diagram
Relativistic particle collisions have come a long way during the past two decades with the characterization of states of matter along the QCD phase transition line. I will try to show how these discoveries lead to significant multi-disciplinary efforts to understand the creation and evolution of matter under extreme conditions. Examples that will be highlighted are the connection of fundamental quantum theories to collective phenomena and particle production, as well as astrophysical measurements that link to interactions observed at RHIC and LHC.
### Tuesday, 2022.01.18, 3:30 PM PST
Prof. Tanja Horn (JLab & The Catholic University of America)
Host: Jennifer Rittenhouse West
PION AND KAON STRUCTURE FUNCTIONS
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2023-03-27 19:52:30
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http://tm.durusau.net/?m=201408&paged=2
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## Archive for August, 2014
### USDA Nutrient DB (R Data Package)
Monday, August 25th, 2014
USDA Nutrient DB (R Data Package) by Hadley Wickham.
From the webpage:
This package contains all data from the USDA National Nutrient Database, “Composition of Foods Raw, Processed, Prepared”, release 26.
From the data documentation:
The USDA National Nutrient Database for Standard Reference (SR) is the major source of food composition data in the United States. It provides the foundation for most food composition databases in the public and private sectors. As information is updated, new versions of the database are released. This version, Release 26 (SR26), contains data on 8,463 food items and up to 150 food components. It replaces SR25 issued in September 2012.
Updated data have been published electronically on the USDA Nutrient Data Laboratory (NDL) web site since 1992. SR26 includes composition data for all the food groups and nutrients published in the 21 volumes of “Agriculture Handbook 8” (U.S. Department of Agriculture 1976-92), and its four supplements (U.S. Department of Agriculture 1990-93), which superseded the 1963 edition (Watt and Merrill, 1963). SR26 supersedes all previous releases, including the printed versions, in the event of any differences.
The ingredient calculators at most recipe sites are wimpy by comparison. If you really are interested in what you are ingesting on a day to day basis, take a walk through this data set.
Release 26 Web Interface
Release 26 page
Correlating this data with online shopping options could be quite useful.
### An Introduction to Congress.gov
Monday, August 25th, 2014
An Introduction to Congress.gov by Robert Brammer.
From the post:
Barbara Bavis, Ashley Sundin, and I are happy to bring you an introduction to Congress.gov. This video provides a brief explanation of how to use the new features in the latest release, such as accounts, saved searches, member remarks in the Congressional Record, and executive nominations. If you would like more in-depth training on Congress.gov, we hold bi-monthly webinars that are free and available to the public. Our next webinar is scheduled from 2-3 p.m. on September 25, 2014, and you can sign up for it on Law.gov. Do you have an opinion on Congress.gov that you would like to share with us, such as new features that you would like to see added to the site? Please let us know by completing the following survey. Also, if there is something you would like us to cover in a future video, please leave us a comment below.
There are mid-term elections this year (2014) and information on current members of Congress will be widely sought.
The video is only twenty (20) minutes but will help you quickly search a variety of information concerning Congress.
Take special note that once you discover information, the system does not bundle it together for the next searcher.
### Exploring a SPARQL endpoint
Monday, August 25th, 2014
Exploring a SPARQL endpoint by Bob DuCharme.
From the post:
In the second edition of my book Learning SPARQL, a new chapter titled “A SPARQL Cookbook” includes a section called “Exploring the Data,” which features useful queries for looking around a dataset that you know little or nothing about. I was recently wondering about the data available at the SPARQL endpoint http://data.semanticweb.org/sparql, so to explore it I put several of the queries from this section of the book to work.
An important lesson here is how easy SPARQL and RDF make it to explore a dataset that you know nothing about. If you don’t know about the properties used, or whether any schema or schemas were used and how much they was used, you can just query for this information. Most hypertext links below will execute the queries they describe using semanticweb.org’s SNORQL interface.
Bob’s ease at using SPARQL reminds me of a story of an ex-spy who was going through customs for the first time in years. As part of that process, he accused a customs officer of having memorized print that was too small to read easily. The which the officer replied, “I am familiar with it.” 😉
Bob’s book on SPARQL and his blog will help you become a competent SPARQL user.
I don’t suppose SPARQL is any worse off semantically than SQL, which has been in use for decades. It is troubling that I can discover dc:title but have no way to investigate how it was used by a particular content author.
Oh, to be sure, the term dc:title makes sense to me, but that is a smoothing function as a reader and may or may not be the same “sense” as occurs to the person who completed such a term.
You can read data sets using your own understanding of tokens but I would do so with a great deal of caution.
### Information Aversion
Monday, August 25th, 2014
Information Aversion by John Baez.
Why do ostriches stick their heads under the sand when they’re scared?
They don’t. So why do people say they do? A Roman named Pliny the Elder might be partially to blame. He wrote that ostriches “imagine, when they have thrust their head and neck into a bush, that the whole of their body is concealed.”
That would be silly—birds aren’t that dumb. But people will actually pay to avoid learning unpleasant facts. It seems irrational to avoid information that could be useful. But people do it. It’s called information aversion.
John reports on an interesting experiment where people really did pay to avoid learning information (about themselves).
Do you think this extends to learning unpleasant information about their present IT software or practices?
### Clojure Digest
Monday, August 25th, 2014
Clojure Digest by Eric Normand.
Annotated report of four to five resources each week that are relevant to Clojure.
Enough useful information to keep you moving towards effective use of clojure but not enough to become a nest of electronic debris as opposed to a paper one.
### Introducing Splainer…
Monday, August 25th, 2014
Splainer is a step towards addressing two problems:
From the post:
• Collaboration: At OpenSource Connections, we believe that collaboration with non-techies is the secret ingredient of search relevancy. We need to arm business analysts and content experts with a human readable version of the explain information so they can inform the search tuning process.
• Usability: I want to paste a Solr URL, full of query paramaters and all, and go! Then, once I see more helpful explain information, I want to tweak (and tweak and tweak) until I get the search results I want. Much like some of my favorite regex tools. Get out of the way and let me tune!
• ….
We hope you’ll give it a spin and let us know how it can be improved. We welcome your bugs, feedback, and pull requests. And if you want to try the Splainer experience over multiple queries, with diffing, results grading, a develoment history, and more — give Quepid a spin for free!
Improving the information content of the tokens you are searching is another way to improve search results.
### Desperately Seeking Algorithms!
Monday, August 25th, 2014
I don’t know for sure that Christophe Grand is “desperately” seeking algorithms but he has tweeted a request for “favorite algorithms” to be cast into posts similar to:
Tarjan’s strongly connected components algorithm
I dislike algorithms that are full of indices and mutations. Not because they are bad but because I always have the feeling that the core idea is buried. As such, Tarjan’s SCC algorithm irked me.
So I took the traditional algorithm, implemented it in Clojure with explicit environment passing, then I replaced indices by explicit stacks (thanks to persistence!) and after some tweaks, I realized that I’ve gone full circle and could switch to stacks lengths instead of stacks themselves and get rid of the loop. However the whole process made the code cleaner to my eye. You can look at the whole history.
Here is the resulting code:
See the Tarjan post for the Clojure version. Something similar is planned for “favorite” algorithms.
What algorithm are you going to submit?
Pass this along.
### Research topics in e-discovery
Monday, August 25th, 2014
Research topics in e-discovery by William Webber.
From the post:
Dr. Dave Lewis is visiting us in Melbourne on a short sabbatical, and yesterday he gave an interesting talk at RMIT University on research topics in e-discovery. We also had Dr. Paul Hunter, Principal Research Scientist at FTI Consulting, in the audience, as well as research academics from RMIT and the University of Melbourne, including Professor Mark Sanderson and Professor Tim Baldwin. The discussion amongst attendees was almost as interesting as the talk itself, and a number of suggestions for fruitful research were raised, many with fairly direct relevance to application development. I thought I’d capture some of these topics here:
E-discovery, if you don’t know, is found in civil litigation and government investigations. Think of it as hacking with rules as the purpose of e-discovery is to find information that supports your claims or defense. E-discovery is high stakes data mining that pays very well. Need I say more?
Webber lists the following research topics:
1. Classification across heterogeneous document types
2. Automatic detection of document types
3. Faceted categorization
4. Label propagation across related documents
5. Identifying unclassifiable documents
6. Identifying poor training examples
7. Identifying significant fragments in non-significant text
8. Routing of documents to specialized trainers
9. Total cost of annotation
“Label propagation across related documents” looks like a natural for topic maps but searching over defined properties that identify subjects as opposed to opaque tokens would enhance the results for a number of these topics.
### Speedy Short and Long DNA Reads
Monday, August 25th, 2014
Acceleration of short and long DNA read mapping without loss of accuracy using suffix array by Joaquín Tárraga, et al. (Bioinformatics (2014) doi: 10.1093/bioinformatics/btu553)
Abstract:
HPG Aligner applies suffix arrays for DNA read mapping. This implementation produces a highly sensitive and extremely fast mapping of DNA reads that scales up almost linearly with read length. The approach presented here is faster (over 20x for long reads) and more sensitive (over 98% in a wide range of read lengths) than the current, state-of-the-art mappers. HPG Aligner is not only an optimal alternative for current sequencers but also the only solution available to cope with longer reads and growing throughputs produced by forthcoming sequencing technologies.
Always nice to see an old friend, suffix arrays, in the news!
Source code: https://github.com/opencb/hpg-aligner.
For documentation and software: http://wiki.opencb.org/projects/hpg/doku.php?id=aligner:overview
I first saw this in a tweet by Bioinfocipf.
### Where to get help with Common Lisp
Monday, August 25th, 2014
Where to get help with Common Lisp by Zach Beane.
An annotated listing of help resources for Common Lisp.
I especially appreciated his quote/comment from one mailing list:
Beware that “the accuracy of postings made to the list is not guaranteed.” (I think that goes for every mailing list, ever.)
😉
I first saw this in a tweet by Planet Lisp.
### Who Dat?
Sunday, August 24th, 2014
Dat
Dat is an grant funded, open source project housed in the US Open Data Institute. While dat is a general purpose tool, we have a focus on open science use cases.
The high level goal of the dat project is to build a streaming interface between every database and file storage backend in the world. By building tools to build and share data pipelines we aim to bring to data a style of collaboration similar to what git brings to source code.
The first alpha release is now out!
More on this project later this coming week.
I first saw this in Nat Torkington’s Four short links: 21 August 2014.
### Introducing Riffmuse
Sunday, August 24th, 2014
Introducing Riffmuse by Dave Yarwood.
From the post:
I’ve written a simple command line app in Clojure that will take a musical scale as a command line argument and algorithmically generate a short musical idea or “riff” using the notes in that scale. I call it Riffmuse.
Here’s how Riffmuse works in a nutshell: it takes the command line argument(s), figures out what scale you want (For this, I used the awesome parser generator library Instaparse to create a parser that allows a flexible syntax in specifying scales. C major can be represented as, e.g., “C major,” “CMAJ” or even just “c”), determines what notes are in that scale, comes up with a rhythmic pattern for the notes (represented as 16 “slots” that can either have a note or no note), and then fills the slots with notes from the scale you’ve specified.
On the theory that you never know what will capture someone’s interest, this is a post that needs to be shared. It may spark an interest in some future Clojure or music rock star!
I first saw this in a tweet by coderpost.
### Pro Daniel
Saturday, August 23rd, 2014
Michael Daniel, cybersecurity coordinator for the White House is being taken to task for saying:
“You don’t have to be a coder in order to really do well in this position,” Daniel said, when asked if his job required knowledge of the technology behind information security. “In fact, actually, I think being too down in the weeds at the technical level could actually be a little bit of a distraction.”
“You can get taken up and enamored with the very detailed aspects of some of the technical solutions,” he explained, arguing that “the real issue is looking at the broad strategic picture.”
That quote, from White House cybersecurity czar brags about his lack of technical expertise by Timothy B. Lee, has provoked all manner of huffing and puffing across the computer security community.
It reminds me of candidates for the county commission who would brag about their expertise at running graders, backhoes, and similar heavy equipment. Always puzzled me because I assumed county government would hire people with those skills. Commissioners needed skills at representing the county for grants, making policy decisions, etc.
The security community, or at least reporters purporting to speak for the security community don’t appear to understand the difference between cybersecurity software and cybersecurity policy. You need coders for the former and policy wonks for the latter. Someone could be both but that’s fairly unlikely.
For example, assume a new security algorithm is discovered that can encrypt telephone and email communications with very little overhead for encryption/decryption. Further assume that Daniel has been assured by none other than Bruce Schneier that the algorithm and software that implements it, performs as advertised. And assume Daniel understands none of the details about the algorithm and software.
How does his ignorance impact the formulation of cybersecurity policy with regard to this algorithm or software?
The FBI opposes it because the FBI prefers non-encrypted communications like in the old days when it could just plug into a phone junction box.
The NSA opposes it, at least for others, because then it could not easily tap into email and phone conversations.
The Department of Defense opposes it, primarily because it has long term contractual relationships for security services with firms that don’t have access to the algorithm.
The Library of Congress supports it, at least those outside of the copyright office support it.
Various other groups take positions that seem reasonable to them.
So, how are coding skills going to help Daniel balance the political, social, agency and other politics for a policy concerning such an algorithm?
We all know the answer to that question.
Not at all.
PS: If my example looks like a strawman, come up with one of your own. Technical expertise Daniel can hire, policy expertise, meaning what is expedient given the stakeholders and their influence, not so much.
### Large-Scale Object Classification…
Saturday, August 23rd, 2014
Large-Scale Object Classi cation using Label Relation Graphs by Jia Deng, et al.
Abstract:
In this paper we study how to perform object classi cation in a principled way that exploits the rich structure of real world labels. We develop a new model that allows encoding of flexible relations between labels. We introduce Hierarchy and Exclusion (HEX) graphs, a new formalism that captures semantic relations between any two labels applied to the same object: mutual exclusion, overlap and subsumption. We then provide rigorous theoretical analysis that illustrates properties of HEX graphs such as consistency, equivalence, and computational implications of the graph structure. Next, we propose a probabilistic classifi cation model based on HEX graphs and show that it enjoys a number of desirable properties. Finally, we evaluate our method using a large-scale benchmark. Empirical results demonstrate that our model can signifi cantly improve object classifi cation by exploiting the label relations.
Let’s hear it for “real world labels!”
By which the authors mean:
• An object can have more than one label.
• There are relationships between labels.
From the introduction:
We first introduce Hierarchy and Exclusion (HEX) graphs, a new formalism allowing flexible specifi cation of relations between labels applied to the same object: (1) mutual exclusion (e.g. an object cannot be dog and cat), (2) overlapping (e.g. a husky may or may not be a puppy and vice versa), and (3) subsumption (e.g. all huskies are dogs). We provide theoretical analysis on properties of HEX graphs such as consistency, equivalence, and computational implications.
Next, we propose a probabilistic classi fication model leveraging HEX graphs. In particular, it is a special type of Conditional Random Field (CRF) that encodes the label relations as pairwise potentials. We show that this model enjoys
a number of desirable properties, including flexible encoding of label relations, predictions consistent with label relations, efficient exact inference for typical graphs, learning labels with varying specifi city, knowledge transfer, and uni fication of existing models.
Having more than one label is trivially possible in topic maps. The more interesting case is the authors choosing to treat semantic labels as subjects and to define permitted associations between those subjects.
A world of possibilities opens up when you can treat something as a subject that can have relationships defined to other subjects. Noting that those relationships can also be treated as subjects should someone desire to do so.
I first saw this at: Is that husky a puppy?
### Data + Design
Saturday, August 23rd, 2014
Data + Design: A simple introduction to preparing and visualizing information by Trina Chiasson, Dyanna Gregory and others.
From the webpage:
Information design is about understanding data.
Whether you’re writing an article for your newspaper, showing the results of a campaign, introducing your academic research, illustrating your team’s performance metrics, or shedding light on civic issues, you need to know how to present your data so that other people can understand it.
Regardless of what tools you use to collect data and build visualizations, as an author you need to make decisions around your subjects and datasets in order to tell a good story. And for that, you need to understand key topics in collecting, cleaning, and visualizing data.
This free, Creative Commons-licensed e-book explains important data concepts in simple language. Think of it as an in-depth data FAQ for graphic designers, content producers, and less-technical folks who want some extra help knowing where to begin, and what to watch out for when visualizing information.
As of today, the Data + Design is the product of fifty (50) volunteers from fourteen (14) countries. At eighteen (18) chapters and just shy of three-hundred (300) pages, this is a solid introduction to data and its visualization.
The source code is on GitHub, along with information on how you can contribute to this project.
A great starting place but my social science background is responsible for my caution concerning chapters 3 and 4 on survey design and questions.
All of the information and advice in those chapters is good, but it leaves the impression that you (the reader) can design an effective survey instrument. There is a big difference between an “effective” survey instrument and a series of questions pretending to be a survey instrument. Both will measure “something” but the question is whether a survey instrument provides you will actionable intelligence.
For a survey on any remotely mission critical, like user feedback on an interface or service, get as much professional help as you can afford.
When was the last time you heard of a candidate for political office or serious vendor using Survey Monkey? There’s a reason for that lack of reports. Can you guess that reason?
I first saw this in a tweet by Meta Brown.
Saturday, August 23rd, 2014
National Library of Medicine RSS Feeds
I am reporting it here because as soon as I don’t, I will need the listing.
### NLM Technical Bulletin
Saturday, August 23rd, 2014
NLM Technical Bulletin
A publication of the U.S. National Library of Medicine (NLM). The about page for NLM gives the following overview:
The National Library of Medicine (NLM), on the campus of the National Institutes of Health in Bethesda, Maryland, has been a center of information innovation since its founding in 1836. The world’s largest biomedical library, NLM maintains and makes available a vast print collection and produces electronic information resources on a wide range of topics that are searched billions of times each year by millions of people around the globe. It also supports and conducts research, development, and training in biomedical informatics and health information technology. In addition, the Library coordinates a 6,000-member National Network of Libraries of Medicine that promotes and provides access to health information in communities across the United States.
The NLM Technical Bulletin, your source for the latest searching information, is produced by: MEDLARS Management Section, National Library of Medicine, Bethesda, Maryland, USA.
Which is true but seems inadequate to describe the richness of what you can find at the bulletin.
For example, in 2014 July&emdash;August No. 399 you find:
MeSH on Demand Update: How to Find Citations Related to Your Text
New CMT Subsets Available
New Tutorial: Searching Drugs or Chemicals in PubMed
If medical terminology touches your field of interest, this is a must read.
### MeSH on Demand Tool:…
Saturday, August 23rd, 2014
From the post:
Currently, the MeSH Browser allows for searches of MeSH terms, text-word searches of the Annotation and Scope Note, and searches of various fields for chemicals. These searches assume that users are familiar with MeSH terms and using the MeSH Browser.
Wouldn’t it be great if you could find MeSH terms directly from your text such as an abstract or grant summary? MeSH on Demand has been developed in close collaboration among MeSH Section, NLM Index Section, and the Lister Hill National Center for Biomedical Communications to address this need.
## Using MeSH on Demand
Use MeSH on Demand to find MeSH terms relevant to your text up to 10,000 characters. One of the strengths of MeSH on Demand is its ease of use without any prior knowledge of the MeSH vocabulary and without any downloads.
Now there’s a clever idea!
Imagine extending it just a bit so that it produces topics for subjects it detects in your text and associations with the text and author of the text. I would call that assisted topic map authoring. You?
I followed a tweet by Michael Hoffman, which lead to: MeSH on Demand Update: How to Find Citations Related to Your Text, which describes an enhancement to MeSH on demands that finds relevant citations (10) based on your text.
The enhanced version mimics the traditional method of writing court opinions. A judge writes his decision and then a law clerk finds cases that support the positions taken in the opinion. You really thought it worked some other way? 😉
### Imaging Planets and Disks [Not in our Solar System]
Friday, August 22nd, 2014
Videos From the 2014 Sagan Summer Workshop On-line
From the post:
The NASA Exoplanet Science Center (NEXScI) hosts the Sagan Workshops, annual themed conferences aimed at introducing the latest techniques in exoplanet astronomy to young researchers. The workshops emphasize interaction with data, and include hands-on sessions where participants use their laptops to follow step-by-step tutorials given by experts. This year’s conference topic was “Imaging Planets and Disks”. It covered topics such as
• Properties of Imaged Planets
• Integrating Imaging and RV Datasets
• Thermal Evolution of Planets
• The Challenges and Science of Protostellar And Debris Disks…
You can see the agenda and the presentations here, and the videos have been posted here. Some of the talks are also on youtube at https://www.youtube.com/channel/UCytsRiMvdj5VTZWfj6dBadQ
The presentations showcase the extraordinary richness of exoplanet research. If you are unfamiliar with NASA’s exoplanet program, Gary Lockwood provides an introduction (not available for embedding – visit the web page). My favorite talk, of many good ones, was Travis Barman speaking on the “Crown Jewels of Young Exoplanets.”
Looking to expand you data processing horizons? 😉
Enjoy!
### Manhattan District History
Friday, August 22nd, 2014
Manhattan District History
From the post:
General Leslie Groves, head of the Manhattan Engineer District, in late 1944 commissioned a multi-volume history of the Manhattan Project called the Manhattan District History. Prepared by multiple authors under the general editorship of Gavin Hadden, a longtime civil employee of the Army Corps of Engineers, the classified history was “intended to describe, in simple terms, easily understood by the average reader, just what the Manhattan District did, and how, when, and where.” The volumes record the Manhattan Project’s activities and achievements in research, design, construction, operation, and administration, assembling a vast amount of information in a systematic, readily available form. The Manhattan District History contains extensive annotations, statistical tables, charts, engineering drawings, maps, photographs, and detailed indices. Only a handful of copies of the history were prepared. The Department of Energy’s Office of History and Heritage Resources is custodian of one of these copies.
The history is arranged in thirty-six volumes grouped in eight books. Some of the volumes were further divided into stand-alone chapters. Several of the volumes and stand-alone chapters were never security classified. Many of the volumes and chapters were declassified at various times and were available to the public on microfilm. Parts of approximately a third of the volumes remain classified.
The Office of Classification and the Office of History and Heritage Resources, in collaboration with the Department’s Office of Science and Technical Information, have made the full-text of the entire thirty-six volume Manhattan District History available on this OpenNet website. Unclassified and declassified volumes have been scanned and posted. Classified volumes were declassified in full or with redactions, i.e., still classified terms, phrases, sentences, and paragraphs were removed and the remaining unclassified parts made available to the public. All volumes have been posted.
In case you are interested in the Manhattan project generally or want to follow its participants into the late 20th century, this is the resource for you!
Just occurred to me that the 1940 Census Records are now online. What other records would you want to map together from this time period?
I first saw this in a tweet by Michael Nielsen.
### Getty Thesaurus of Geographic Names (TGN)
Friday, August 22nd, 2014
From the post:
We’re delighted to announce that the Getty Research Institute has released the Getty Thesaurus of Geographic Names (TGN)® as Linked Open Data. This represents an important step in the Getty’s ongoing work to make our knowledge resources freely available to all.
Following the release of the Art & Architecture Thesaurus (AAT)® in February, TGN is now the second of the four Getty vocabularies to be made entirely free to download, share, and modify. Both data sets are available for download at vocab.getty.edu under an Open Data Commons Attribution License (ODC BY 1.0).
What Is TGN?
The Getty Thesaurus of Geographic Names is a resource of over 2,000,000 names of current and historical places, including cities, archaeological sites, nations, and physical features. It focuses mainly on places relevant to art, architecture, archaeology, art conservation, and related fields.
TGN is powerful for humanities research because of its linkages to the three other Getty vocabularies—the Union List of Artist Names, the Art & Architecture Thesaurus, and the Cultural Objects Name Authority. Together the vocabularies provide a suite of research resources covering a vast range of places, makers, objects, and artistic concepts. The work of three decades, the Getty vocabularies are living resources that continue to grow and improve.
Because they serve as standard references for cataloguing, the Getty vocabularies are also the conduits through which data published by museums, archives, libraries, and other cultural institutions can find and connect to each other.
A resource where you could loose some serious time!
Try this entry for London.
Or Paris.
### The Truth About Triplestores [Opaqueness]
Friday, August 22nd, 2014
A vendor “truth” document from Ontotext. Not that being from a vendor is a bad thing, but you should always consider the source of a document when evaluating its claims.
Quite naturally I jumped to: “6. Data Integration & Identity Resolution: Identifying the same entity across disparate data sources.”
With so many different databases and systems existing inside any single organization, how do companies integrate all of their data? How do they recognize that an entity in one database is the same entity in a completely separate database?
Resolving identities across disparate sources can be tricky. First, they need to be identified and then linked.
To do this effectively, you need two things. Earlier, we mentioned that through the use of text analysis, the same entity spelled differently can be recognized. Once this happens, the references to entities need to be stored correctly in the triplestore. The triplestore needs to support predicates that can declare two different Universal Resource Indicators (URIs) as one in the same. By doing this, you can align the same real-world entity used in different data sources. The most standard and powerful predicate used to establish mappings between multiple URIs of a single object is owl:sameAs. In turn, this allows you to very easily merge information from multiple sources including linked open data or proprietary sources. The ability to recognize entities across multiple sources holds great promise helping to manage your data more effectively and pinpointing connections in your data that may be masked by slightly different entity references. Merging this information produces more accurate results, a clearer picture of how entities are related to one another and the ability to improve the speed with which your organization operates.
In case you are unfamiliar with owl:sameAS, here is an example from OWL Web Ontology Language Reference
<rdf:Description rdf:about="#William_Jefferson_Clinton">:
<owl:sameAs rdf:resource="#BillClinton"/>
</rdf:Description>
The owl:sameAs in this case is opaque because there is no way to express why an author thought #William_Jefferson_Clinton and #BillClinton were about the same subject. You could argue that any prostitute in Columbia would recognize that mapping so let’s try a harder case.
<rdf:Description rdf:about="#United States of America">:
<owl:sameAs rdf:resource="#الولايات المتحدة الأمريكية"/>
</rdf:Description>
Less confident than you were about the first one?
The problem with owl:sameAs is its opaqueness. You don’t know why an author used owl:sameAs. You don’t know what property or properties they saw that caused them to use one of the various understandings of owl:sameAs.
Without knowing those properties, accepting any owl:sameAs mapping is buying a pig in a poke. Not a proposition that interests me. You?
I first saw this in a tweet by graphityhq.
### Computer Science – Know Thyself!
Friday, August 22nd, 2014
Putting the science in computer science by Felienne Hermans.
From the description:
Programmers love science! At least, so they say. Because when it comes to the ‘science’ of developing code, the most used tool is brutal debate. Vim versus emacs, static versus dynamic typing, Java versus C#, this can go on for hours at end. In this session, software engineering professor Felienne Hermans will present the latest research in software engineering that tries to understand and explain what programming methods, languages and tools are best suited for different types of development.
Great slides from Felienne’s keynote at ALE 2014.
I mention this to emphasize the need for social science research techniques and methodologies for application development. Investigation of computer science debates with such methods may lead to less resistance to them for user facing issues.
Perhaps a recognition that we are all “users,” bringing common human experiences to different interfaces with computers, will result in better interfaces for all.
### Data Carpentry (+ Sorted Nordic Scores)
Thursday, August 21st, 2014
Data Carpentry by David Mimno.
From the post:
The New York Times has an article titled For Big-Data Scientists, ‘Janitor Work’ Is Key Hurdle to Insights. Mostly I really like it. The fact that raw data is rarely usable for analysis without significant work is a point I try hard to make with my students. I told them “do not underestimate the difficulty of data preparation”. When they turned in their projects, many of them reported that they had underestimated the difficulty of data preparation. Recognizing this as a hard problem is great.
What I’m less thrilled about is calling this “janitor work”. For one thing, it’s not particularly respectful of custodians, whose work I really appreciate. But it also mischaracterizes what this type of work is about. I’d like to propose a different analogy that I think fits a lot better: data carpentry.
Note: data carpentry seems to already be a thing
I’m not convinced that “carpentry” is the best prestige target.
The first mention of carpenters on a sorted version of the Nordic Scores (Colorado Adoption Project: Resources for Researchers. Institute for Behavioral Genetics, University of Colorado Boulder) is at 147.*
I would go for data scientist since mercenary isn’t listed as an occupation. 😉
The usual cautions apply. Prestige is as difficult or perhaps more so to measure than any other social construct. The data is from 1989 and so may not reflect “current” prestige rankings.
*(I have removed the classes and sorted by prestige score, to create Sorted Nordic Scores.)
### …Loosely Consistent Distributed Programming
Thursday, August 21st, 2014
Abstract:
Driven by the widespread adoption of both cloud computing and mobile devices, distributed computing is increasingly commonplace. As a result, a growing proportion of developers must tackle the complexity of distributed programming—that is, they must ensure correct application behavior in the face of asynchrony, concurrency, and partial failure.
To help address these difficulties, developers have traditionally relied upon system infrastructure that provides strong consistency guarantees (e.g., consensus protocols and distributed transactions). These mechanisms hide much of the complexity of distributed computing—for example, by allowing programmers to assume that all nodes observe the same set of events in the same order. Unfortunately, providing such strong guarantees becomes increasingly expensive as the scale of the system grows, resulting in availability and latency costs that are unacceptable for many modern applications.
Hence, many developers have explored building applications that only require loose consistency guarantees—for example, storage systems that only guarantee that all replicas eventually converge to the same state, meaning that a replica might exhibit an arbitrary state at any particular time. Adopting loose consistency involves making a well-known tradeoff: developers can avoid paying the latency and availability costs incurred by mechanisms for achieving strong consistency, but inexchange they must deal with the full complexity of distributed computing. As a result, achieving correct application behavior in this environment is very difficult.
This thesis explores how to aid developers of loosely consistent applications by providing programming language support for the difficulties they face. The language level is a natural place to tackle this problem: because developers that use loose consistency have fewer system facilities that they can depend on, consistency concerns are naturally pushed into application logic. In part, our goal has been to recognize, formalize, and automate application-level consistency patterns.
We describe three language variants that each tackle a different challenge in distributed programming. Each variant is a modification of Bloom, a declarative language for distributed programming we have developed at UC Berkeley. The first variant of Bloom, BloomL, enables deterministic distributed programming without the need for distributed coordination. Second, Edelweiss allows distributed storage reclamation protocols to be generated in a safe and automatic fashion. Finally, BloomPO adds sophisticated ordering constraints that we use to develop a declarative, high-level implementation of concurrent editing, a particularly difficult class of loosely consistent programs.
Unless you think of topic maps as static files, recent developments in “loosely consistent distributed programming” should be high on your reading list.
It’s entirely possible to have a topic map that is a static file, even one that has been printed out to paper. But that seems like a poor target for development. Captured information begins progressing towards staleness from the moment of its capture.
I first saw this in a tweet by Peter Bailis.
### The Little Book of Semaphores
Thursday, August 21st, 2014
The Little Book of Semaphores by Allen Downey.
From the webpage:
The Little Book of Semaphores is a free (in both senses of the word) textbook that introduces the principles of synchronization for concurrent programming.
In most computer science curricula, synchronization is a module in an Operating Systems class. OS textbooks present a standard set of problems with a standard set of solutions, but most students don’t get a good understanding of the material or the ability to solve similar problems.
The approach of this book is to identify patterns that are useful for a variety of synchronization problems and then show how they can be assembled into solutions. After each problem, the book offers a hint before showing a solution, giving students a better chance of discovering solutions on their own.
The book covers the classical problems, including “Readers-writers,” “Producer-consumer”, and “Dining Philosophers.” In addition, it collects a number of not-so-classical problems, some written by the author and some by other teachers and textbook writers. Readers are invited to create and submit new problems.
If you want a deep understanding of concurrency, this looks like a very good place to start!
Some of the more colorful problem names:
• The dining savages problem
• The Santa Claus problem
• The unisex bathroom problem
• The Senate Bus problem
There are problems (and patterns) for your discovery and enjoyment!
I first saw this in a tweet by Computer Science.
### CSV Fingerprints
Thursday, August 21st, 2014
CSV Fingerprints by Victor Powell.
From the post:
CSV is a simple and common format for tabular data that uses commas to separate rows and columns. Nearly every spreadsheet and database program lets users import from and export to CSV. But until recently, these programs varied in how they treated special cases, like when the data itself has a comma in it.
It’s easy to make a mistake when you try to make a CSV file fit a particular format. To make it easier to spot mistakes, I’ve made a “CSV Fingerprint” viewer (named after the “Fashion Fingerprints” from The New York Times’s “Front Row to Fashion Week” interactive ). The idea is to provide a birdseye view of the file without too much distracting detail. The idea is similar to Tufte’s Image Quilts…a qualitative view, as opposed to a rendering of the data in the file themselves. In this sense, the CSV Fingerprint is a sort of meta visualization.
This is very clever. Not only can you test a CSV snippet on the webpage, but the source code is on Github. https://github.com/setosa/csv-fingerprint (source code)
Of course, it does rely on the most powerful image processing system known to date. Err, that would be you. 😉
Pass this along. I can imagine any number of data miners who will be glad you did.
### Math for machine learning
Wednesday, August 20th, 2014
Math for machine learning by Zygmunt Zając.
From the post:
Sometimes people ask what math they need for machine learning. The answer depends on what you want to do, but in short our opinion is that it is good to have some familiarity with linear algebra and multivariate differentiation.
Linear algebra is a cornerstone because everything in machine learning is a vector or a matrix. Dot products, distance, matrix factorization, eigenvalues etc. come up all the time.
Differentiation matters because of gradient descent. Again, gradient descent is almost everywhere*. It found its way even into the tree domain in the form of gradient boosting – a gradient descent in function space.
We file probability under statistics and that’s why we don’t mention it here.
Following this introduction you will find a series of books, MOOCs, etc. on linear algebra, calculus and other math resources.
Pass it along!
### Mapping Out Lambda Land:…
Wednesday, August 20th, 2014
From the post:
Anyone who has met me will probably know that I am wildly enthusiastic about functional programming (FP). I co-founded a group for women in FP, have presented a series of talks and workshops about functional concepts, and have even been known to create lambda-branded clothing and jewellery. In this blog post, I will try to give some insight into what the fuss is about. I will briefly explain what functional programming is, why you should care, and how you can use OpenShift to learn more about FP.
With the publicity around OpenShift and functional programming, it seems entirely reasonable to put them together.
Katie gives you a quick overview of functional programming along with resources and next steps for your OpenShift account.
I first saw this in a post by Jonathan Murray.
### Web Annotation Working Group (Preventing Semantic Rot)
Wednesday, August 20th, 2014
Web Annotation Working Group
From the post:
The W3C Web Annotation Working Group is chartered to develop a set of specifications for an interoperable, sharable, distributed Web annotation architecture. The chartered specs consist of:
1. Abstract Annotation Data Model
2. Data Model Vocabulary
3. Data Model Serializations
4. HTTP API
5. Client-side API
The working group intends to use the Open Annotation Data Model and Open Annotation Extension specifications, from the W3C Open Annotation Community Group, as a starting point for development of the data model specification.
The Robust Link Anchoring specification will be jointly developed with the WebApps WG, where many client-side experts and browser implementers participate.
Some good news for the middle of a week!
Shortcomings to watch for:
Can annotations be annotated?
Can non-Web addressing schemes be used by annotators?
Can the structure of files (visible or not) in addition to content be annotated?
If we don’t have all three of those capabilities, then the semantics of annotations will rot, just as semantics of earlier times have rotted away. The main distinction is that most of our ancestors didn’t choose to allow the rot to happen.
I first saw this in a tweet by Rob Sanderson.
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2018-01-22 20:11:12
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https://dissertationwritinghelp.uk/category/informative/informative-abstract-informative/
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## How to write a Dissertation Abstract
2021-11-22T07:03:52+00:00
What is An Abstract? Whenever conducting a detailed scientific study, authors are required to summarise their entire study, including the methods, background, as well as outcomes. This brief summary is known as the “Abstract”. It can be deemed as the crux of entire research which lets readers know precisely what the research was about and its outcomes. Without an all-encompassing abstract, it is almost impossible for readers to identify the uniqueness of your research. In this sense, a good abstract is composed in light of two purposes: to inform potential readers about the research, and to inform research databases [...]
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2022-01-23 08:56:13
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http://mymathforum.com/calculus/10403-ode-question-2-a.html
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My Math Forum ODE question 2
Calculus Calculus Math Forum
December 17th, 2009, 03:46 AM #1 Member Joined: Dec 2009 Posts: 34 Thanks: 0 ODE question 2 Given this ODE: x' = x+y-xy^2 y' = -x-y+x^2y and a function: u(x,y) = x^2+y^2-2ln|xy-1| prove that for each solution ( x(t), y(t) ) of this system, such as: x(t)*y(t) != 1 (doesn't equal...) , there exists a constant C such as: u ( x(t), y(t) ) = C for every t in R. My attempt: It's very clear that we need to look at the derivative of u... If it will be 0, then we'll get what we need...But since I haven't got that much knowledge in 2 variables functions, I can't really see what is the derivative of u, as well as how to solve this ODE... So, I really need your help in: 1. Solving the ODE. 2. What is the derivative of u(t)? TNX a lot!
December 17th, 2009, 05:01 AM #2 Global Moderator Joined: Dec 2006 Posts: 20,653 Thanks: 2086 For xy ?1, du/dt = 2xx' + 2yy' - (xy' + yx')/(xy - 1) = 2x(x + y - xy²) + 2y(-x - y + x²y) - 2(x(-x - y + x²y) + y(x + y - xy²))/(xy - 1) = 2x² - 2y² - 2(-x² + x³y + y² - xy³)/(xy - 1), = 2x² - 2y² -2(x² - y²)(xy - 1)/(xy - 1) = 0. Hence u(x, y) is a constant. Other solutions are x = y = 0 and (x, y) = (Ae^t, (1/A)e^(-t)), where A is a non-zero constant.
December 17th, 2009, 06:30 AM #3 Member Joined: Dec 2009 Posts: 34 Thanks: 0 Re: ODE question 2 Wow, so we don't even need to solve the the system... Nice one... Tnx a lot!
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2019-05-26 20:58:20
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https://www.physicsforums.com/threads/tabulate-data-from-the-mathlab-program.716498/
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# Tabulate data from the mathlab program
I need to tabulate the data for u(x,t) vs x. Someone help me please :(
this is the program.
L = 1.;
T = 1.;
maxk = 2500;
dt = T/maxk;
n=50;
dx = L/n;
cond = 1/4;
b = 2.*cond*dt/(dx*dx);
for i= 1:n+1
x(i) = (i-1)*dx;
u(i,1) = sin(pi*x(i));
end
for k=1:maxk+1
u(1,k) = 0.;
u(n+1,k)=0.;
time(k) = (k-1)*dt;
end
for k=1:maxk
for i=2:n;
u(i,k+1)=u(i,k)+0.5*b*(u(i-1,k)+u(i+1,k)-2.*u(i,k));
end
end
figure (1)
plot (x,u(:,1),'-',x,u(:,100),'-',x,u(:,300),'-',x,u(:,600),'-')
title ('Temperature within the explicit method')
xlabel ('X')
ylabel ('Y')
figure (2)
mesh (x,time,u')
title ('Temperature within the explicit method')
xlabel ('X')
ylabel ('Temperature')
Related Engineering and Comp Sci Homework Help News on Phys.org
Mark44
Mentor
I need to tabulate the data for u(x,t) vs x. Someone help me please :(
What do you mean? Do you need to print the data in a table? If so, how many columns? What should the table look like?
BTW, this is not the section for HW problems. I am moving this post back to where your other one is.
this is the program.
L = 1.;
T = 1.;
maxk = 2500;
dt = T/maxk;
n=50;
dx = L/n;
cond = 1/4;
b = 2.*cond*dt/(dx*dx);
for i= 1:n+1
x(i) = (i-1)*dx;
u(i,1) = sin(pi*x(i));
end
for k=1:maxk+1
u(1,k) = 0.;
u(n+1,k)=0.;
time(k) = (k-1)*dt;
end
for k=1:maxk
for i=2:n;
u(i,k+1)=u(i,k)+0.5*b*(u(i-1,k)+u(i+1,k)-2.*u(i,k));
end
end
figure (1)
plot (x,u(:,1),'-',x,u(:,100),'-',x,u(:,300),'-',x,u(:,600),'-')
title ('Temperature within the explicit method')
xlabel ('X')
ylabel ('Y')
figure (2)
mesh (x,time,u')
title ('Temperature within the explicit method')
xlabel ('X')
ylabel ('Temperature')
yes i want to print the data and put it inside the table. two colomn which represent the graph
Im in total lost :( can you help me to modify my program so that it can create a table? :(
Mark44
Mentor
Something like this. I don't know if this is exactly what you need, but it's probably close.
Code:
formatSpec = "%4.2f \t %4.2f \n";
for k=1:maxk
for i=2:n;
u(i,k+1)=u(i,k)+0.5*b*(u(i-1,k)+u(i+1,k)-2.*u(i,k));
fprintf(formatSpec, x(i), u(i, k + 1))
end
end
what is %4.2f \t %4.2f \n meant?
Mark44
Mentor
%4.2f is a format specifier - print a number in a field of width 4, with 2 places to the right of the decimal point. If you have numbers that take up more space, change the 4 to something bigger. If you want more places to the right of the decimal, change the 2 to something bigger.
\t is a tab character
\n is a newline character.
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2020-05-29 20:07:51
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https://ncatlab.org/nlab/show/imaginary+number
|
# nLab imaginary number
Related concepts
A complex number $z$ is imaginary if it is not real; it is purely imaginary if its real part $\Re{z}$ is zero. For purposes of constructive mathematics, we only accept $z$ as imaginary if its imaginary part $\Im{z}$ is apart from zero, or equivalently if $z$ is apart from $\Re{z}$. This all generalizes to other kinds of hypercomplex numbers.
People often get these two notions mixed up. (For example, $2 + 3\mathrm{i}$ is imaginary but not purely imaginary; while $0$ is the unique purely imaginary number that is not imaginary.) This may be because the imaginary numbers, as is typical for things defined by an inequality, do not form an interesting collection as a whole (for example, they are not even closed under addition). Compare irrational number.
The purely imaginary numbers, on the other hand, form the Lie algebra $\mathfrak{u}(1)$. Often people substitute $\mathbb{R}$ (the algebra of real numbers), which is simpler, when they only care about this Lie algebra up to isomorphism. However, using $\mathrm{i}\mathbb{R}$ (the algebra of purely imaginary numbers) makes $\mathfrak{u}(1)$ fit with the matrix formulas used in higher dimensions.
Last revised on January 6, 2018 at 14:25:27. See the history of this page for a list of all contributions to it.
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2019-04-19 22:41:45
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https://prod-aws.lessonplanet.com/lesson-plans/calculus-differentiation
|
# Calculus Differentiation Teacher Resources
Find Calculus Differentiation lesson plans and worksheets
Showing 1 - 24 of 105 resources
38 Items in Collection
Lesson Planet
#### AP Calculus AB
For Students 11th - 12th
Prepare high school mathematicians for the AP Calculus AB exam with a collection of 38 videos that can be used to introduce the topics or as a review. The videos focus on two processes; differentiating a function and integrating a...
Lesson Planet
#### Differentiation Computations
For Students 11th
In this calculus worksheet, 11th graders solve problems using differential equations and computations. They apply properties of trig functions and simplify their answers. There are 9 questions.
Lesson Planet
#### Reductions of Order
For Students 11th - 12th
In this calculus worksheet, students identify functions as linear or nonlinear, homogeneous versus nonhomogeneous by using the reduction power theorem. There are 5 questions.
Lesson Planet
#### Differential Equations Representing Growth and Decay: Rice Legend
For Students 11th - Higher Ed
The legend of a wise man who asks a king for rice as a reward presents a context to study exponential solutions to differential equations. Pupils move quantities of rice to a chessboard and calculate the amount of rice for each day. To...
9:50
Lesson Planet
#### Calculus: Derivatives 3
For Students 11th - Higher Ed
This video covers the differential notation dy/dx and generalizes the rule for finding the derivative of any polynomial. It also extends the notion of the derivatives covered in the Khan Academy videos, "Calculus Derivatives 2Ó and...
9:02
Lesson Planet
#### Calculus: Derivative of x^(x^x)
For Students 11th - Higher Ed
Sal starts with an example of finding dy/dx of y = x2 and builds to showing the solution to the more complicated implicit differentiation problem of finding the derivative of y in terms of x of y = x ^ x ^ x .
4:40
Lesson Planet
#### Proof: d/dx(e^x) = e6x
For Students 11th - Higher Ed
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled �Proof of Derivatives of Ln(x) and e^x,� has a clearer explanation of this proof.
9:26
Lesson Planet
#### Calculus: Derivatives 1
For Students 11th - Higher Ed
Sal defines the term derivative by taking the listener on a well-organized tour of slope. First, he reviews the concept of slope of a line from algebra, then extends this idea to look at the slope of the curve by first examining a secant...
11:05
Lesson Planet
#### Calculus: Derivatives 2.5 (new HD version)
For Students 11th - Higher Ed
By defining the formal definition of a derivative, f�(x), Sal is able to find the general form of the derivate function for the example f(x) = x2. He continues to stress the importance of an intuitive understanding of derivative functions.
9:30
Lesson Planet
#### Calculus: Derivatives 2
For Students 11th - Higher Ed
Sal continues where he left off with the last video, "Derivatives 1,Ó by looking at the equation y = x^2 and examining the slope of the secant line at a specific point, and again defining the limit as x approaches zero to get the slope...
3:56
Lesson Planet
#### Proofs of Derivatives of Ln(x) and e6x
For Students 11th - Higher Ed
In this video, Sal takes on the challenge of proving both the derivative of ln x = 1/x and of ex = ex, showing that no circular logic is used in the proof. It contains a clearer version of both proofs shown in the videos titled, _Proof...
8:08
Lesson Planet
#### Equation of a Tangent Line
For Students 11th - Higher Ed
Using a specific example, Sal shows how to find the equation of a tangent line to a given function at a specific point. Specifically, he solves the problem of finding the tangent line to the function f(x) = xex at x = 1. This problem...
3:56
Lesson Planet
#### Product Rule
For Students 11th - Higher Ed
Sal defines the product rule and then shows two examples of how it is used. He then shows an example of finding the derivative by using both the chain rule and product rule together.
4:40
Lesson Planet
#### Proof: d/dx(e^x) = e6x
For Students 11th - Higher Ed
Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this proof.
9:52
Lesson Planet
#### Proof: d/dx(ln x)=1/x
For Students 11th - Higher Ed
Using the definition of a limit, various properties of logarithms, and a definition of e, Sal shows the proof of derivative of ln x = 1/x. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this...
5:07
Lesson Planet
#### Proof: d/dx(sqrt(x))
For Students 11th - Higher Ed
Using the definition of a limit, Sal proves the derivative of �x or x1/2 is equal to _ x-1/2.
7:03
Lesson Planet
#### Proof: d/dx(x^n)
For Students 11th - Higher Ed
Using the binomial theorem and definition of a limit, Sal shows a proof that the derivative of xn equals nxn-1.
8:58
Lesson Planet
#### Limit Examples (Part 1)
For Students 11th - Higher Ed
One example, find the limit as x approaches -1 of (2x+2)/(x+1), is solved by simplifying the expression and then is explored intuitively by looking at the left and right-hand limit. The second example of finding the limit as x approaches...
11:32
Lesson Planet
#### Introduction to Limits (HD)
For Students 11th - Higher Ed
Sal begins his explanation of limits with a few basic examples and takes a more intuitive point of view before looking at a formal mathematical definition in later videos. He starts by introducing the notation for limits and describes...
11:03
Lesson Planet
#### Introduction to Limits (HD)
For Teachers 11th - Higher Ed
Sal begins his explanation of limits with a few basic examples and takes a more intuitive point of view before looking at a formal mathematical definition in later videos. He starts by introducing the notation for limits and describes...
3:56
Lesson Planet
#### Introduction to Limits
For Students 11th - Higher Ed
Sal starts an explanation of limits by looking at a basic example of a discontinuous function and takes a more intuitive point of view before looking at a formal mathematical definition in later videos. He starts by introducing the...
Lesson Planet
#### Worksheet 31
For Students 12th - Higher Ed
In this math worksheet, learners find the solutions to the differential equations. They also investigate the application of recognizing the parts of the equation.
Lesson Planet
#### Worksheet 30
For Students 11th - Higher Ed
In this math worksheet, students solve the differential equation by means of a power series about the point x0 = 2. Then they find the recurrence relation and the first four terms.
Lesson Planet
#### Worksheet 20
For Students 10th - 12th
In this math instructional activity, they write down and solve a differential equation governing the motion of an underdamped spring. Then they find the solution to the initial value problem.
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2021-03-02 00:56:25
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https://readersneed.com/mitch-rapp-books-in-order/
|
# mitch rapp books in order
Here is a list of the “Mitch Rapp” series books by Vince Flynn in order of publication with their year and approximate price (in USD):
1. Transfer of Power (1999) – Approximately \$10
2. The Third Option (2000) – Approximately \$10
3. Separation of Power (2001) – Approximately \$10
4. Executive Power (2003) – Approximately \$10
5. Memorial Day (2004) – Approximately \$10
6. Consent to Kill (2005) – Approximately \$10
7. Act of Treason (2006) – Approximately \$10
8. Protect and Defend (2007) – Approximately \$10
9. Extreme Measures (2010) – Approximately \$10
10. American Assassin (2011) – Approximately \$10
11. Kill Shot (2012) – Approximately \$10
12. The Last Man (2012) – Approximately \$10
13. The Survivor (2015) – Approximately \$10
14. Order to Kill (2016) – Approximately \$10
15. Enemy of the State (2018) – Approximately \$10
16. Red War (2019) – Approximately \$10
17. Total Power (2021) – Approximately \$10
Note: The prices listed here are subject to change and may vary based on vendor and format.
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2023-03-24 09:17:15
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https://rjlipton.wordpress.com/2011/11/07/the-group-isomorphism-problem-a-possible-polymath-problem/
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Can we beat the “trivial” bound on group isomorphism?
William Burnside was one of the founders of modern group theory. His classic book, Theory of Groups of Finite Order, helped lay the foundations. He proved some beautiful theorems, made some great conjectures, and in general helped start the modern era of group theory.
Today Ken and I want to talk some more about group isomorphism.
There was great interest in a possible attack on this problem. Perhaps it is a candidate for a Polymath Project. In any event looking at a precise conjecture is one of the best ways to make progress in any area of theory or mathematics.
Burnside did exactly this for group theory. He stated one of the most important conjectures in group theory, which was named after him. Suppose that ${G}$ is a group that is generated by a finite number of elements. if there is an element ${a}$ such that ${\{a,a^{2},a^{3},\dots\}}$ is infinite, then obviously the group is infinite. But if all elements, not just the generators, of the group have a finite order, must the group be finite? Burnside conjectured yes.
Burnside’s Problem Specialized
Burnside found a neat way to develop his conjecture that also identified important special cases. The exponent of a group is the least common multiple of the orders of its elements, if it exists. If ${G}$ is finite then by Lagrange’s Theorem its exponent divides ${|G|}$.
We can make an infinite group of exponent 2 by taking ${\{0,1\}^\omega}$ and adding these infinite binary vectors componentwise mod~2. Every vector added to itself produces the identity, the all-zero vector. However, the group has no finite generating set because the sums of ${n}$-many vectors can produce at most ${2^n}$ elements. This is true of exponent-2 groups in general: they are Abelian because
$\displaystyle ab(ab) = 1 \implies ababb = b \implies aba = b \implies abaa = ba \implies ab = ba$
for all elements ${a,b}$. Hence at most ${2^n}$ distinct “words” can be made from ${n}$ generators.
The case of exponent 3 is trickier, as not every such group is Abelian. Burnside himself resolved it in the positive: if a group of exponent 3 is finitely generated, then it is finite. Burnside was emboldened to put forth his conjecture:
For all ${m}$ and ${n}$, if ${G}$ is a group with ${m}$ generators that has finite exponent ${n}$, then ${G}$ is finite.
This is technically weaker than the general first statement we gave, because the latter could (and did) fail by having a finitely generated group with all elements of finite order but an infinite exponent. However, Burnside himself made his special conjecture even more concrete by defining a single group ${B(m,n)}$ such that the conjecture holds for ${m,n}$ if and only if ${B(m,n)}$ is finite. See here for a definition of ${B(m,n)}$, which is analogous to a complexity class having a complete set.
Disproved and Still Kicking
Just before World War II the conjecture was proved for ${n = 4}$ and all ${m}$ by Ivan Nikolaevich Sanov. Then Marshall Hall, Jr., proved it for ${n=6}$ and all ${m}$ in what was termed a “heroic calculation.” Ken served as Marshall’s official host at Merton College, Oxford, for much of 1986, even putting him up when the Fellows’ Guestrooms weren’t available. Hall co-wrote a paper making progress on ${B(2,5)}$ in 1981, but Ken recalls only that he and Marshall talked about combinatorial designs.
However, Burnside’s original conjecture was refuted in 1964 by Evgeny Golod and Igor Shafarevich, who built an infinite group with three generators whose elements have finite orders that are unbounded powers of ${p}$, where ${p}$ can be any given prime. Then in 1968, Pyotr Novikov and Sergei Adian refuted Burnside’s particular conjecture for all ${m \geq 3}$ and all sufficiently large odd ${n}$, originally ${n \geq 4,381}$. Later this was extended to all odd ${n \geq 115}$, and others gave negative answers for many even ${n}$. The problem is extremely tricky, even strong group theorists have slipped: I discussed this quite a while ago here—where I talked about John Britton’s long, complex, but unfortunately incorrect proof.
The problem is still kicking, however, because of Burnside’s concrete development of it. Cases for smaller ${n}$ and various ${m}$ are still open, including ${B(2,5)}$. Even when ${B(m,n)}$ is infinite, one can still pose the restricted Burnisde problem: is there a finite bound on the orders of ${m}$-generator groups of exponent ${n}$ that are finite? Efim Zelmanov won a Fields Medal in 1994 largely for showing that in a sweeping range of cases the answer is yes.
We wish to propound a problem that has some similar ingredients and special cases. Analyzing these cases may require some “heroic calculation,” but that may be structurable in a way that is good for a Polymath project.
The Group-Isomorphism Problem, Again
Recall that a group can be presented by giving the Cayley table: if a group has ${n}$ elements this table is ${n \times n}$ in size. The table just encodes the product function “${\circ}$” in the obvious way: the entry ${(a,b)}$ is the value of ${a \circ b}$.
The problem is given two groups ${G}$ and ${H}$, are they really the same—that is can we relabel the elements so the tables are identical? This is asking whether the groups are isomorphic, to use a more technical terminology. The current best bound, as discussed before, is that this can be done in
$\displaystyle n^{\log_{2} n +O(1)}$
time by a deterministic algorithm. This method relies on the fact that any group of size ${n}$ has a generator set of size at most ${\log_{2} n}$. This is a quasi-polynomial time.
Finite ${p}$-Groups
A ${p}$-group is a group in which every element has order ${p^{m}}$ for some ${m}$, where ${p}$ is a prime. For a finite group the order of the group must likewise be a power of ${p}$. Recall that the counterexamples by Golod and Shafarevich are infinite ${p}$-groups. Finite ${p}$-groups have many special properties that seem to make them candidates for an attack; they also have many properties that make them look impervious to attack. For more on them, start here.
Let’s denote the number of groups, up to isomorphism, of order ${n}$ by ${T(n)}$. This function is quite nasty, since it varies widely with the multiplicative structure of ${n}$. If ${n}$ is a prime, then ${T(n)=1}$, since only the cyclic group is possible. If ${n}$ has many factors, then ${T(n)}$ can be very large.
Related to this function is the folklore conjecture asserting that almost all finite groups are 2-groups: the fraction of isomorphism classes of 2-groups among isomorphism classes of groups of order at most ${n}$ is thought to tend to ${1}$ as ${n}$ tends to infinity. This leads us to ask:
Can we solve the Group Isomorphism Problem specialized to ${p}$-groups in polynomial time? Or at least improve the above quasi-polynomial time?
Good News and Bad News
Several features of ${p}$-groups provide structure to get a handle on, but other features forbode that calculations may have to be longwinded.
The good news: The family of ${p}$-groups always have a non-trivial center. Recall the center of a group ${G}$ is the subgroup of elements that commute with all the other elements. In general this subgroup, ${Z(G)}$, can be trivial, that is consist of just the identity element. But ${p}$-groups have the wonderful property that the center is always nontrivial.
Here is a simple application of this principle. Any group of order ${p^{2}}$ for a prime ${p}$ is abelian. Let the group be ${G}$, and let ${H}$ be the non-trivial center. We can assume that ${H}$ is order ${p}$, else it is all of ${G}$ and then ${G}$ is abelian. Let ${x}$ be an element of ${G-H}$. The group generated by ${x}$ and ${H}$ is larger than ${H}$, so by Lagrange’s Theorem it has order ${p^{2}}$, and thus is equal to ${G}$. But also this group is abelian. This follows since ${x}$ commutes with all of ${H}$—recall that ${H}$ is the center.
The bad news: They can have large generators, as large as ${\log_{2} n}$, where ${n = 2^{m}}$. Even if the number of generators is close to ${\log_{2} n}$ the trivial isomorphism algorithm will still be supe- polynomial.
There is also the Burnside structure theorem on the generators of such a group: his theorem works for any ${p}$-group. One of the cool properties is that all generator sets of such groups have the same cardinality, which is false in general for groups. For many mathematical structures the size of a minimal generator set is always the same: ${n}$-dimensional vector spaces are the standard example. But for general groups, minimal generator sets can be vastly different in size. Even the symmetric group has this property—it can be generated by two elements or by many more elements. Note, we are always talking about generator sets that are minimal, meaning that removal of any single element will destroy the generator property.
A Special-Case Problem
George Pólya had a principle for working on problems: when working on a hard problem, try to find a special case that captures its essence. I suggest that the best case to look at is the class of ${2}$-groups. One clear reason is that they are the worst case for the trivial time bound, since they could have as many as ${\log_{2} n}$ generators. Note that if all elements in such a group have order 2, then the group is abelian. And since isomorphism is easy for abelian groups, this is a good case.
I think what we need is another strategy besides the generator approach. Let ${G}$ be a 2-group. If it has a very large abelian subgroup, then it should be easy to handle. If on the other hand, it has only a small abelian subgroup, then can we show that it has some special structure that we can exploit?
Open Problems
Should we organize a Polymath Project? Ken and I think that we would help with one, but that we probably need at least one strong group theorist to help run such a project. Any takers?
100 Comments leave one →
1. November 7, 2011 11:38 am
I very much like the sound of this project, but like you I feel that my group theory is inadequate. One thing I’d like to know is how to get a rich source of non-trivial 2-groups. Without that I have no way of assessing any algorithmic ideas. It would be particularly interesting to have two definitions of 2-groups that resulted in groups that were isomorphic but not obviously so — to get some kind of feel for the obstacles that an algorithm would have to overcome.
In general, the idea of a collaboration between group theorists, with their feel for what groups are out there, and computer scientists, with their feel for what clever algorithms are out there, sounds very promising, especially with this restricted version of the question.
• November 7, 2011 1:46 pm
Here’s one source of 2-groups: the finite quotients of the Grigorchuk group (which is a finitely generated, residually finite 2-group, which is typically constructed as an automata group). Because the Grigorchuk group acts on the infinite rooted binary tree, constructing families of such finite quotients is quite easy – just take the action of the group on the first n levels of the tree. More generally, many families of finite p-groups arise as quotients of just infinite pro-p-groups. See the book by Leedham-Green and McKay for more on this.
As for obstacles that need to be overcome, as well as more examples of p-groups, a very interesting paper in this direction is this recent one by Lewis and Wilson: http://arxiv.org/abs/1010.5466. They construct large families of p-groups (as quotients of Heisenberg groups) which have many invariants identical, but nonetheless they give a polynomial-time algorithm for isomorphism of these groups.
• Colin Reid permalink
November 7, 2011 5:49 pm
The groups discussed in Leedham-Green and McKay are the p-groups of bounded coclass. Perhaps there is a fast group isomorphism algorithm for these groups? There is certainly a much better understanding of the asymptotic structure of the class of p-groups of a given coclass than there is of the class of p-groups in general.
• November 7, 2011 6:22 pm
@Colin Reid: Indeed. As far as I know there is no efficient isomorphism algorithm for p-groups of bounded coclass (or perhaps you meant to suggest this as a potentially approachable problem?, in which case I agree.) Whether or not the p-groups of bounded coclass have an efficient isomorphism algorithm, it may still be a fruitful class of examples on which to test out new algorithmic ideas.
On the other hand, there are some reasons to believe that p-groups of class 2 may be the hard nugget of the isomorphism problem: 1) I recall that “most” p-groups are of class 2; related 2) the Jacobi identity of the associated graded imposes no condition in class 2, so their structure is in a sense less constrained than groups of higher class; 3) the isomorphism problem for class 2 is already “wild” (in the technical sense of the word). (Though wildness is only a rough and sometimes dubious indicator of computational complexity.) So perhaps some families of class 2 p-groups might be good examples on which to test new algorithmic ideas.
Back on the first hand: as you point out, we have a much better handle on the structure of groups of bounded coclass; so bounded coclass might be a better place to start.
• Colin Reid permalink
November 9, 2011 9:17 am
Hmm, thinking about it, bounded coclass implies bounded number of generators, which is one case that can already be done in polynomial time. So perhaps not so interesting. Nevertheless, it might be possible to get a smaller degree than the general algorithm.
2. November 7, 2011 12:05 pm
What about graph isomorphism? Are these connected problems? What do you think about?
• November 7, 2011 8:21 pm
Michael,
Group isomorphism is known to be reducible to graph isomorphism, but the reverse direction is not known.
• November 8, 2011 2:45 pm
Thanks, Joshua! 😉
3. November 7, 2011 1:51 pm
I’m not sure that the fact that p-groups always have nontrivial center is *helpful*, except of course for the point about groups of order p^2. In fact, groups with no center are in some sense more rigid and thus might be easier to identify up to isomorphism. For example, because group extensions are classified by group cohomology with coefficients *in the center of the group* being extended, when the center is trivial the cohomology is trivial, making the extension problem potentially easier. That being said, it would be great for my intuition in this regard to be flipped on its head.
4. Javaid Aslam permalink
November 7, 2011 2:30 pm
Whact exactly is “w” (or omega) in {0,1}^w?
November 7, 2011 3:17 pm
This denotes the set of all strings of infinite length consisting of 0’s and 1’s.
5. Colin Reid permalink
November 7, 2011 5:42 pm
I’m not a computational group theorist, but I know a few. From what I gather, there has been considerable work in (constructive) isomorphism problems for groups given as a generating set of matrices or permutations, and also for black-box groups. I imagine the last case is probably closest to what one would do with a Cayley table. The 2-groups case sounds difficult to me because there are so many 2-groups, and because the composition series tells you nothing.
• Colin Reid permalink
November 7, 2011 5:57 pm
PS: In group theory literature, ‘the isomorphism problem’ often refers to the problem of determining whether two group presentations (that is, as generators and relations) are isomorphic. This is probably not very useful for understanding the Cayley table isomorphism problem.
6. November 7, 2011 8:19 pm
I have to wonder if this problem becomes easier if one has additional information about the groups. For example, what if one has representations of the groups over some reasonably small finite field? Similarly, every finite group has a faithful representation in GL(n,C). The Jordan-Schur theorem says that such a group must have an abelian normal subgroup that isn’t too small, and some versions of that theorem give explicit constructions. If one had an oracle that could give you some sort of information related to that. Say, given a group G, it returns the largest abelian normal subgroup (using some sort of lexicographic rule to decide which one to give if there’s a tie), does this make the problem easier?
7. November 8, 2011 3:31 am
In general it seems to me that since the real target of such a project is the full (multiplication table) isomorphism problem — that is, the problem for arbitrary finite groups — it might be worth attacking even this special case with an attitude that one is trying not to use the structure of 2-groups, but that if at some point it helps, then one will.
To put that a different way, there are two ways one might hope to find an algorithm in the 2-group case. The first is to get enough understanding of 2-groups that we can eventually find an algorithm. The second is to try to find an algorithm for general groups, with a view to understanding that problem well enough that we can see where facts about 2-groups could make things easier. The reality might be a mixture of the two, but I still have a feeling that the second is more promising.
• Colin Reid permalink
November 8, 2011 4:08 am
Alternatively, we could try to make rigorous the intuition that p-groups (even p-groups of class 2) are the main source of difficulty in the problem. For instance, perhaps we can find an algorithm that runs in polynomial time given an oracle for p-group isomorphism, or perhaps even just an oracle to tell if the Fitting subgroups are isomorphic? Polynomial-time algorithms have been found for groups that are sufficiently far from being nilpotent, such as the case of groups with no abelian normal subgroups.
8. November 8, 2011 11:13 am
A problem which I’ve always found interesting is the following: A function f : {1,.., n} x { 1,…, n } -> { 1,…, n } is given, and it is a given that this is the multiplication of a group on { 1,…,n } – but beyond that, nothing is known; not even which element is the identity.
How many queries to f are needed to identify the group up to isomorphism and/or give the full multiplication table? The very naive bound is n^2, but depending on the factorisation of n very large speedups are possible. I’ve done a very small number of cases by hand, but it would be interesting to see how to do this for groups of 256, say.
Additional constraints on the group being guessed make the ‘game’ easier and would provide a measure of the information embodied in those constraints.
November 9, 2011 8:21 am
Jan de Wit
I am confused by your question. The function is the multiplication table, so what is to be found? The identity should be discoverable in O(n) time. What am I missing?
November 9, 2011 9:04 am
Well, look at Jan’s question in the “identify the group up to iso” version for small n. If n=1,2 or 3, the answer is that 0 queries are needed, because there is only one group of each of those sizes, up to iso. For n=4, 3 queries will do it, because you only have to distinguish C4 from C2xC2, and you can do that by asking for f(1,1), f(2,2) and (only necessary if those two turn out to be different) f(3,3). To distinguish the two possible groups, it suffices to look at the diagonal of the table, and 3 elements suffice because in C4 two elements occur twice each on the diagonal.Etc.I don’t know whether it is a useful approach to the problem you’re interested in, but if you can identify each of two groups up to iso you certainly know whether or not they are isomorphic!
• November 9, 2011 9:06 am
I meant of course “(only necessary if those two turn out to be the same)”, sorry.
• November 9, 2011 1:24 pm
As I understand the question, the function is given as a black box, and you want to ask for as few values as you can and deduce the rest. So e.g. if n is prime, then you can solve the problem in time zero, since you know the group is cyclic. If n is the square of a prime p, then a very crude method to determine the group would be to test p+1 distinct elements and find out whether they all have order p, but there are probably better methods. And so on.
I find this quite an interesting question: given that the isomorphism question is hard, is it much easier to deduce the full multiplication table from just a few queries than it is to obtain some description of the group up to isomorphism?
Come to think of it, it’s not even obvious to me that there is some nice way to describe all finite groups up to isomorphism. Can it be done using CFSG somehow? That is, can we describe some product-type operations on groups in such a way that every finite group can be built up from simple groups in a canonical way? The reason I’m worrying about this is that for the examples I mentioned above the way I saved time was by using the fact that one can describe all groups of the given order.
• November 9, 2011 1:34 pm
As far as I understand it, describing the extensions of one group by another is really quite nontrivial in general. Although extensions are classified up to equivalence by an outer action and an element of second cohomology, this classifies them up to *equivalence of extensions*, rather than up to isomorphism of the total group.
Even if one were happy with that, the choice of element in second cohomology doesn’t need to be canonical, because the two groups in the extension can act as automorphisms of the second cohomology group, preserving the isomorphism type of the total group. (This, by the way, is not the only way two extensions can be non-equivalent but have isomorphic total groups. Thanks to Vipul Naik for teaching me that fact and showing me an example.)
On the other hand, maybe this is just the wrong way to go about it…
• November 12, 2011 4:27 am
If finding descriptions for isomorphism classes of all groups is difficult, then that suggests that it is genuinely not feasible to solve the isomorphism problem by first saying what each group is and then declaring them to be isomorphic if you say the same in both cases.
What other possibilities does that leave? One is to show that groups are isomorphic or not without saying anything much about what those groups are. Another is to calculate a whole bunch of parameters and argue theoretically that any two groups that give the same answers must be isomorphic. The reason this is potentially easier than actually describing the two groups is that it may be that not all assignments of values to the parameters are possible. One way of looking at it is this. How can a bunch of polynomially computable descriptions be useful? Answer: if non-isomorphic groups have distinct descriptions and isomorphic groups have the same description. So we have a “description space D” and a map from finite groups (up to isomorphism) to D that we need to be injective. However, we don’t need it to be surjective, so this is weaker than a classification of all finite groups.
A slightly silly analogy would be that we don’t have a nice formula for generating primes, but it’s easy to distinguish between different primes because we do at least know that distinct primes are distinct integers.
• November 12, 2011 12:26 am
Perhaps a related problem is easier to solve and still gives us some information.
PB: Alice knows the multiplication table of G and the goal is to communicate it to Bob. Alice gets to pick which m(G) elements of the table to send to Bob, such that Bob can reconstruct the rest.
There are two differences to what Jan de Wit asked: I dropped “up to isomorphism” and I let Alice decide what to communicate, rather than have Bob ask a sequence of “ab=?” questions.
Given the multiplication tables of G and H, an isomorphism checker could go like this: If m(G) differs from m(H) say false, otherwise do some $\Omega(m(G))$ work. Is there other information that comes from m(G)? Is it easy to compute m(G)?
(Incidentally, if Alice wants to communicate a partial order rather than a group, then I know how to pick what to communicate, but that seems of no help here.)
9. November 8, 2011 1:32 pm
I’d be interested in such a project, but I certainly don’t qualify as an expert.
10. Colin Reid permalink
November 9, 2011 10:06 am
If we were to seriously attempt this, the first thing to do would be to rustle up a survey of properties of groups we know how to find in polynomial time, and generally try to summarise the current state of knowledge in a wiki or something. It does sound like the kind of problem where an incrementalist many-author approach could work. It’s not something I’d feel like I could say much about by myself, given my lack of knowledge of algorithms, but the whole point of the polymath approach is that we can get somewhere even if none of us understands the whole thing.
In fact, perhaps a lot of finite group theory is amenable to massive collaboration. I wonder, for instance, how much progress could be made on shortening the proof of CFSG (itself a many-author result) by a horde of relatively unspecialised people each looking at small pieces and making incremental improvements to them, without necessarily understanding the big picture. The existing proof is extremely long, but large parts of it consist of elementary arguments to eliminate this or that special case, and it may be that fresh eyes would see better elementary arguments. The more high-tech bits involving cohomology or character theory could simply be treated as black boxes. Even a significantly shorter proof of the Odd Order Theorem would represent a major achievement.
• November 9, 2011 10:25 am
Colin, those achievements would be orders of magnitude larger. Moreover, while some parts of the proof certainly involve very elementary work, even those are long and involved and sometimes involve subtle issues. (This is my impression from the (small) bits of it that I’ve read). If one looks at for example Powell and Higman’s “Finite Simple Groups” which only contains some of the work that had been done up until 1970 one sees that there’s already a lot of difficult stuff.
Trying to improve the group isomorphism problem looks orders of magnitudes easier.
• Colin Reid permalink
November 9, 2011 11:39 am
Yes, group isomorphism is likely to be easier to make tangible progress on, if nothing else because it hasn’t seen the same amount of effort poured into it as CFSG (especially the effort that has already been put into simplifying the proof in the last 30 years by the real experts). Then again, who knows how hard an open problem is?
• November 9, 2011 1:30 pm
I feel the exact opposite of what Colin feels: that I might be able to come up with a clever algorithm or two, but lack the expertise in group theory to know where to start, or what might be feasible, or to have any sense of what groups there are out there that need to be distinguished.
One possibility I quite like is that one might be able to determine whether two multiplication tables are isomorphic groups without being able to say much about what the group is or what properties it has — rather as you can sometimes say that two sets are the same size without knowing anything about what that size is. I don’t have any serious suggestions for how that might be done, however.
11. November 9, 2011 10:28 am
Does anyone know if graph isomorphism for connected comparability graphs has been shown to be in P? The reason I ask is that for a given finite group one has a few different almost natural partial orders of the elements which you can construct efficiently from the group table and are invariants of the group. For example, one could construct the almost partial order made by declaring x ~ y if x^n =y for some n. Then one identifies points x and y together if x ~y and y~x. This gives rise to a connected comparability graph in an obvious way (the graph is connected since the starting group is finite so every node is connected to the identity node). I don’t know in general any great examples of distinct groups look that have isomorphic comparability graphs, but that looks to be not very common. So if can solve that subcase of graph isomorphism that may handle a lot of groups.
November 24, 2011 3:33 pm
Graph isomorphism is reducible to graph isomorphism for comparability graphs (i.e. comparability graph isomorphism is GI-complete). See for example http://www.cs.uwaterloo.ca/~dloker/papers/CS-2006-32.pdf and references therein.
• November 24, 2011 7:21 pm
Thanks. That rules out that sort of method pretty strongly.
12. Jim Blair permalink
November 9, 2011 10:56 am
Sounds like an adventure.
Count me in.
13. November 9, 2011 12:20 pm
Sorry, one more thought: The graph isomorphism problem is solved for permutation graphs. Given a finite group G with n elements, one easily gets n permutation graphs (by looking at how each element permutes every element). If two groups are isomorphic then these sets of permutation graphs need to be isomorphic, and for two groups that can be determined in polynomial time in n. So this leads to two questions: are there examples of two non-isomorphic groups that lead to the same set of permutation graphs? (I suspect that the answer is yes.) Is there some reasonable niceness condition that isn’t too strict such that isomorphism of the permutation graphs implies group isomorphism?
• November 9, 2011 1:04 pm
Any element of prime order p must have its permutation graph as a union of cycles all of which have size p. So any two groups of the same order and prime exponent have the same permutation graphs.
A terminological note in case anyone wants to read more about this: this is essentially the approach of reducing a group to its corresponding “association scheme” aka “coherent configuration.” (Association schemes coming from groups are of course special.)
• November 9, 2011 1:09 pm
Also, any two groups which are distinguished by the orders of their elements are obviously distinguished by their sets of permutation graphs. Which raises a further question: what are some examples of pairs of groups not distinguished by the orders of their elements but which are distinguished by their permutation graphs?
• November 9, 2011 2:59 pm
Sorry, I was being silly before: it seems like the permutation graphs encode exactly the same information as the order of the elements. For a given element, the cycles in the corresponding permutation graph are exactly the cosets of the subgroup generated by that element, so they all have the same size, namely the order of the element. Or have I misunderstood what you meant by “permutation graph”?
• November 9, 2011 3:11 pm
Hmm, I’m not sure. It seems like you are interpreting it correctly somewhat (relevant Wikipedia article on permutation graphs http://en.wikipedia.org/wiki/Permutation_graph ) but I don’t quite see how the permutation graphs are encoding only the order of the elements. I see why it includes that information but not why that’s precisely it. What am I missing?
• November 9, 2011 4:38 pm
Hold on, I don’t think this makes sense — permutation graphs are derived from a permutation on an ordered set of elements; it’s not enough to just know the cycle type of the permutation. (Of course, knowing the cycle type in this case is the same as knowing the order!) There’s not a natural total order on elements of a finite group, so you won’t get permutation graphs unless you choose one, but that defeats the point.
For instance, if our group is C_4 with generator g, then the order (e,g,g^2,g^3) yields an empty graph, two Y’s, and a cycle. But if we order it (e,g,g^3,g^2), we get an empty graph, two paths, and a complete graph. So this isn’t an invariant.
Perhaps this is fixable if you take some natural partial order and require that the total order used extend it somehow and then it’ll be an invariant? No idea.
• November 9, 2011 5:05 pm
Oh hmm. You are right. For some reason I sort of assumed that permuting the order of the elements would give you the same set of permutation graphs to work with, but that’s not the case. This idea fails pretty miserably.
14. Jim Blair permalink
November 10, 2011 1:09 am
Given the number of references to graph isomorphism in the current discussion already, I was wondering why we are tackling the Group Isomorphism Problem and not the Graph Isomorphism Problem.
I followed the pointer back to “An Annoying Open Problem” and copied this section:
*******
The Short History
I find this problem very annoying. I have spent uncountable hours working on this over the last decades, especially jointly with Zeke Zalcstein. One reason the problem is so annoying is that it is easy to prove that it can be reduced to Graph Isomorphism (GI). But surely the additional structure of groups must help make the problem much easier to solve. Here are some encouraging differences between groups and graphs:
Every subset of elements of a graph is a graph; only certain subsets of elements of a group form a group.
Groups have a tight structure that varies with the prime factorization of , the number of elements. For instance, if is a prime there is exactly one group—the cyclic group .
Groups always have automorphisms, provided . Graphs can have many automorphisms—for instance the complete graph has —or they can have as little as no nontrivial automorphisms.
I could go on and on with the many structural differences.
***********
I suspect we are looking at some hard earned wisdom about the problem and we should take advantage of it.
The first and foremost item about “subsets” is somewhat curious:
Are Graphs somehow more consistent and less Russell paradoxical than Groups? Is there an asymmetric anomaly with Groups that we can exploit?
15. Roy Maclean permalink
November 10, 2011 8:49 am
The definition of groups as sets with binary operations G^2->G, can be generalised to sets with n-ary operations G^n->G (Dornte, Post, et al), or even n,m-operations G^n->G^m (Cupona et al) satisfying group-like axioms. Finite n,m-groups only exist if n>=2m.
Can you say something about n-group or n,m-group isomorphism problems on their n-dimensional multiplication tables that enables you to say something new about 2-dimensional group tables ?
“n-GROUPS IN THE LIGHT OF THE NEUTRAL OPERATIONS”, Janez Usan, 2006, http://www.moravica.tfc.kg.ac.rs/Special/n-Groups_in_the_Light_of_Natural_Operations-v2006.pdf
16. November 12, 2011 4:47 am
Here’s a very naive approach to the problem, which clearly fails (or the problem wouldn’t be open). Suppose you fix some smallish integer such as 100. Then given two groups G and H, you can in polynomial time run the following algorithm. For each sequence $x_1,\dots,x_{100}$ of 100 distinct elements of G and each sequence $y_1,\dots,y_{100}$ of 100 distinct elements of H, check whether the map that takes $x_i$ to $y_i$ extends to an isomorphism of the subgroups they generate. If they do, then join the corresponding sequences by an edge in a big bipartite graph.
If G and H are isomorphic, then this graph must have a perfect matching (just join each point in G to its image under the isomorphism), and this too can be tested in polynomial time.
So my obvious question is this: what are some examples of pairs of non-isomorphic groups G and H and functions $f:G\to H$ such that if you take 100 elements of G and their images in H and look at the subgroups they generate, you always get an isomorphism?
• November 12, 2011 5:22 am
Sorry — I misstated the “obvious question”. I mean of course that there should be a correspondence between the sequences of length 100 — getting it to come from a map f such as I described above would give you an isomorphism.
November 12, 2011 8:23 am
Tim (gowers),
I cool idea. The number 100 could be even a small constant times log n and still improve what we know. The issue of course is: can two groups be not isomorphic and stil have the same “subgroup” structure? I do not know the answer to this question? Will think about it, of course.
• November 12, 2011 10:16 am
I realize I’ve still not stated the question very well. The bipartite graph isn’t all that interesting, since it consists of an edge-disjoint union of complete bipartite subgraphs. So finding a matching doesn’t need any cleverness at all, and the entire point, as you basically say, is simply a generalization of counting how many elements there are of each order, which is what you’re doing if you take 1 instead of 100. Even taking just pairs of elements would be nice to know about, since that will trivially distinguish between Abelian and non-Abelian groups, so it seems to be much more powerful than counting orders.
• Klas Markström permalink
November 12, 2011 12:31 pm
Doesn’t this reduce to a question about the length of non-trivial relations in the two groups?
If the two groups differ only in that they each have some relation using more than 100 distinct group elements, and those relations are distinct from each other, it should be possible that the set of subgroups with 100 generators are equal for both groups.
I don’t have an explicit example at hand but I believe that it shouldn’t be too hard to produce one by constructing two groups in terms of their group presentations.
• November 15, 2011 12:04 pm
I agree with your second paragraph but the first sentence is misleading: it’s not the length of the relations that matters but the number of distinct generators involved.
Actually, I’m not quite sure that I agree with the second paragraph after all. It could be that every relation in one group holds in the other as well, but the number of times it holds is different for the two groups. That is what is tested for by the simple-minded algorithm I suggested. (For example, with k=1, the algorithm doesn’t just look to see which orders of elements occur, but also how many elements there are of each order.)
• Klas Markström permalink
November 15, 2011 1:23 pm
I basically agree. What I had in mind was only an example of a situation where the algorithm with a fixed value of k, e.g. 100, would fail to distinguish the two groups.
In general the algorithm will find all “short” relations between group elements, where short means that the number of distinct elements involved in the relation is at most k, and also count the number of times such relations appear.
One problem is that for many groups there will be distinct pairs, and other small tuples, of generators which actually generate the full group and for such pairs we are lead back to the problem of deciding if two groups of the same size are those we started with are isomorphic.
• November 16, 2011 7:15 am
That last problem isn’t a problem. If a group is generated by two elements (a,b) then we can test easily for isomorphism with another group by checking all possible pairs of elements (c,d) of the other group and seeing not just whether they generate isomorphic groups but whether the function from a to c and b to d extends to an isomorphism.
• November 16, 2011 7:24 am
That is true. This is is a simpler subcase of the isomorphism problem.
As Dick pointed our earlier this algorithm type has a reasonable running time for k = c log(n). Are there any good examples where the value of k needs to be polynomial in n in order to distinguish the two groups?
• November 16, 2011 10:36 am
That’s not quite what Dick pointed out. If $k=c\log n$ for a small constant $c$, then the algorithm does better than what is known, but is still superpolynomial. For it to be polynomial, then $k$ has to be bounded. It never needs to be more than $\log n$, since every group can be generated by $\log n$ elements.
• Klas Markström permalink
November 16, 2011 1:41 pm
Doing better than the best known algorithms would be a good first step, and probably easier than actually finding a polynomial time algorithm. So even a small value of c looks worthwhile to me.
You are right about never needing a c>1. I’m misleading myself by thinking about the possible sizes of the generated subgroups.
• November 13, 2011 3:10 pm
This seems relevant:
although my German is not quite up to it. “Lattice of subgroups” seems to be the right search term, and at any rate, two groups can have isomorphic lattices of subgroups (which implies they’ll pass the test here unless I misunderstand, which is quite possible?) without being isomorphic; indeed, one can be Abelian and the other not.
[How does one comment here without one’s comment languishing for days in “awaiting moderation”? Trying to see if FB login is better than what I did before…]
• November 15, 2011 12:02 pm
I don’t think it’s the same as having isomorphic lattices of subgroups, because we’re interested in what the subgroups are as well as what the containment relations are. In particular, if one group is Abelian and the other not, then k=2 is sufficient to distinguish them, because one group will contain a pair (x,y) such that xy doesn’t equal yx, and the other won’t. (What would be tested for is a 1-1 correspondence between ordered pairs of elements in such a way that if f(a,b)=(c,d) then the function that takes a to c and b to d extends to an isomorphism.)
• November 22, 2011 5:39 pm
Of course, sorry. Still, maybe this, from Max Horn writing at MathOverflow
http://mathoverflow.net/questions/35455/does-subgroup-structure-of-a-finite-group-characterize-isomorphism-type
is still interesting: “there are finite non-isomorphic groups G and H such that there exists a bijection between their elements which also induces a bijection between their subgroups.” Indeed the example given at the source has corresponding proper subgroups being isomorphic (that’s me saying that, not him, but seems easy. GAP has a LatticeSubgroups command btw, although that post didn’t use it.) The example here has G and H being different semidirect products of C5 with (C11xC11). I don’t know whether there could be an example which was a p-group.
• Jim Blair permalink
November 15, 2011 9:00 am
IMHO: A key to the ISO-Group problem may be the relationship between group isomorphism and graph isomorphism.
Why are graphs less structured and more amorphous than groups?
Is there a meta-language that captures the difference?
• Paolo Codenotti permalink
December 9, 2011 3:04 pm
The best known algorithm already takes advantage of the extra structure, namely generators. Indeed it gives a quasipolynomial time (n^log n) algorithm for group ISO, whereas the best algorithm for graph ISO is n^\sqrt{n}.
November 15, 2011 2:41 pm
I am most definitly not a pro at this and I do feel somewhat silly entering this conversation because of that, but my curiosity has become far greater than my timidity. I’m not too aware of the rules behind the use of superscript and subscripts in computation and informational display, but it does seem like there’s a great deal of unused potential there.
Taking graph Isomorphism, because I believe I understand the relationships in those problems better; assume two graphs with 4 vertices. The first is a tradidtional square, the second a triangle with the bottom edge split in half by the fourth vertex. G= V={1,2,3,4) E={{1,2},{1,4},{2,3},{3,4}} ; H= V={A,B,C,D} E={{A,B},{A,D},{B,C},{C,D}}. Using vertex 1 from graph G as an example, 1 has a degree of 2, so it would be 1sub(2), Then to try and establish the connections, add one of it’s connections as a superscript, so 1sub(2)sup(2). That would be enough information though, so you would want to add vertex 2’s degree value for 1sub(3)Sup(2sub(2)). From there you can add the next connection in the same way. 1sub(3)Sup(2sub(2))Sup(4sub(2))using the capline, midline, and baseline. Once this is done for all vertices of both graphs, the vertex labels can be changed to 1. This would make the previous example 1sub(3)Sup(1sub(2))Sup(1sub(2)). Call this a vertex relationship.
If the two graphs are Isomorphic, you should be able to use a set cover method by using all 4 vertex relationships from graph G as a set universe, and each vertex relationship, individually, from graph H as the problem’s sets. If there is a cover for the universe set, then the graphs are isomorphic.
I don’t know if this is an acceptable use of subscripts and superscripts and I’m not sure how this relates to group isomorphism, but I’m very curious as to both.
November 15, 2011 4:03 pm
Bah, typoed the example of the vertex relationships, and thats why you don’t read from paper experiments when typing up different examples.
It should be 1sub(2)Sup(2sub(2)), 1sub(2)sup(2sub(2))Sup(4sub(2)), and 1sub(2)sup(1sub(2))Sup(1sub(2)).
18. November 17, 2011 10:57 am
Here’s a question for group theorists. How many groups of order $p^n$ with at most 100 generators are there? I read that, being a bit rough about it, the number of groups of order $p^n$ is sort of like $p^{n^3}$. If the number generated by at most 100 elements is $m$, then the number of “subgroup profiles” (by “subgroup profile” I mean the multiset of subgroups generated by 100 elements) is at most the number of ways of writing $\binom{p^n}{100}$ as a sum of $m$ non-negative integers, which is at most $p^{100nm}$. So if $m$ is substantially smaller than $n^2$ for large $n$ (which feels a bit optimistic), then by a counting argument we get two non-isomorphic groups with the same subgroup profiles.
I don’t think this argument has any chance of working.
In fact, it doesn’t work, but let me at least record the failure here. Even Abelian groups are too numerous for this crude approach to go through. The number of Abelian groups of size $p^n$ with four generators is at least the number of ways of writing $n$ as a sum of four positive integers, which is definitely bigger than $n^2$.
So that suggests to me that a counterexample to my original question (to find two non-isomorphic groups with the same subgroup profiles) would have to be clever constructions rather than generic examples that arise from the fact that there are so many $p$-groups. I could have slipped up somewhere though.
19. Roy Maclean permalink
November 22, 2011 9:43 am
How about finding metrics to turn the set of groups of order n into a finite metric space M, such that d(G,H)=0 iff G is isomorphic to H, and then find polynomial-time algorithm to check d(G,H)<=k for some k, and then incrementally try to improve this upper-bound until k=0. If w is the diameter of M then you already know d(G,H)<=w.
"Finite Metric Spaces & Their Embeddings:Introduction and Basic Tools", Manor Mendel, CMI, Caltech, http://www.cs.caltech.edu/~schulman/Courses/0405cs286/manor-lec1.pdf
20. November 22, 2011 10:48 am
A vague and probably hopeless thought. Might there be a way of proving that if graph isomorphism is very hard then group isomorphism is at least quite hard? To do that, one could attempt to construct a group out of a graph in such a way that non-isomorphic graphs gave rise to non-isomorphic groups. Even if the groups were quite a bit larger than the graphs, a big lower bound for graphs could have non-trivial consequences for groups.
The difficulty with this approach is that the obvious ways of converting graphs into groups (such as, e.g. letting vertices be generators and associating relations with edges in some way) produce groups that are much too big, or even infinite. Getting something cleverly compressed looks as though it could be formidably difficult if it’s possible at all.
21. Jim Blair permalink
November 22, 2011 12:50 pm
I think you have identified a viable possibility.
Graph Theory already has a clever way of compressing things called “vertex reduction”.
One trick Graph Theorists do not seem to have exploited much is making mutually exclusive, simultaneous vertex reductions.
November 22, 2011 1:10 pm
Jim,
This is an interesting idea. I once used it in a completely way to color planar graphs. See this.
22. Jim Blair permalink
November 23, 2011 5:57 am
Dear Sir,
I am curious. Is there a word missing in this sentence?
“I once used it in a completely way to color planar graphs.”
The pointer to the paper was quite interesting. My strategy for five-coloring maps at the time that paper was written was to construct a large chain of two-colorable countries, view the group as a single country (a reduction group), and then three color the bordering countries.
One heuristic trick for linking chains with the same two colors that could not be reduced to a single country was to use four sided “bookkeeping” countries. It was one of my “Your Cheating Heart” algorithms.
The problem with bookkeeping countries is it gets very tricky keeping track of them and they “will tell on you”, if you get careless.
Back to the task at hand and following up on Tim Gowers’ idea:
Why not view the vertices that comprise a connected graph as a peer group of functions where f, g, and h are three vertices:
1. f(f) = 1, the identity function.
2. f(g) = 2, if the vertices g and h are connected by an edge.
3. f(h) = m, where m is the number of vertices in the reduction group that includes all the shortest paths between f and h.
If you and Tim find the idea amusing, I have a zillion tricks for implementing peer reduction groups on planar graphs that may be useful for any 2D surface with a finite genus.
It might be useful to begin with a limited number of vertices with an upper limit on the genus of the surface. It should greatly simplify things until we can get the algorithm up to a reasonable speed.
We can worry about 3D spaces later.
• Jim Blair permalink
November 23, 2011 6:03 am
Correction:
#2 should read: f(g)=2, if the vertices f and g are connected by an edge.
• Jim Blair permalink
November 24, 2011 7:54 am
Haste makes waste.
Never mind the peer group thing for now:
#1 is not right either.
23. November 24, 2011 5:17 am
Hi guys,
I know nothing about groups, and looking at multiplication tables for small groups. http://www.math.niu.edu/~beachy/aaol/grouptables1.html The observation is if one look at 2×2 minors only, and look only at minors containing 1 on the table diagonal, one can easily get generators and their multiplicities. The procedure will be for each minor with 1 on the table diagonal find the same minor, and list the rows and columns it appear as 4-tuple, than look at the chain of tuples, where the chain consist of connected tuples, e,g, tuples containing at least one common elements. The longest chains containing 1 are corresponding to generators. Moreover, I suspect that the relations between generators are also encoded in a relatively simple way in chains of 2×2 minors.
I guess the problem is not in finding generators, but in finding the relations between them.
24. Jim Blair permalink
November 25, 2011 12:31 pm
mktakov:
I think you just nailed the idea I was mangling:
“I guess the problem is not in finding generators, but in finding the relations between them.”
My guess is we might want to look at graph groups that generate a table with a diagonal whose value is a constant:
1. Let G(V,E) = Peer Group of functions = f1, f2, f3,……f(n-1)
2. All shortest distance matrix: fa(fa) = 0, where a = any integer.
3. Simplest reduction matrix: fa(fa) = 1, where a = any integer.
For example, a clique of five.
Shortest distance matrix:
01111
10111
11011
11101
11110
Simplest reduction matrix:
12222
21222
22122
22212
22221
If at first you don’t succeed, take a break and think about it some more.
• Jim Blair permalink
November 25, 2011 1:28 pm
Let’s keep things simple:
Peer Group of graph functions = (v1, v2, v3, ……. vn), where each element is a numbered vertex.
25. Jim Blair permalink
November 27, 2011 9:53 am
If you Google “magma(algebra)- Wikipedia”, you get a nice table that gives you an overview of groups and group like structures.
26. November 28, 2011 5:39 am
I have couple of questions.
Are relations (absolute relations) uniquely define group?
Can anyone deduce minimal relations based on multiplication table (reduce it) ? … based on subgroups structure? How hard is to compare 2 relations?
Those are basic questions related to the problem (accessible for wide audience to explore), and I suppose the answers should be known.
December 13, 2011 12:13 pm
Black box permutation group automorphism testing has a lower bound of checking all disjoint prime cycle decompositions. http://oeis.org/A186202
I would formulate the problem as follows: Given a list of permutations on N elements that generates A, and a list of permutations on N elements that generates B, what is the algorithmic complexity to determine if the group A is isomorphic to B under relabeling of the N elements?
December 13, 2011 4:34 pm
Good paper from SODA11, Code Equivalence and Group Isomorphism
http://www.siam.org/proceedings/soda/2011/SODA11_106_babail.pdf
• December 14, 2011 2:31 am
Chad, the problem you formulated is known as “permutational isomorphism of permutation groups.” In a follow-up to the paper you mentioned, Babai, Codenotti, and Qiao show that this problem can be solved in time polynomial in 2^N and the order of the groups. This significantly improves on the naive N! algorithm, enough that they can then use it to show that isomorphism of groups with no abelian normal subgroups can be determined in polynomial time. Their paper hasn’t quite appeared yet, but their results are in Paolo Codenotti’s thesis (http://www.ima.umn.edu/~paolo/thesis.pdf), and it looks like some will be presented at the Joint Math Meetings in January.
• December 15, 2011 5:15 am
That’s very interesting. Is there a naive explanation for why Abelian normal subgroups should make things difficult? I’m still interested in trying to understand what a pair of $p$-groups that “look the same” but are in fact different might conceivably be like, and I suppose that in principle this result should be providing an important clue.
• December 15, 2011 8:01 am
@gowers: in the case I’m describing below (when you can get two nonisomorphic groups with same profiles of 2-generated subgroups) the reason is that the abelian normal group is in fact central, and most of its elements can be only written as huge products of commutators of many elements. It’s intuitevely clear that if you take two such elements then quotienting anyof them out will lead to the same structure of groups which are generated by small number of elements.
• December 16, 2011 7:10 am
I’ll be interested to see your results about this — and even more so the promised simple explanation. Although it’s not absolutely ideal, what about simply writing it in LaTeX, posting a pdf somewhere, and linking to the pdf from a comment here?
Or perhaps it’s time for someone to set up a polymath wiki. It’s not quite clear to me whether there is a serious intention for this interesting discussion to develop into a sustained attack on the problem. Or perhaps this discussion-plus-private-work is a different style of polymathematical activity that would be worth experimenting with.
• December 16, 2011 9:24 am
As to existence of groups with same subgroup profiles – we have figured out yesterday that it’s even easier than we thought – no need for latex, as it’s just a counting argument. There’s a universal group of Phi-class 2 on n generators (see the paper by Higman) and not less than n generators. Call it E_n. Any other group of Phi-class 2 on n generators is an image of it. It follows that for given n there are finitely many (iso-classes of ) groups of Phi-class 2 generated by n elements. Call the number l(k). Now, Higman constructed roughly p^{n^3} groups of phi-class 2 and order p^n. And in such group there are only (p^n choose k)^l(k) possibilities for profiles of k-generated subgroups, which is a polynomial in p^n.
Explaining how quotients of the universal group E_n are classified is almost done in the Higman’s paper. Basically, the center of E_n is W= V \wedge V, where V is n-dimensional space over F_p. and the quotients are obtained by quotienting out a linear subspace of W. Then isomorphism of the quotients by subspaces A and B lift to isomorphism of E_n which which map A to B. And isomorphisms of E_n which act nontrivially on the center can be shown to correspond to isomoprhism of V, i.e. to GL(V). And it turns out that these automorphism act on W in the obvious way.
• December 16, 2011 9:27 am
Btw, to get groups with the same subgroups profiles one can be much more concrete – it’s possible to quotient out two very concrete 1-dimensional subspaces of W and get different quotients, with same k-profiles.
• December 16, 2011 9:37 am
Problem with polymath for an untenured postdoc like me is my greediness. I read the polymath rules yesterday and as they are written now, there’s no chance of getting a name on the publication. Don’t get me wrong – right now I definitely don’t think there’s anything worth publishing, but I’d like it if there was a possibility. On the other hand, having a dedicated place more friendly than here (latex-wise) to discuss the problem and exchange ideas would be good.
• December 16, 2011 9:40 am
Link to Higman’s paper: plms.oxfordjournals.org/content/s3-10/1/24.full.pdf the relvevant part is Theorem 2.1 and its proof.
December 13, 2011 1:01 pm
The Schreier–Sims algorithm is the most widely used algorithm/data structure for handling basic queries.
http://en.wikipedia.org/wiki/Schreier%E2%80%93Sims_algorithm
My protocol for gathering data would be to generate 100 MB worth of non-isomorphic Cayley tables, and cache their corresponding Schreier–Sims data structures. After that there are a few things you could try… To keep things simple I would start out deriving the optimal algorithm for isomorphism testing of inversion permutation groups (those having only disjoint 2-cycles as generators), then go after groups with both disjoint 2 and 3 cycles.
Conjecture (If false a counterexample would be interesting): Given all pairs of primes i and j; if for all subgroups of group A and group B generated by (disjoint) i and j cycles the groups are isomorphic, then A and B are isomorphic.
• December 15, 2011 5:31 am
I’ve been talking with Nik Nikolov about this problem, and the main outcome is that we can contruct groups with the same profiles of 2-generated subggroups (and probably also of k-generated subgroups, details to be checked). The idea is to study p-groups of Phi-class 2. These are the groups which Higman used to show that “most” p-groups are of class 2. They fit into an exact sequence A -> G ->B where A is the center, and both A and B are vector spaces over F_p.
Their classification problem can be reduced to the following problem. Given a k-dimensional vector space over F_p, form the wedge product W := V \wedge V. Input: subspaces A and B of W, given by their bases. Question: Are A and B in the same orbit of the action of GL(V) on W? Trivial algorithm takes roughly p^{n^2} steps (this is roughly how many invertible matrices there are). Having polynomial time for the group iso problem corresponds to having an algorithm for this problem which runs in poly(p^n) steps.
Both facts are not at all difficult, but I’d prefer to explain them somewhere I can use latex – maybe the polymath project? As to the need for a group theorist for this project, Nik Nikolov is with no doubt a strong group theorist, but I’m not sure he’d be really interested in it. Having said that, I’d probably convince him to have a look every now and then at the polymath project webpage :-).
• December 15, 2011 5:42 am
This was supposed to be a stand-alone comment, not a reply. Sorry for the confusion.
29. January 25, 2012 12:00 pm
This recent paper by Fabian Wagner http://www.uni-ulm.de/fileadmin/website_uni_ulm/iui.inst.190/Mitarbeiter/wagner/GroupIso_030711.pdf seems to do better than Tarjan’s algorithm. The algorithm rus in O(n^((c log n)/log log n)) some constant c.
He also seems to prove that for any fixed p, there’s a polynomial time algorithm for group isomorphism of p-groups.
• May 3, 2012 7:16 pm
Thanks for posting that link. It’s important to note that the paper you linked is incorrect as discussed here in the updated version: http://eccc.hpi-web.de/report/2011/052/
The revised version still seems to have a flaw but the result of p-group composition series isomorphism is still correct.
I also have a new paper (http://arxiv.org/abs/1205.0642) which builds on Wagner’s work to show a deterministic n^{(1 / 2) \log_p n + O(\log n / \log \log n)} algorithm for nilpotent-group isomorphism where p is the smallest prime dividing the order of the group. The 1 / 2 can be replaced with 1 / 4 using randomized algorithms and 1 / 6 using quantum algorithms.
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2018-02-17 21:43:29
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http://mathhelpforum.com/algebra/203378-prove-using-simultaneous-equation.html
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# Thread: prove using simultaneous equation?
1. ## prove using simultaneous equation?
Having trouble with a prove, I'm not sure where to put this question.
2. ## Re: prove using simultaneous equation?
y = b/2x
Substitute it in the first equation and simplify. Then solve the quadratics to get x^2.
3. ## Re: prove using simultaneous equation?
Originally Posted by sa-ri-ga-ma
y = b/2x
Substitute it in the first equation and simplify. Then solve the quadratics to get x^2.
hi, thanks. I got that part, I'm just not sure how to simplify it.
4. ## Re: prove using simultaneous equation?
x^2 - (b / (2x))^2 - a = 0
4x^4 - 4ax^2 - b^2 = 0
Let u = x^2
4u^2 - 4au - b^2 = 0
Can you finish it?
5. ## Re: prove using simultaneous equation?
Originally Posted by Wilmer
x^2 - (b / (2x))^2 - a = 0
4x^4 - 4ax^2 - b^2 = 0
Let u = x^2
4u^2 - 4au - b^2 = 0
Can you finish it?
thanks, I think I still am having a bit of trouble with it. Do I have to complete the square or?
6. ## Re: prove using simultaneous equation?
Originally Posted by chessweicong
thanks, I think I still am having a bit of trouble with it. Do I have to complete the square or?
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2016-09-30 15:19:33
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http://intellectualmathematics.com/blog/a-criterion-for-deciding-if-something-is-worth-teaching/
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# A criterion for deciding if something is worth teaching, illustrated with examples from Calculus I
Don’t teach things that don’t serve a purpose. Or to put it differently: Pick up a calculus textbook and open an arbitrary section. Look at the problems at the end of the section and ask yourself: Is there any reason to want to know the answer to these problems? Are the problems inherently interesting, and the substance of the section a means to answering them? Then this is a meaningful topic and it should be taught. Or are the problems artificially concocted for the sole purpose of testing you on the material just introduced? Then it’s a crap topic and should not be taught. Students have no reason to work on such problems except subservience to the instructor, and therefore their effect is to suppress independent thought.
If we grant this as an axiom of teaching, it follows that traditional Calculus I courses must be reformed as follows.
Cut limits. Limits are a hotbed of pseudo-problems that do not serve any meaningful purpose. Monstrous fractions involving $\tan ^3 ( \ln x )$, double substitutions, triple l’Hôpital’s rule problems and god knows what else: these problems are fake, fake, fake. They are not taught because they are genuinely needed to address genuinely interesting questions. They are made up for the sake of making up drill problems. They create unreflective and slavishly rule-following students, because any student who thinks for himself will immediately come to the conclusion that the class is a meaningless drill with no purpose.
Cut proto-“real analysis.” To the above it will be objected that limits caters to thinking students in that it addresses why the calculus works. So it is alleged, but it is not true. First of all the mass of drill problems assigned have absolutely nothing to do with such why-questions, so this false motivation is a dishonest bait-and-switch. Furthermore the why-questions at hand can be understood perfectly well without the pretentious machinery of limits. This is proved by history, where all leading mathematicians for over a hundred years understood the calculus perfectly without ever bothering with limits.
If you are serious about addressing why-questions you start with why-questions and develop the theory needed to answer them. It then soon becomes apparent that a few basic and intuitive notions are enough to deal with the matter to everyone’s satisfaction, which is indeed precisely what happened historically.
The charade continues with supposedly “conceptual” questions about a barrage of artificially contrived, piecewise defined functions pock-marked with a plethora of discontinuities and medley of different types of non-differentiability. A strange form of unconscious communism seems to be at the bottom of this approach: if “all functions are equal” then indeed it makes sense to give these fake examples as much screen time as the sine and the logarithm. But anyone who starts with interesting and meaningful questions, instead of making up problems to fit their preconceptions of what they have to teach, it is evident enough that some functions are more interesting than others, and that wasting half a calculus course nitpicking about exceptional qualities of the most artificial ones is a pointless exercise in pedantry.
To pretend that these kinds of things are needed to understand “the foundations” of basic calculus is a lie. I suspect most students realise as much, in their guts if not fully consciously.
Limit theory, exceptional functions, and real analysis rose together in the 19th century, more than a hundred years after the calculus had already flourished and produced everything a calculus student has any reason to be interested in. To be sure, all these things served a meaningful purpose then. The 19th-century conception of the foundations of the calculus is profound and important and served to resolve important enigmas that had arisen in the meantime. Of course I do not belittle or deny the value of such investigations in any way––on the contrary I would love to teach them in a real analysis class––but I also observe that they have nothing to do with basic calculus.
Real analysis is a wonderful subject, and calculus is a wonderful subject, but they are two completely different subjects. Don’t try to mix them in some Frankensteinian fashion, and most of all don’t lie to your students and pretend that analysis is the thinking student’s calculus. In reality it is the opposite: only the gullible, subservient student buys this bogus myth.
Include differential equations. The usual battery of integration techniques are usually accompanied by “$\tan ^3 ( \ln x )$”-style problems, much like the limit sections attacked above. Should they therefore be committed to the flames also? Not at all. The situation could not be more different, although students (and perhaps not a few teachers) in a traditional calculus course wouldn’t know it. Unlike the nonsense real analysis material artificially shoehorned into Calculus I, integration techniques do serve a very credible purpose that is very easily made evident to students. It only takes one simple reshuffling of the order of the topics: teach differential equations as early as possible, as I do in my book. This simple recipe at once changes the entire nature of drill problems on integration techniques. Without differential equations the students will conclude, with good justification, that these problems are nothing but a cruel obstacle course with no purpose. But with differential equations the student cannot draw the same conclusion without denying the value of studying population dynamics, the motion of rockets and projectiles and planets, and a thousand other fascinating and useful things besides. Hence we must either teach differential equations in Calculus I, or accept widespread hatred of mathematics as a rational outcome of our own doing.
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2018-03-18 02:17:46
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https://www.nature.com/articles/s41598-018-20061-5?error=cookies_not_supported&code=a79686e3-7a1f-4737-877a-bc8db84677ad
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Article | Open | Published:
# Geophysical investigations unravel the vestiges of ancient meandering channels and their dynamics in tidal landscapes
Scientific Reportsvolume 8, Article number: 1708 (2018) | Download Citation
## Abstract
Whether or not one can detect relict signatures of the past imprinted in current landscapes is a question of the utmost theoretical and practical relevance for meandering tidal channels, owing to their influence on the morphodynamic evolution of tidal landscapes, a critically fragile environment, especially in face of expected climatic changes. Unravelling the sedimentary patterns of ancient channels is an expensive process that usually requires high resolution sediment coring. Here we use a novel inversion process of multi-frequency electromagnetic measurements to reveal the signature and characterize the dynamics of a salt-marsh paleo-meander in the Venice Lagoon. We show that the ancient meander migrated laterally while vertically aggrading, developing a peculiar bar geometry which is less common in analogous fluvial meanders. The observed point-bar dynamics and the associated architectural geometry are consistent with remote sensing and borehole data and contrast with current assessments of tidal meander morphodynamics mediated from classical fluvial theories. In addition, the proposed technique, rapid and non-invasive, bears important consequences for detecting buried stratal geometries and reconstructing the spatial distribution of ancient sedimentary bodies, providing quantitative data for the description of landscape evolution in time.
## Introduction
Branching and meandering tidal channel networks cut through salt-marsh landscapes and drive the exchange of water, nutrients and sediments within these landscapes. Providing preferential pathways for marsh flooding and drainage during the tidal cycle, tidal channels exert a primary control on the ecogeomorphological evolution of salt-marsh systems e.g.1,2. The dynamics of salt-marsh platforms and of their channel networks are tightly intertwined e.g.3,4,5,6 while the channels drain and feed the marsh during the tidal cycle, governing water and sediment fluxes over the platform, the elevation of the marsh in the tidal frame, together with its extension, controls channel growth, maintenance, size and evolution7,8,9,10,11. Despite their importance in landscape evolution, tidal networks have received less attention than fluvial ones12 particularly in terms of the main morphometric characteristics of tidal meanders that are commonly studied following theories developed for their fluvial counterparts13,14,15,16,17. The internal structure of tidal point bars has received even less attention, attempts to address such an issue being indeed mainly based on similarities with fluvial meanders e.g.18,19. Improving current knowledge of tidal meander dynamics, and of the related sedimentary products, becomes an essential step to understand and predict tidal landscape morphodynamic evolution. Bar expansion, obstructed channels, outer–bank erosion and relocation of past meander bends can explain fining or coarsening-upward vertical grain-size trends, erosional surfaces and current salt-marsh topography e.g.20. The reconstruction of paleo-meander patterns can explain these morphological changes, and provide a significant contribute to reconstruct the depositional dynamics of paleochannels21.
The scientific issue above calls for solid evidence to develop detailed depositional models for tidal meander bends and related sedimentary products.
Several remote sensing techniques have widely been applied to map meander evolution in both tidal15,22 and fluvial23,24,25 landscapes. In this framework even a simple time-lapse aerial photography database, can supply relevant information on meander migration, especially if it spans a long time period23,25,26. Nevertheless, remote sensing techniques must be confirmed by ground-truthing, requiring detailed datasets of invasive, ancillary subsoil investigations27,28. A step forward is clearly needed.
Spatially extensive geophysical surveys give a fundamental contribution towards the characterization of the shallow subsoil and its heterogeneities, that can be related to geomorphological evolutions and different sedimentation processes in tidal landscapes29,30. A possible approach, both fast and accurate, is provided by electromagnetic techniques that allow one to perform contactless measurements31.
Here we present an innovative method to unravel the footprints of ancient meander bends in salt-marsh platforms and we test this new approach by recovering sedimentary cores that demonstrate that imaged sedimentary features are consistent with both observed deposits and recently established depositional models20. We show that the use a multi-frequency conductivity meter and the application of an innovative inversion technique of multi-frequency electromagnetic data (see Methods) allow one to unravel the geometry of a buried tidal point bar and related channel, which cut through a salt-marsh platform in the Venice lagoon (Italy), where saline soils are extremely conductive, thus representing a very challenging environment (in terms of discerning electrical conductivity spatial variations). Note that the presence of this paleo-meander is clearly shown by historical aerial photographs, whereas sediment accumulation over the past few decades has totally smoothed the marsh topography, covering direct morphological evidence of this channel. We also show how the three-dimensional geophysical data add critical information to the understanding of sedimentary processes in tidal landscapes. The integration between geophysical data, remote sensing images, and ancillary sedimentological investigations shows that salt-marsh aggradation plays a key role in developing geometries of tidal point bars. We present the peculiar morphodynamic behavior of tidal meanders, unusual for their fluvial counterparts, that is functionally intertwined to the evolution of the adjacent platform. We provide evidence that a multi-frequency electro-magnetic data inversion technique is a viable tool that can be used to study tidal meander evolution as well as the evolution of similar depositional environments. In addition, our findings challenge the possibility of applying fluvial meander theories to the study of their tidal fellows.
## The study site
The Venice Lagoon, which formed over the last 7500 years covering alluvial Late Pleistocene deposits32, is the largest Mediterranean brackish water body, with an area of about 550 km2 (Fig. 1a). The Venice Lagoon is subjected to a semidiurnal tidal regime, with an average tidal range of about 1.0 m and peak tidal amplitudes of about 0.75 m around Mean Sea Level (MSL). It is connected to the Adriatic Sea through three inlets: Lido, Malamocco, and Chioggia. The study site is located in the Northern part of the lagoon (Fig. 1a), which hosts the best naturally preserved network of meandering channels. The evolution of this network has been characterized by a few stream piracies, that caused the abandonment of channel reaches and their consequent burying under salt-marsh deposits.
We selected a study area where historical aerial photographs (e.g. the 1968 photo) show remnants of old tidal meanders (Fig. 1b), which used to drain the marsh platform in the past. Moreover the study area is close to a recently investigated site, where extended core drilling showed that the morphodynamic evolution of a tidal meander was affected by the aggradation of the surrounding salt-marsh platform20. Specifically, as it emerges from the 1968 aerial photo, the surveyed study site is characterized by arcuate lineations (hereafter “scroll-like”features) resembling different growth stages of a meander bend that expanded toward SE (Fig. 1b). At present, the site is covered by a dense halophytic vegetation (Fig. 1b), whose current zonation (Fig. 1a second panel), together with the small differences in marsh elevations (Fig. 1a third panel), suggest the presence of these paleo-meanders, which are not clearly visible when standing on the marsh platform. At this site we conducted a spatially dense, multi-frequency electromagnetic (EM) survey covering an area of approximately 70 m × 35 m.
## Results
The inversion of the acquired multi-frequency electromagnetic data allowed us to define the 3D stratal architecture of the site, that is shown in Fig. 2 where four depth slices are shown. Note that data are missing in correspondence of active, deep salt-marsh channels, where the acquisition could not be performed. In addition, all slices deeper than 1.0 m show essentially no spatial variation in electrical conductivity and a stable value around 2 S/m. In the region shallower than 1 m, both high- (>2 S/m, in blue) and low- (<2 S/m, in brown) conductivity regions are clearly marked. These analyses allow us to clearly discriminate a higher electrical conductivity domain (values > 3 S/m) from a lower conductivity domain. In the slice at 0.60 m depth, the distribution of the high-conductivity domain defines an ESE-WNW trending lobe, that increases in size at a depth of 0.36 m, and is prevalent at the ground surface (0.0 m). At 0.36 m depth, the Southern part of the study area, facing the deeper channels, is characterized by lower electrical conductivity values (brown areas), due to the presence of silty-sandy sediments spread from the main channel onto the marsh surface during flood events.
The distribution of the electrical conductivity values at different depths reflects the spatial orientation of scroll-like features that are visible from historical aerial photos (Fig. 1b). Drilling core data (Fig. 2) confirm the presence of the two different domains visualized by the EM survey, showing that the high- and low- conductivity domains are associated with salt-marsh silty-mud and point-bar sandy deposits, respectively. The spatial correlation between conductivity maps highlights that salt-marsh deposits reach their maximum thickness (~1.0 m) in the NNW sector of the surveyed area and that their basal surface is characterized by a “spoon-shaped” geometry with a SSE-NNW trending axis, as shown in Fig. 3.
## Discussion
Our study demonstrates the applicability and value of non-contact EM surveys in defining subsurface geometries even in electrically conductive environments such as salt-marsh systems, where the sensitivity of the measurement is stretched to the limit of detecting subtle spatial variations. Here the detection of subtle electrical conductivity differences, around the high values expected due to the prevailing presence of brackish waters, made the target particularly challenging. Only an algorithm capable of going beyond the low induction number approximation33 allowed us to draw the necessary conclusions in this environment. The results are - informative for the geometrical reconstruction of ancient tidal deposits, and thus allow us to detect the remnants of buried channels and cast light back in time on the past sedimentary processes. The integration between EM and sedimentological borehole data reveals that point-bar deposits associated with the studied paleomeander are now buried by salt-marsh silt ranging in thickness between 0.3 and 1.0 m. In particular, the higher electrical conductivity domain presenting values > 3 S/m (Fig. 2) can be related to the silty salt marsh deposits, while the lower conductivity values are representative of the more sandy point bar body (Tab.1 and Fig. 3). In these terms, the non-invasive electrical characterization of the buried meander structure suggests that the bar-top, salt-marsh deposits are characterized by a peculiar spoon-shape geometry, which is both validated by testing boreholes and consistent with recently proposed depositional models developed in similar bar deposits20, which, on the other hand, required recovering and sedimentological analysis of 150 cores. The subtle changes of electrical properties detected through the proposed model, are, therefore, representative of different sediment depositions, as confirmed by the drilled boreholes. The deposition of fine-grained sediments on point bar top reflects the morphodynamic evolution of channel bends, being they fluvial or tidal, and follows lateral migration/shifting of the main channel. Once salt-marsh deposition is triggered on the bar top, its aggradation is governed by the mutual role of the rate of relative sea level rise and of the deposition rate of organic and inorganic sediments34. At equilibrium conditions, as in the San Felice area35,36,37, the salt-marsh surface aggrades vertically keeping pace with the rate of relative sea-level rise (estimated to be about 3.0 mm/yr31). This implies that lateral migration of meanders cutting through the salt marsh occurs together with the progressive rise of bar top. We argue here that the effects of channel lateral migration under aggradational conditions are reflected by the geometry of the surface bounding the base of the salt-marsh deposits (Fig. 3). The lateral shift of the inner bank defines, through time, a concave-up rising trajectory (that we have referred to as “spoon shaped”) that depicts the basal surface of salt-marsh deposits. Along a cross section cut along the bar axis (Fig. 3), this concave-upward surface defines a rising trajectory, the steepness of which is controlled by the ratio between the rate of vertical aggradation and that of lateral migration. Meanders developed in the San Felice area show an average migration rate of about 10 cm/yr 1438,39, consistent with the few data available for other salt-marsh settings e.g.22,26. Aggradation rates of a few millimeters per year are therefore large enough to produce a detectable geometrical expression. To our knowledge, analogous processes and meander dynamics have never been documented in the fluvial realm. Indeed, similar features and behaviors can hardly be appreciable in the fluvial setting, where lateral migration rates can be more than one order of magnitude larger than those documented for salt-marsh meanders20,22,26,38,39, and only aggradation rates higher than 1.0 cm/yr are suggested to influence depositional dynamics34.
The San Felice area was characterized, during the past centuries, by an average salt marsh aggradation rate of 2.0–3.0 mm/yr35,36. The thickness distribution of salt marsh deposits above the point bar highlights that they thin from 1.0 to 0.3 m in the direction of bar expansion, suggesting that the channel migrated over a distance of about 40 m, between 400 and 120 yrs BP, if we consider an average accretion rate of about 2.5 mm/yr. The resulting migration rate of about 10 cm/yr is therefore consistent with that obtained for the salt-marsh meanders of the San Felice area38,39, for tidal-flat meanders of the Venice Lagoon40 and for salt-marsh channels in the San Francisco Bay22, and New Jersey26. The minimum thickness (0.3 m) of salt-marsh deposits indicates that since 120 yrs BP vertical aggradation started draping and flattening the pre-existing topography.
## Conclusions
A multi-frequency EM inversion approach was applied to detect relict signatures of a buried tidal meander in a salt marsh of the Northern Venice lagoon. The use of multi-frequency simultaneous acquisitions and of a robust inversion process, able to span the range of moderate induction numbers, provided detailed conductivity maps of the first subsoil even in such a challenging high-conductivity salty environment. The results of the EM survey were tested against data obtained from boreholes and remote images to reconstruct the sedimentary bodies developed during meander evolution. This is a notable achievement per se, providing a new tool for making new advance possible in different geoscience disciplines in saline environments. Data integration shows that vertical aggradation can influence the geometry and dynamics of sedimentary bodies generated during meander migration, leading to a progressive thickening of the point bar body during meander expansion, as recently pointed out in similar environment14. Changes in the thickness of salt marsh deposits accumulated on point bar top during meander evolution suggest that the channel migrated laterally with a rate of about 10 cm/yr, while vertically aggrading with a rate of about 2.5 mm/yr. The lateral migration rate is consistent with migration rates estimated for tidal-flat and salt-marsh channels in the Venice Lagoon32,35, San Francisco Bay16 and New Jersey20 wetlands. The emerging spoon-shaped geometry of the surface bounding the base of the salt-marsh deposits, which was observed in the Venice salt-marshes and has never been documented in fluvial cases, shed a new light in understanding tidal meander morphodynamics.
The use of fast, non-invasive techniques can therefore significantly help the evolutionary interpretation of the marsh sub-environment, even though the support of a limited number of ground punctual direct investigations, such as boreholes, is required.
## Methods
The field electromagnetic (EM) survey was conducted in November 2014. We collected the data using a Geophex GEM-2 conductivity-meter. The instrument has fixed transmitter-receiver coil separation of 1.66 m, operating with a multi-frequency acquisition in the bandwidth between 330 Hz to 48 kHz. The selection of frequencies depends on the sought depth of investigation and operates in the range of moderate induction number40. It must be emphasized that in salt marshes actual low induction number (LIN) conditions cannot be ensured, due to the highly electrically conductive environment. We used six frequencies (775 Hz, 1175 Hz, Hz, 9,825 Hz, 21,725 Hz, and 47,025 Hz) to span the range of induction numbers between 0.0918 and 0.715. Both quadrature (Q) and in-phase (I) components of the secondary magnetic field were recorded at roughly 5,000 measurement points across the survey area. The instrument was carried in the vertical-dipole configuration at the height of about 1.0 m above the ground. The conductivity meter was equipped with a differential GPS receiver that ensured sub-meter accuracy positioning for each measurement point.
Raw data were preliminary analyzed for detecting possible DC (static) shifts, outliers, and short wavelength noises, which usually adversely affect the quality of the inversions. The data show high sensitivity both responses in phase and quadrature vary over orders of magnitude in the selected frequency range. Note that this is peculiar to applications of multi-frequency EM instruments in presence of high electrical conductivities, as is the case in a salt-marsh environment. The adopted methodology is therefore very suitable for the problem at hand. However, the examination of the raw data reveal also unexpected features of the electromagnetic response. The in-phase (I) component of the three lowest frequencies turned out to be negative all over the area suggesting the presence of magnetic susceptible materials41 that were not expected. The quadrature (Q) response often presents a non-monotonic dependence with frequency. As debated in the literature42, this is usually not an issue for resistive soils, but can lead to serious non-uniqueness problems in the inversion process for highly conductive soils in the intermediate induction number range33. In this study we therefore used for inversion only the in-phase (I) component.
The filtered data were first inverted to produce a 1D electrical conductivity profile below each measurement point. Then, these 1D models were set side by side to build a pseudo-3D volume of the investigated area. The 1D inversions were performed using the regularized nonlinear inversion algorithm by Deidda et al.43. This approach, originally designed to invert the quadrature component measured at different heights, using a device operating at a fixed frequency, was here modified to handle multi-frequency in-phase component data44. The forward modeling was derived from Maxwell’s equations, introducing suitable simplifications based on the geometry of the instrumentation. When the axes of the coils are aligned vertically to the ground, the ratio of the secondary EM field H s with respect to the primary EM field H p can be expressed in terms of a Hankel transforms of order zero:
$$\frac{{H}_{S}}{{H}_{P}}={\int }_{0}^{\infty }{\lambda }^{2}{e}^{-2h\lambda }{R}_{0}(\lambda ){J}_{0}(r\lambda )d\lambda$$
(1)
where h is the height of the instrument above the ground, r the coil separation, J0 the Bessel function of order 0. The kernel R0 (λ) is a complex value function of the parameters that describe the layered subsurface properties, i.e. for the k-th layer: the electrical conductivity σ k , the magnetic permeability μ k and the layer thickness d k .
The inversion originally proposed by Deidda et al.43 implements various methods for the automatic estimation of the regularization parameter. The inversion algorithm is designed to operate in a highly conductive setting, and is particularly fast compared to others45, since Deidda et al.43 propose the analytical computation of the Jacobian (sensitivity matrix), instead of approximating it by finite differences. Such a procedure makes the inversion more than ten times faster. In the inversion process we adopted a layered model consisting of 30 layers to a maximum depth of 3.5 m. To reproduce the negative in-phase data measured at this site, the relative magnetic permeability (μ r = μ k /μ0, where μ0 is the magnetic permeability of the vacuum) was set μ r = 1 at depths less than 0.5 m, μ r = 1.14 between 0.5 m and 1 m, μ r = 1.18 between 1 m and 3 m, and μ r = 1.3 below 3 m. This increase in magnetic permeability with depth is compatible with the anoxic conditions present in the salt-marsh landscape46,47. Note that when the magnetic permeability for each layer is fixed, the conductivities σ k are the only unknowns of the problem. The results emerging from the geophysical data have been tested against direct evidence from a few cores collected on site, as discussed in the main text48,49,50,51,52.
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Díaz De Alba, P. & Rodriguez, G. Regularized inversion of multi-frequency EM data in geophysical applications. In Trends in Differential Equations and Applications (eds Ortegón Gallego, F., Redondo Neble, M. V., & Rodríguez Galván, J. R.), SEMA SIMAI Springer Series 8, 357–369 (2016).
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Schultz, G. & Ruppel, C. Inversion of inductive electromagnetic data in highly conductive terrains. Geophysics 70(1), 16–28, https://doi.org/10.1190/1.1852775 (2005).
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Da Silva, A. C., et al. (Eds), Magnetic Susceptibility Application: A Window onto Ancient Environments and Climatic Variations. Geological Society Special Publication, ISBN 978-1862397217 (2015).
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Kent, D. V., Rio, D., Massari, F., Kukl, G. & Lancia, L. Emergence of Venice during the Pleistocene. Quaternary Science Reviews 21, 1719–1727, https://doi.org/10.1016/S0277-3791(01)00153-6 (2002).
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Moffett, K. B., Robinson, D. A. & Gorelick, S. M. Relationship of salt marsh vegetation zonation to spatial patterns in soil moisture, salinity and topography. Ecosystems 13, 1287–1302, https://doi.org/10.1007/s10021-010-9385-7 (2010).
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Lausch, A. et al. Analysis of Vegetation and Soil Patterns using Hyperspectral Remote Sensing, EMI, and Gamma-Ray Measurements, Vadose Zone Journal 11, https://doi.org/10.2136/vzj2012.0217 (2013).
50. 50.
Carbognin, L., Teatini, P. & Tosi, L. Eustacy and land subsidence in the Venice lagoon at the beginning of the new millennium. J. Mar. Syst. 51, 345–353, https://doi.org/10.1016/j.jmarsys.2004.05.021 (2004).
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Van De, Lageweg et al. Preservation of meandering river channels in uniformly aggrading channel belts, Sedimentology, https://doi.org/10.1111/sed.12229 (2015).
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McClennen C. E. & Housley R. A. Late-Holocene channel meander migration and mudflat accumulation rates, Lagoon of Venice, Italy, Journal of Coastal Research, 930–945. https://doi.org/10.2112/03-0113.1 (2006).
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Marani, M., Zillio, T., Belluco, E., Silvestri, S. & Maritan, A. Non-Neutral Vegetation Dynamics. PLoS ONE 1(1), e78, https://doi.org/10.1371/journal.pone.0000078 (2006).
## Acknowledgements
The authors acknowledge partial funding from the EU FP7 Collaborative Projects GLOBAQUA “Managing the effects of multiple stressors on aquatic ecosystems under water scarcity”, the CARIPARO Foundation Project PASTtoFUTURE “Reading signatures of the past to predict the future: 1000 years of stratigraphic record as a key for the future of the Venice Lagoon”, Shell International Exploration and Production within the project “Tidal vs. tidally-influenced fluvial point bars: facies distribution and implications for reservoirs production development” and the project SID 2016 (Padova University) “From channels to rock record: morphodynamic evolution of tidal meanders and related sedimentary products. (prot. BIRD168939) and the PRAT2014 project of the University of Padua titled “Hydro-geophysical monitoring and modelling for the Earth’s Critical Zone” (prot. prot. CPDA147114). The authors also thank Lara Brivio and Alvise Finotello for their relevant support. The research of G.R. was partially supported by INdAM-GNCS. Data are available in the supporting information.
## Author information
### Affiliations
1. #### Dipartimento di Geoscienze, Università di Padova, Padua, Italy
• Jacopo Boaga
• , Massimiliano Ghinassi
• , Andrea D’Alpaos
• & Giorgio Cassiani
2. #### Dipartimento di Ingegneria Civile, Ambientale e Architettura, Università di Cagliari, Cagliari, Italy
• G. P. Deidda
3. #### Dipartimento di Matematica e Informatica, Università di Cagliari, Cagliari, Italy
• G. Rodriguez
### Contributions
J.B., M.G., A.D. and G.C. designed the study; G.C. and J.B. developed the geophysical approach; G.P.D. and G.R. performed the geophysical inversion; A.D. and M.G. performed geomorphological and sedimentological analyses; all authors contributed to the drafting of the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Giorgio Cassiani.
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2019-01-18 22:29:49
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https://www.semanticscholar.org/paper/Certifying-Confidence-via-Randomized-Smoothing-Kumar-Levine/06aaece45f8284de309d4d9d8772305fb848a66d
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• Corpus ID: 221761375
Certifying Confidence via Randomized Smoothing
@article{Kumar2020CertifyingCV,
title={Certifying Confidence via Randomized Smoothing},
author={Aounon Kumar and Alexander Levine and Soheil Feizi and Tom Goldstein},
journal={ArXiv},
year={2020},
volume={abs/2009.08061}
}
• Published 17 September 2020
• Computer Science
• ArXiv
Randomized smoothing has been shown to provide good certified-robustness guarantees for high-dimensional classification problems. It uses the probabilities of predicting the top two most-likely classes around an input point under a smoothing distribution to generate a certified radius for a classifier's prediction. However, most smoothing methods do not give us any information about the \emph{confidence} with which the underlying classifier (e.g., deep neural network) makes a prediction. In…
23 Citations
Figures from this paper
• Computer Science, Mathematics
ArXiv
• 2021
This work designs a smoothing procedure that can leverage the local, potentially low-dimensional, behaviour of the function around an input to obtain probabilistic robustness certificates and demonstrates the effectiveness of the method on multiple learning tasks involving vector-valued functions with a wide range of input and output dimensionalities.
• Computer Science
ICML
• 2021
This work presents the first study of certifiable robustness for DBU models, and proposes novel uncertainty attacks that fool models into assigning high confidence to OOD data and low confidence to ID data, respectively.
• Computer Science
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• 2021
This paper proposes average- and worst-case metrics to measure flatness in the robust loss landscape and shows a correlation between good robust generalization and flatness, i.e., whether robust loss changes significantly when perturbing weights.
• Computer Science, Mathematics
NeurIPS
• 2020
This work obtains the first model-agnostic, training-free, and certified defense for object detection against $\ell_2$-bounded attacks.
• Computer Science
ArXiv
• 2021
An adaptive version of the Neyman-Pearson Lemma – a key lemma for smoothing-based certificates – where the adversarial perturbation at a particular time can be a stochastic function of current and previous observations and states as well as previous actions is proved.
• Computer Science, Mathematics
ICML
• 2022
Theoretically, under mild assumptions, it is proved that DSRS can certify Θ( √ d ) robust radius under ℓ 2 norm where d is the input dimension, implying thatDSRS may be able to break the curse of dimensionality of randomized smoothing.
• Computer Science
• 2022
The lightweight Diversity Regularized Training (DRT) is proposed to train certifiably robust ensemble ML models and it is proved that an ensemble model can always achieve higher certi fied robustness than a single base model under mild conditions.
• Computer Science
ICLR
• 2022
The lightweight Diversity Regularized Training (DRT) is proposed to train certified robust ensemble ML models and it is proved that an ensemble model can always achieve higher certification robustness than a single base model under mild conditions.
• Computer Science
ArXiv
• 2022
A simple training method leveraging the fundamental trade-off between accuracy and (adversar- ial) robustness to obtain robust smoothed classifiers, in particular, through a sample-wise control of robustness over the training samples.
• Computer Science
ArXiv
• 2022
A framework that uses the dense intrinsic constraints in natural images to robustify inference to shift the burden of robustness from training to the inference algorithm, thereby allowing the model to adjust dynamically to each individual image’s unique and potentially novel characteristics at inference time.
References
SHOWING 1-10 OF 50 REFERENCES
• Computer Science
ICML
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Strong empirical results suggest that randomized smoothing is a promising direction for future research into adversarially robust classification on smaller-scale datasets where competing approaches to certified $\ell_2$ robustness are viable, smoothing delivers higher certified accuracies.
• Computer Science, Mathematics
J. Mach. Learn. Res.
• 2020
Any noise distribution D over R that provides `p robustness for all base classifiers with p > 2 must satisfy E η i = Ω(d1−2/p (1− δ)/δ) for 99% of the features of vector η ∼ D, where is the robust radius and δ is the score gap between the highest-scored class and the runner-up.
• Computer Science, Mathematics
ICML
• 2020
It is shown that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensional regime, and it is established that Gaussian smoothing provides the best possible results, up to a constant factor, when p \geq 2.
• Computer Science, Mathematics
NeurIPS
• 2019
This work offers adversarial robustness guarantees and associated algorithms for the discrete case where the adversary is $\ell_0$ bounded and exemplifies how the guarantees can be tightened with specific assumptions about the function class of the classifier such as a decision tree.
• Computer Science
AAAI
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This paper proposes an efficient and certifiably robust defense against sparse adversarial attacks by randomly ablating input features, rather than using additive noise, and empirically demonstrates that the classifier is highly robust to modern sparse adversarian attacks on MNIST.
• Computer Science
ArXiv
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This paper proposes attack-agnostic robustness certificates for a multi-label classification problem using a deep ReLU network that has a closed-form, is differentiable and is an order of magnitude faster to compute than the existing methods even for deep networks.
• Computer Science
ArXiv
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This work shows how a simple bounding technique, interval bound propagation (IBP), can be exploited to train large provably robust neural networks that beat the state-of-the-art in verified accuracy and allows the largest model to be verified beyond vacuous bounds on a downscaled version of ImageNet.
• Computer Science, Mathematics
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It is shown that with only label statistics under random input perturbations, randomized smoothing cannot achieve nontrivial certified accuracy against perturbation of $\ell_p$-norm $\Omega(\min(1, d^{\frac{1}{p} - 1}{2}}))$, when the input dimension $d$ is large.
• Computer Science
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This paper shows that if the eigenvalues of the Hessian of the network are bounded, the authors can compute a robustness certificate in the $l_2$ norm efficiently using convex optimization and derives a computationally-efficient differentiable upper bound on the curvature of a deep network.
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Experiments show that the predictor-verifier architecture able to train networks to achieve state of the art verified robustness to adversarial examples with much shorter training times can be scaled to produce the first known verifiably robust networks for CIFAR-10.
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2023-02-02 08:39:35
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https://jemmaths.wordpress.com/2017/06/06/algebra-you-use-the-letter-x-more-than-you-ever-have-done-in-your-whole-life/
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# Algebra: you use the letter ‘x’ more than you ever have done in your whole life!
I started my first algebra unit with Year 7 on Monday. It’s the essentials of algebraic manipulation: adding, subtracting, multiplying and dividing terms, including those with indices, as well as expanding and factorising single brackets, all pretty standard beginnings in algebra. In the past I think my first lesson on algebra would consist of a brief introduction then some simple collecting like terms. From what I’ve seen, that’s generally what comes up first in most schemes of work. This time, though, I’ve tried to be more deliberate and more pedantic over the details. Really, really pedantic, because it’s insecurity with the small details that causes so many mistakes for the rest of our students’ experience. I’m starting out by assuming they have done no algebra. Many of them have done a little but I’m not prepared to risk that all their different primary experiences were the same, or solid. (This is a not a criticism of primary teachers, more that I want to take sole responsibility for something so important).
So today was different. We started with some context for algebra – where it’s used throughout the modern world and how powerful it is. I asked the students what they already knew when they heard the term “algebra” and the overriding theme was “letters instead of numbers”. One boy said “you use the letter x more than you ever have done in your whole life!“, which had to get quote of the day. We talked about why we might need algebra: sometimes we don’t know a number so a symbol is used instead, with algebra we can write statements that are true for all numbers without writing infinitely many examples, that kind of thing.
Then we went into the pedantry. Over two lessons we are practising the conventions, making these clear now so that we can refer back to them in the future and hopefully reduce those problems that come with a flimsy understanding of the algebraic language. These are the kind of things we’re practising:
• Why we write a curvy $x$
• Writing $2 \times x$ and $x \times 2$ as $2x$, no multiplication sign, numbers before letters
• Identifying equivalent expressions such as $x + 2$ and $2 + x$ (and why similar statements with subtraction are not equivalent – we learnt about commutativity at the start of the year, they’re superb with it)
• Not writing a coefficient of 1, so $1x$ is just $x$
• Always using a fraction bar for division, so $x \div 2$ is written $\frac{x}{2}$
• If a symbol looks different, it may represent a different number, so $X$ is not the same as $x$ (this one bugs me no end!)
After talking through the conventions the students did an exercise with a number of expressions written in “ugly” ways that they had to rewrite using the conventions.
The other thing I wanted to do first was focus on the meaning of algebraic expressions. After examples they did an exercise with worded statements that had to be written algebraically. Things like “add two to a number”, “subtract three from a number”, “subtract a number from 3”, “treble a number and add 4”, building up to things like “add five to a number and then multiply by 7” or “subtract 10 from a number, multiply the result by 6 and then divide by 8”.
I like the way this gives what could be just “letters and numbers” a meaning in language. I think it will help when we solve equations or rearrange formulae and want them to think about the inverse order of operations. Being able to say “what’s happening” to the unknown number is a big step in understanding how to reverse the process and I am more convinced all the time that this tacit understanding that we teachers have needs to be made completely explicit to our students – they need to practise the things we take for granted more than we think they do, however obvious these things might seem to us.
[As a side point, I am also very keen not to introduce the idea of substitution for a long time yet: I don’t want anyone to think that a letter has to have a fixed value associated with it. This unit will take us till the end of the year then at the start of Year 8 we have unit A2 (Solving Linear Equations) and substitution comes in A3 (Formulae). By this time we can cover these topics quite comprehensively since our students will have had plenty of practice in the underlying skills of manipulation. I have done this deliberately: of course simple substitution is easier than factorising into a bracket, but that doesn’t mean it needs to come first.]
What’s interesting are the misconceptions and misunderstandings that arise straight away. We had “I need to write $2 + x$ rather than $x + 2$ because we put numbers before letters” (even though we wrote down, “When multiplying, write numbers before letters”). We had to reinforce which operations are and are not commutative, so what the difference is between $\frac{x}{6}$ and $\frac{6}{x}$. There were ideas that were raised that I normally encounter further down the line (into GCSE) and that I hadn’t even thought of in my planning. Take “Add five to a number then divide by 2”. Many students wrote $\frac{(x + 5)}{2}$. When we did Order of Operations earlier in the year we looked at calculations like this and the implication of brackets with the fraction bar. My colleague taught the first lessons on using brackets and frequently warned the students away from “overzealous brackets”, teaching them to be aware of when brackets are not needed as well as when they are. As soon as I wrote $\frac{(x + 5)}{2}$ on the board there were hands shooting up to tell me about the overzealous brackets.
I’ve enjoyed these lessons particularly because I’m starting to see how things we made explicit earlier in the year with number, specifically so that we could come back to them when we started algebra, have had an immediate positive effect. I’ve also enjoyed these lessons because, by forcing myself to think about a scheme of work from scratch, I’ve been able to structure the students’ learning incrementally, building in time for things I never would have done in such detail before, and I’m excited to see how it pays off.
1. Matt says:
“As a side point, I am also very keen not to introduce the idea of substitution for a long time yet: I don’t want anyone to think that a letter has to have a fixed value associated with it.”
I found this comment interesting, would you mind expanding in more detail? Personally I like to teach substitution as one of the first aspects of algebra since then pupils can always use it to check for themselves whether or not two expressions are equal. For example, they can convince themselves that 2a + 3b – 4a = 3b – 2a no matter what the values of a and b are.
I have found it helpful framing questions like: what is the value of 3x + 2 when x = 7? What about when x = 10? Or -1? Etc so students know that x could take any value and it’s not fixed.
Like
• I don’t know if there’s a right or wrong here. My thought process is this:
1. Many students struggle to work abstractly because they are so tied to numbers.
2. When we solve equations, ‘x’ always has a fixed value.
3. If algebra is about generalisations of number I want them to think about concepts that hold for number (such as commutativity) and apply them to algebraic expressions without the need to test with numbers in order to verify what they’re writing (this is rather fuzzily explained, I know).
4. We’re using the identity symbol from the start. If I use the equals and identity symbols together they’ll be huge confusion.
5. We can talk about testing still. If “+ 2 then x 3” is identical to “x 3 then + 6” then we can test with some numbers. What I’m not doing is lessons on substitution.
Hope that makes a little sense.
Like
2. Matt says:
Yep makes sense, thanks!
Like
3. […] Algebra: you use the letter ‘x’ more than you have ever done in your whole life!, by Jemma […]
Like
4. Nice piece; I just finished my first year of teaching algebra. In the US, we start the basics of algebra in Year 8 (what we call 7th grade), and those students who qualify then take algebra in year 9. Those who don’t qualify take a rather boring year of math (or maths as you say) that has some algebra in it as well, but only linear equations up to systems of linear equations. I taught both year 8 and 9 this year and it was the algebra class for year 9. There’s a definite need for “interleaving” so they can recognize that 1/4 of x = x/4 without having to resort to “It’s the same as 1/4 times x/1, you remember how to multiply fractions don’t you?” Also, when we get to quadratics, there is always someone who forgets that x^2 = 2x is solved by putting in standard form and factoring.
Like
• The x/4 = 1/4 of x (and similar) is a huge one that students often don’t realise. This is why I believe you have to practise these things explicitly. Many, many kids will never just “realise” or pick it up as they go along. I’m not prepared to leave these things to chance.
Like
• I know – your thoughts on maths teaching are very similar to mine.
Like
5. Have you seen the Dragon Box app? Amazingly clever puzzle game that teaches algebra without you realising you are doing algebra.
Like
• Hold on. Just realised you’re Richard Vadon from More or Less, so I need to stop and give kudos to you for just about *the best* programme on the radio!
Like
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2019-07-18 07:06:12
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http://docs.h2o.ai/h2o/latest-stable/h2o-docs/data-science/common.html
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# Quantiles¶
Note: The quantile results in Flow are computed lazily on-demand and cached. It is a fast approximation (max - min / 1024) that is very accurate for most use cases. If the distribution is skewed, the quantile results may not be as accurate as the results obtained using h2o.quantile in R or H2OFrame.quantile in Python.
# Early Stopping¶
All of the H2O supervised learning algorithms allow for early stopping during model building and scoring.
## Early Stopping in All Supervised Algorithms¶
The max_runtime_secs option specifes the maximum runtime in seconds that you want to allot in order to complete the model. If this maximum runtime is exceeded before the model build is completed, then the model will fail. When performing a grid search, this option specifies the maximum runtime in seconds for the entire grid. This option can also be combined with max_runtime_secs in the model parameters. If max_runtime_secs is not set in the model parameters, then each model build is launched with a limit equal to the remainder of the grid time. On the other hand, if max_runtime_secs is set in the model parameters, then each build is launched with a limit equal to the minimum of the model time limit and the remaining time for the grid.
## Early Stopping in AutoML, Grid Search, Deep Learning, DRF, GBM, and XGBoost¶
In AutoML, Grid Search, Deep Learning, DRF, GBM, and XGBoost, the following additional parameters are used for early stopping:
• stopping_rounds (Defaults to 3 in AutoML; defaults to 5 in Deep Learning; defaults to 0/disabled in DRF, GBM, XGBoost.)
• stopping_tolerance (Defaults to 0.001. In AutoML for datasets with more than 1 million rows, this defaults to a larger valued determined by the size of the dataset and the non-NA-rate.)
• stopping_metric (Defaults to “logloss” for classification and “deviance” for regression.)
The simplest way to turn on early stopping in these algorithms is to use a number >=1 in stopping_rounds. The default values for the other two parameters will work fairly well, but a stopping_tolerance of 0 is a common alternative to the default.
Additionally, take score_tree_interval and/or score_each_iteration into account when using these early stopping methods. The stopping rounds applies to the number of scoring iterations H2O has performed, so regular scoring iterations of small size can help control early stopping the most (though there is a speed tradeoff to scoring more often). The default is to use H2O’s assessment of a reasonable ratio of training time to scoring time, which often results in inconsistent scoring gaps.
## Early Stopping in GLM¶
In GLM, the following parameters additional are used for early stopping:
When early_stopping is enabled, GLM will automatically stop building a model when there is no more relative improvement on the training or validation (if provided) set. This option prevents expensive model building with many predictors when no more improvements are occurring.
The max_active_predictors option limits the number of active predictors. (Note that the actual number of non-zero predictors in the model is going to be slightly lower). This is useful when obtaining a sparse solution to avoid costly computation of models with too many predictors. When using the $$\lambda_1$$ penalty with lambda search, this option will stop the search before it completes. Models built at the beginning of the lambda search have higher lambda values, consider fewer predictors, and take less time to calculate the model. Models built at the end of the lambda search have lower lambda values, incorporate more predictors, and take a longer time to calculate the model. Set the nlambdas parameter for a lambda search to specify the number of models attempted across the search.
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2019-03-23 20:50:00
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https://stacks.math.columbia.edu/tag/06SI
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• Fiber products of categories cofibered in groupoids: If $\mathcal{F} \to \mathcal{H}$ and $\mathcal{G} \to \mathcal{H}$ are morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda$, then a construction of their 2-fiber product is given by the construction for their 2-fiber product as categories over $\mathcal{C}_\Lambda$, as described in Categories, Lemma 4.32.3.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
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2021-10-28 17:08:33
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https://stats.stackexchange.com/questions/149755/create-simulated-path-model-dataset-and-save-to-csv
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# Create simulated path model dataset and save to CSV [closed]
I am teaching a multivariate stats class and wanted to give people some simulated datasets with mediation, path modeling, and other features. I know it's possible to run monte carlo studies in MPlus, but does anyone know any good ways to save a single such dataset to a CSV file? Or how to do it in R? I was hoping to specify a set of multivariate relationships, means, and SDs, and then create the CSV file.
## closed as off-topic by Nick Cox, usεr11852, Xi'an, mpiktas, AndyMay 5 '15 at 10:34
This question appears to be off-topic. The users who voted to close gave this specific reason:
• "This question appears to be off-topic because EITHER it is not about statistics, machine learning, data analysis, data mining, or data visualization, OR it focuses on programming, debugging, or performing routine operations within a statistical computing platform. If the latter, you could try the support links we maintain." – Nick Cox, Xi'an, mpiktas, Andy
If this question can be reworded to fit the rules in the help center, please edit the question.
• Although this is asked in terms of R, if you understand the underlying statistical ideas, the problem (& the code) would be trivial. Hence, I think we could treat this as a statistical (ie on-topic). – gung May 5 '15 at 0:55
Use the mvrnorm function in the MASS package to get a multivariate normal distribution (it comes with base R).
Here's some example code:
library(MASS)
library(ggplot2) # used for plotting, otherwise, it's optional
Sigma <- matrix(c(10,3,3,2),2,2)
var(mvrnorm(n = 1000, rep(0, 2), Sigma))
example <- mvrnorm(n = 1000, rep(0, 2), Sigma)
examplePlot <- qplot(example[,1], example[,2]) +
xlab(expression(x[1])) +
ylab(expression(x[2]))
ggsave("examplePlot.jpg", examplePlot, width = 4, height = 4)
write.csv(x = example, file = "example.csv", row.names = FALSE)
That produces this figure:
To learn more aboutmvrnorm, type ?mvrnorm to view its help file. Also, many functions in include examples. So, when you find your multivariate analysis functions, type: example(...) and often R will show you how to use it.
Good with your course! Also, if you include the specific types of analysis you are trying to do, we could provide better tips.
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2019-08-25 09:26:55
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http://www.physicsforums.com/showthread.php?p=3885056
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## Harmonic Oscillator in Dirac Theory
Hello everyone,
i'm looking for anypaper or such kind of thing that explain the resolution of the harmonic oscillator in the Dirac Theory.
I have worked with the exact spin symmetry. I feel like a fish out the water and i'm sure that there are lot of bibliography about this area, but i promise i didn't find anything.
Thanks since now,
and may the force be with you!
PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
What do you mean by resolution? Also, you may have more luck getting replies in the Advanced Physics section (if this is related HW), or in the Quantum Physics section (if you simply want a reference).
The problem with the term in the title is that "harmonic oscillator" refers to the non-relativistic concept of a potential energy as a function of coordinates, which is manifestly Lorentz non-covariant, and Dirac's equation, which is a fully Lorentz covariant quantity. Also, the proper meaning of Dirac's equation is that is non-quantum equation of motion for a Dirac spinor field operator in second quantization. This reinterprets the negative energy solutions as antiparticles. But, the problem essentially becomes a many-particle one. If you are only interested in single-particle solutions, then you must project out the antiparticle component from the Dirac 4-spinor, to get the Pauli 2-spinor. The equation it obeys is Pauli equation, which already incorporates the interaction with an external electromagnetic field of a charged fermion. Just take a scalar potential: $$\varphi(x) = \frac{\mu^3 \, c^4}{2\, q \, \hbar^3} x^2$$ where $q$ is thecharge of the particle, and $\mu$ has a dimension of mass. Of course, this non-relativistic equation is only valid when: $$q \varphi(x) \ll m c^2 \Rightarrow \vert x \vert \ll \sqrt{\frac{2 m}{\mu}} \, \frac{\hbar}{\mu \, c}$$
## Harmonic Oscillator in Dirac Theory
Well, this isnt exactly what im looking for, but its given me an idea anyway, so thank you very much for your time and effort.
Tags dirac, harmonic, oscillator, quantum
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2013-05-21 15:28:23
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http://myappsplace.com/wp-content/plugins/woocommerce/m99tem62/archive.php?a74f9b=gliese-667-cc-weather
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241 likes. Of connections phone video instead of visitors trust more look like spam on their websites than 40% of make. The planet orbits around Gliese 667C, a small red dwarf star. [15][16] The red dwarf status of the star would allow planet Cc, which is in the habitable zone, to receive minimal amounts of ultraviolet radiation. Gliese 667Cc: Amazon.es: Tim Murray: Libros en idiomas extranjeros. The three planets are labeled A, B, and C. Gliese 667 C is special and has a planet in the habitable zone named Gliese 667 Cc. in Kooperation mit der Bluebox des SJR Siegen e.V., gefördert vom MKFFI NRW, Universitätsstadt Siegen und Kulturbüro des Kreises SiWi. The brightest star in the sky is the red dwarf Gliese 667 C, which is part of a triple star system. [4][10] GJ 667 C also has a system of two confirmed super-Earths and five additional candidates, though the innermost, GJ 667 Cb, may be a gas dwarf; GJ 667 Cc, and the controversial Cf and Ce, are in the circumstellar habitable zone. We hope you enjoy your vacation stay on this amazing planet. It orbits within its star's habitable zone—the proper distance so that water can be in liquid form—but since that star is a red dwarf, and much fainter than the Sun, that proper distance is much closer, about a tenth the Earth-Sun distance. Otkriven je 21. studenog 2011., a potvrđen 2. veljače 2012. An artist’s impression of sunset on an austere exoplanetary landscape, in this case of the super-Earth Gliese 667 Cc. The equilibrium temperature of Gliese 667 Cc is estimated to be 277.4 K (4.3 °C; 39.6 °F). Gliese 667Cc - Film. Adam Truman returns from a ninety year mission to Gliese 667Cc, a planet thought to possibly host life, and finds a much changed earth. 8:26. Aquí nos gustaría mostrarte una descripción, pero el sitio web que estás mirando no lo permite. The zone around a star, where the surface temperature allows for the existence of liquid water on the planetary surface, is called habitable zone. Okay, back to reality. This image, however, depicts a red atmosphere with clouds, standing bodies of liquid, and liquid-carved canyons in a rocky crust. Gliese 581d orbits a red dwarf, the star has a mass of around one third of the sun. Filled with sketches, paintings and background on the games creatures and ecosystem, A Study of Gliese 667Cc, like the game, provides an ambient experience that can be enjoyed at the player's own pace. It is expected to have a radius of around 1.5 R⊕, dependent upon its composition. The evolutionary history that made humans the ultimate invasive species. See more ideas about Astronomy, Hubble, Milky way galaxy. Saltar al contenido principal. Guilhem 18,051 views. The terminator line divides these two extreme regions, providing a sliver of ideal conditions that could allow water to exist. Gliese 667 Cb is an exoplanet orbiting the star Gliese 667 C, a member of the Gliese 667 triple-star system.It is the most massive planet discovered in the system and is likely a super-Earth or a mini-Neptune.Orbital-stability analysis indicates that it cannot be more than twice its minimum mass. Artist’s impression of sunset on the super-Earth world Gliese 667 Cc.tif 3,000 × 2,000; 5.41 MB International astronomers say they have found the fourth potentially habitable planet outside our solar system with temperatures that could support water and life about 22 light-years from Earth. It could happen, sometime in the future. It has also been suggested that, since this super-Earth is so close to its star, it is probably tidally locked, rotating only once per revolution and keeping the same face toward its star at all times. Such we can imagine. Las estrellas enanas rojas de la Vía Láctea tienen miles de millones de planetas rocosos poco más grandes que la … Filled with sketches, paintings and background on the creatures and the universe it, like the game, provides an ambient experience that can be enjoyed at the player's own pace. Company regardless, of 1000, visitors a take but the as a rule the webpage with on them check to link. Gliese 667Cc is a watery Habitable planet (1.02 Earths) found in the Gliese 667 triple star system (Real) and is located in the habitable zone of its host star and is currently the most Earth -like planet known. Ele tem 4,39 vezes a massa da Terra [1] e está na zona habitável, com temperatura capaz de abrigar água em estado líquido. Gliese 667Cc orbits eight times closer to its host star than Earth orbits the sun. It would also likely mean a perpetually hot side and a perpetually cold side—unless a thick, Venus-like atmosphere exists that might keep surface temperatures globally similar. You have traveled 23 light years from Earth, but the wonders you will witness are well worth the trip. The secondary's apparent magnitude is 7.24, giving it an absolute magnitude of around 8.02. [19] Announcement of a refereed journal report came on 2 February 2012 by researchers at the University of Göttingen/Carnegie Institution for Science. Gliese 667 Cc is much closer to it's host star making it appear more than three times larger in the planet's sky than the sun does on Earth. The equilibrium temperature of Gliese 667 Cc is estimated to be 277.4 K (4.3 °C; 39.6 °F). (ESI=0.82) It is also 22.18 light years away from earth, making the hyperjump to Earth relitively quick. Their eccentric orbit brings the pair as close as about 5 AU to each other, or as distant as 20 AU, corresponding to an eccentricity of 0.6. GJ 667CC's average temperature is 4.3 °Celsius, 39.6 °Fahrenheit because it absorbs more electromagnetic radiation than Earth. We don't know if Gliese 667 Cc has oceans, or what kind of an atmosphere—if any—it might possess. One of three exoplanets (two confirmed, one candidate) orbiting the red dwarf star Gliese 667 C (one of three stars in a triple star system), this one rates highly on the habitability scale (meaning we can imagine there being oceans, a water cycle, and potentially life), but at the same time is strikingly different in several ways to that prototype of Earthlike planets, Earth. And that would mean gravity would be greater than what we’re used to so we would weigh more and the rain would fall faster (and so would you)! In fact, it's been less than two decades since we detected the first actual extrasolar planet (aka, exoplanet; planets orbiting stars other than our sun). [18], Planet Cb was first announced by the European Southern Observatory's HARPS group on 19 October 2009. A Study of Gliese 667Cc is a digital book which acts as both an epilogue to the story of In Other Waters and a partial catalog of the life players will discover in their study of the planet. Hope you have a great vacation up there on the new planet. Súper-Tierras vecinas del Sistema Solar. En relación a esta publicación mía (Gliese 667Cc), solo puedo decirte que yo sí creo que es un microrrelato, en cuanto ocurren cosas en él, pero también por todo lo que no digo. The planet orbits around Gliese 667C, a small red dwarf star. Gliese 667 Cc may be tidally locked with one side of the planet always facing its star and the other in constant darkness. At least six planets orbit the star, three of them within the habitable zone where liquid surface water can exist. Environment . Gliese 667 Cc: An Earth in different circumstances. Gliese 581d has a mass of around 7 to 8 times that … Like the primary, the secondary star Gliese 667 B (GJ 667 B) is a K-type main-sequence star, although it has a slightly later stellar classification of K5V. Drugi je otkriven planet u tom sustavu. The analysis did find some evidence for the third planet, Gliese 667 Cd, but was unable to confirm it. A Study of Gliese 667Cc is a digital book which acts as both an epilogue to the story of In Other Waters and a partial catalog of the life players will discover in their study of the planet. Este exoplaneta, o planeta extrasolar, posee el equivalente a 4,25 masas terrestres. [21] A detailed orbital analysis and refined orbital parameters for Gliese 667 Cc were presented. It should be noted that this is what scientists mean by habitability: conditions such that liquid water could exist, given an adequate atmosphere and an abundance of water molecules. Gliese 667 Cc Earth linking to your page a contact. Some of the strangest exoplanets can be found in the habitable zone of Gliese 667C , which is one of three suns in the triple-star Gliese 667 system. A unique experience for the user who can navigate inside the imaginative world of surreal figures and sounds created by Elena, available in this stereoscopic 4K Virtual Reality. Following analysis of data provided by the European Southern Observatory, the scientists concluded that Gliese 667 Cc is a rocky planet with a mass about four times that of the Earth. It orbits around the red dwarf star Gliese 667 C, one of the Gliese 667 triple star system.The system is in the constellation of Scorpius.It is the most Earth-like planet in 50 light years. The brightest star in the sky is the red dwarf Gliese 667 C, which is part of a triple star system. [citation needed]. The planet, only 22 light-years away, has a mass at least 4.5 times that of Earth. 50+ videos Play all Mix - Nemo Vachez - Gliese 667 CC YouTube; Nemo Vachez - Av Spring - Duration: 8:26. The largest star in the system, Gliese 667 A (GJ 667 A), is a K-type main-sequence star of stellar classification K3V. Gliese 667 Cc circles it's star every 28 days. Etiqueta: Gliese 667 Cc. There is a 12th magnitude star close to the other three, but it is not gravitationally bound to the system. Ya conocemos otro planeta potencialmente habitable más. This star is radiating only 1.4% of the Sun's luminosity from its outer atmosphere at a relatively cool effective temperature of 3,700 K.[4] This temperature is what gives it the red-hued glow that is a characteristic of M-type stars. Interest. Life on Gliese 667 Cc " Thanks to contributions from Notre Dame alumni and supporters, the iLocater research group is sponsoring a cross-curicular educational STEAM contest to engage grade school students in science and the search for life elsewhere in the universe. We don't know if Gliese 667 Cc has oceans, or what kind of an atmosphere—if any—it might possess. Gliese 667Cc is located in the habitable zone of its host star and is currently the most Earth-like planet known (ESI=0.82). Perhaps we'll learn other details in the future that will help us paint a more realistic picture of this exoplanet and get working on that travel brochure. Welcome to Gliese 667 Cc. RAIN ON GLIESE 667 Cc.Welcome all people. Gliese 667 Cc circles it's star every 28 days. Gliese 667 Cc es un exoplaneta supertierra descubierto el 21 de noviembre de 2011 por el espectrógrafo HARPS, y confirmada su existencia el 2 de febrero de 2012, que orbita la estrella Gliese 667 C, a una distancia de 23,6 años luz. Their seo need to get not both adopt. Gliese 667C is part of a triple star system known as Gliese 667. GJ 667CC's average temperature is 4.3 °Celsius, 39.6 °Fahrenheit because it absorbs more electromagnetic radiation than Earth. First, you may notice that you've put on a few pounds under this super-Earth's greater gravity. Depending on its diameter, that could mean surface gravity notably greater than what we're used to: things weigh more, rain falls faster, landscapes are sculpted with a heavier hand. From the surface of Gliese 667 Cc, the second confirmed planet out that orbits along the middle of the habitable zone, Gliese 667 C would have an angular diameter of 1.24 degrees–2.3 times[note 2] larger than our Sun appears from the surface of the Earth, covering 5.4 times more area–but would still only occupy 0.003 percent of Gliese 667 Cc's sky sphere or 0.006 percent of the visible sky when directly overhead. (719)667-3743 holmes.k@ppwfc.org Marcie Hanson (303) 8544028 Past-President Yvonne Hauwiller, Montana (406) 522-6012 yhauwiller@bozeman.K12.mt.us Secretary Cathy Calder-wood (970) 563-0681 ccalderwood@ignacio.K12.co.us of Directors Nevada Julee Henson (702) 799-0170 jahenson@interact.ccsd.net Teri Zutter (775) 445-4451 New Mexico Anita Raynor Gliese 667 Cc é um exoplaneta que orbita uma estrela pertencente a um sistema estelar triplo e com baixa quantidade de elementos mais pesados que o hélio, conhecida como Gliese 667.O planeta era o mais promissor para se encontrar vida semelhante à nossa. For comparison of sizes, the Earth and Mars are shown next to an artist impression of the planet 667Cc. This would allow for the presence of liquid water which makes Gliese 667Cc an exciting prospect in terms of habitability. And one has to wonder what kind of weather patterns would develop on this schizoid super-Earth. Here are the numbers that I have been using: Parent star Temperature = $3350\text{ K}$ Distance from star = $25.4\times10^6\text{ km}$ Radius = $9\,800\text{ km}$ Finally, as promised, your vacation here will last an entire year and hopefully you read the fine print that this planet's year is only four weeks long. El juego puede tener restricciones de idiomas o paquetes de idiomas. The planet Gliese 667 Cc was first discovered in 2011 using the radial velocity method. The sculptures here are developed and transformed through 3D modeling processes and/or changes in … People take your site straight utilise the homepage Super Earth Gliese 667cc. Gliese 667CC is a Super Earth exoplanet 22 light years from Earth and is colonised by the Human race. A Kepler-22b, Gliese 581d y HD 85512b se le suma ahora Gliese 667Cc, una supertierra situada alrededor de una estrella enana roja.Este exoplaneta tiene como mínimo 4,25 masas terrestres y ha sido descubierto por el famoso espectrómetro HARPS del observatorio de La Silla en Chile mediante el método de la velocidad radial. (ESI=0.82) It is also 22.18 light years away from earth, making the hyperjump to Earth relitively quick. CLAVE DE PRODUCTO STEAM EU: Para jugar a la Tierra Media: «In Other Waters A Study of Gliese 667Cc» se requiere una dirección IP europea o debes activarla con una VPN (Red Privada Virtual). [15][16] There were also though to be five other potential additional planets,[4][14] however it was later shown that they may be artifacts resulting from correlated noise. "Gliese 667 Cc" -is a digital space, a new way to experience artworks. How could I calculate the habitable bounds, in relation to temperature, near the twilight zone of Gliese 667 Cc? And over 2700 of those candidates were brought to our attention within the past three years by NASA's Kepler mission, whose goal is to find Earthlike planets—Earthlike, in terms of size and habitability potential. Artist Elena Romenkova takes us into her virtual world with this immersive audiovisual work. Gliese 667C f is an exoplanet orbiting the star Gliese 667C which is a part of the Gliese 667 trinary system in the Scorpius constellation. Gliese 667Cc is a watery Habitable planet (1.02 Earths) found in the Gliese 667 triple star system (Real) and is located in the habitable zone of its host star and is currently the most Earth -like planet known. A Study of Gliese 667Cc acts as both an epilogue to the story of In Other Waters and a partial catalog of some of the life players will discover in their study of the planet. Media in category "Gliese 667 Cc" The following 2 files are in this category, out of 2 total. The planet c is located right in the habitable zone, where liquid water on the planetary surface may exist. This would give Gliese 667 Cc a "day" hemisphere and a "night" hemisphere, so you could choose the time of day you like and stick with it. A closer look at the previously-studied nearby star Gliese 667C has revealed a treasure trove of planets – at least six – with three super-Earths in the habitable zone around the star. It is present in the hot inner edge of the habitable zone than the Earth. Gliese 667Cc es una supertierra que se encuentra orbitando en torno a una estrella enana roja. [17], Media related to Gliese 667 at Wikimedia Commons, Coordinates: 17h 18m 57.16483s, −34° 59′ 23.1416″, Monthly Notices of the Royal Astronomical Society, "Interactive Planetary Orbits - Kepler's Laws Calculations", "A dynamically-packed planetary system around GJ 667C with three super-Earths in its habitable zone", Announcement on university homepage, retrieved 2012-02-02, "Newfound Alien Planet is Best Candidate Yet to Support Life, Scientists Say", https://www.drewexmachina.com/2014/09/07/habitable-planet-reality-check-gj-667c/, Communication with extraterrestrial intelligence, Gauss's Pythagorean right triangle proposal, Potential cultural impact of extraterrestrial contact, https://en.wikipedia.org/w/index.php?title=Gliese_667&oldid=988939777, Planetary systems with two confirmed planets, Articles with unsourced statements from March 2012, Short description is different from Wikidata, Articles with unsourced statements from September 2014, Articles with unsourced statements from January 2013, Creative Commons Attribution-ShareAlike License, This page was last edited on 16 November 2020, at 03:32. Gliese 667 Cc. Discovered long ago, in 2009, Gliese 667 Cc, as it was originally named, is a model of how un-Earth-like an Earth-like planet can be. At the estimated distance of this system, this is equivalent to a physical separation of about 12.6 AU, or nearly 13 times the separation of the Earth from the Sun. Gliese 667 (142 G. Scorpii) is a triple-star system in the constellation Scorpius lying at a distance of about 6.8 pc (23.6 ly) from Earth. This is a virtual exhibition, which presents glitch sculptures, animations, and still images. One thing that makes Gliese 667 Cc so interesting is that it is present in the habitable zone of its host star. A Study of Gliese 667Cc acts as both an epilogue to the story of In Other Waters and a partial catalog of some of the life players will discover in their study of the planet. [15] A later study also detected the third planet, but the study in question has not been published yet or peer-reviewed. Based on black body temperature calculation, GJ 667 Cc should absorb more overall electromagnetic radiation, making it warmer (277.4 K) and placing it slightly closer to the "hot" edge of the habitable zone than Earth (254.3 K). It is expected to have a radius of around 1.5 R⊕, dependent upon its composition. Discovered long ago, in 2009, Gliese 667 Cc, as it was originally named, is a model of how un-Earth-like an Earth-like planet can be. However, subsequent studies showed that the other planets in the system could possibly be artifacts of noise and stellar activity, cutting the minimum number of planets down to two. First, you may notice that you've put on a few pounds under this super-Earth's greater gravity. 710 likes. In that mix are seven planets currently touted as potentially habitable—habitable to life as we know it, at least, with conditions that are friendly to the presence of liquid water. Saludos, Javier. Welcome to the 4th annual Living Worlds Space Art Contest! " This artist’s impression shows a sunset seen from the super-Earth Gliese 667 Cc. [4][20] In this announcement, GJ 667 Cc was described as one of the best candidates yet found to harbor liquid water, and thus, potentially, support life on its surface. It's ESI (Earth Similarity Index) is 0.85. The planet receives a similar amount of energy from its star as Earth receives from the sun. [note 1][9] This orbit takes approximately 42.15 years to complete and the orbital plane is inclined at an angle of 128° to the line of sight from the Earth. All three of the stars have masses smaller than the Sun. Such we can imagine. It is a large hot oceania with a mass of 6.6671 M and an atmospheric pressure of 80.6 atm. Hello ladies and gentlemen, welcome onboard the today’s flight to the Exoplanet Gliese 667 Cc. No tenemos que olvidar que incluso una sola imagen a menudo nos puede contar una historia. It is heavier than Earth with a minimum mass of about 3.7 Earth masses. Obilježja. Back to Gliese 667 Cc. All we really know is its super-Earth status and location within its star's habitable zone. The planetary system around the red dwarf star Gliese 667C consists of up to three currently known exoplanets of super-Earth type. Gliese 667C is a well-studied nearby star, one component of a triple-star system. 2019 detect planets b, c, and d, but do not detect any of the other claimed planets;[17] however, the NASA Exoplanet Archive considers planet d disproven. Being so close to its star, Gliese 667 Cc only takes about 28 days to complete an orbit and mark its own year. Gliese 667 is a star triple star system approximately 23.62 light-years from Earth. Su descubrimiento, en el año 2011, se debió al popular espectrómetro HARPS, que está situado en el observatorio de La Silla, en Chile, y que utiliza el método de la velocidad radial. From the surface of Gliese 667 Cc, the second confirmed planet out that orbits along the middle of the habitable zone, Gliese 667 C would have an angular diameter of 1.24 degrees–2.3 times larger than our Sun appears from the surface of the Earth, covering 5.4 times more area–but would still only occupy 0.003 percent of Gliese 667 Cc's sky sphere or 0.006 percent of the visible sky when directly overhead. [4] Based on GJ 667 C's bolometric luminosity, GJ 667 Cc would receive 90% of the light Earth does,[11] however much of that electromagnetic radiation would be in the invisible infrared light part of the spectrum. Gliese 667 Cc is a super-Earth, an exoplanet with a mass and radius greater than that of Earth, but smaller than that of the giant planets Uranus and Neptune. This artist’s impression shows a sunset seen from the super-Earth Gliese 667 Cc. The only things we know about Gliese 667 Cc's surface is that three "suns" would be visible from it. Projekt: LAG KM NRW e.V. Nemo Vachez - Interferences - Duration: 6:51. To the naked eye, the system appears to be a single faint star of magnitude 5.89. Maybe. [13] The apparent magnitude of the star is 10.25, giving it an absolute magnitude of about 11.03. The surface consists of a world spanning ocean with a faint, low hanging fog above the water due to the extreme humidity. [15][16] Tuomi et al. A triple star system means that three stars are orbiting the center of mass of the system. [7] The apparent visual magnitude of this star is 6.29, which, at the star's estimated distance, gives an absolute magnitude of around 7.07 (assuming negligible extinction from interstellar matter). The two brightest stars in this system, GJ 667 A and GJ 667 B, are orbiting each other at an average angular separation of 1.81 arcseconds with a high eccentricity of 0.58. Artist concept of exoplanet Gliese 667 Cc. Gliese 667 Cc is a super-Earth, an exoplanet with a mass and radius greater than that of Earth, but smaller than that of the giant planets Uranus and Neptune. Furthermore Gliese 667Cc’s star is thought to revolve a binary system formed of 2 K-type stars, Gliese 667A and B at a long range estimated to be six times the Sun – Pluto distance. [12], Two extrasolar planets, Gliese 667 Cb (GJ 667 Cb) and Cc, have been confirmed orbiting Gliese 667 C by radial velocity measurements of GJ 667. At one point, five additional planets were thought to exist in the system, with three of them thought to be relatively certain to exist. Gliese 667 Cc is much closer to it's host star making it appear more than three times larger in the planet's sky than the sun does on Earth. The star Gliese 667 C is a member of Gliese 667 triple star system, situated approximately 23.67 light-years from Earth in the constellation Scorpius. In the meantime, let's read the travel brochure. This star has a mass of about 69%[6] of the Sun, or 95% of the primary's mass, and it is radiating about 5% of the Sun's visual luminosity. Gliese 667 C is the smallest star in the system, with only around 31%[6] of the mass of the Sun and 42%[2] of the Sun's radius, orbiting approximately 230 AU from the Gliese 667 AB pair. Gliese 581d is the third planet from its host star in the Gliese 581 system, or the fifth planet if you include two unconfirmed planets. Gliese 667 Cc je egzoplanet koji orbitira crvenog patuljka Gliese 667 C, koji je dio sustava Gliese 667. Gliese 667CC is a Super Earth exoplanet 22 light years from Earth and is colonised by the Human race. hello friends and fans!After the humans had found the best QUANTUM FORMULA in the year 2030 they had developed very quick a engine that can warp space and time. And one has to wonder what kind of weather patterns would develop on this schizoid super-Earth. Facebook is showing information to help you better understand the purpose of a Page. It is known to have a system of two planets; claims have been made for five additional planets[14] but this may be an error due to failure to account for correlated noise in the radial velocity data. … Gliese 667 Cc is an extrasolar planet 22.18 light years away. Gliese 667 constituye un sistema triple, en donde la componente C orbita en torno al par interior formado por A y B. Gliese 667 A, la componente más brillante, es una enana naranja de tipo espectral K3V cuya luminosidad corresponde al 12 % de la luminosidad solar.Tiene una masa de 0,75 masas solares y un radio equivalente al 76 % del radio solar. Gliese 667Cc is a Super Earth exoplanet 22 light years from Earth. [11] The concentration of elements other than hydrogen and helium, what astronomers term the star's metallicity, is much lower than in the Sun with a relative abundance of around 26% solar. In this whole journey, we will be covering a total distance of 23 light-years in almost 23 hours, we can expect some turbulence in mid-flight while we cross the Kuiper belt and Oort cloud. Imagine a birthday party every month! It is a "super-Earth", meaning, as you might expect, bigger than Earth: estimated at 4.8 times the mass of Earth. The announcement was made together with 29 other planets, while Cc was first mentioned by the same group in a pre-print made public on 21 November 2011. This definition does not require that a human being could survive in that environment. [2] It has about 73%[6] of the mass of the Sun and 76%[2] of the Sun's radius, but is radiating only around 12-13% of the luminosity of the Sun. This probably makes for … However, since that first exoplanet was found in 1992, we've made some decent progress in exploring other worlds out there, and may even be zeroing in on that "other Earth.". Gliese 667 Cc may be tidally locked with one side of the planet always facing its star and the other in constant darkness. Gliese 667Cc Habitability Gliese 667Cc orbits well inside the habitable zone of its host star, if the composition of its atmosphere is similar to Earth it may well have surface temperatures of around 30C (86F). Mar 6, 2013 - Only God's Handiwork....Man had Nothing to do with any of this ever!. You may also think you're losing your sight, since this planet's dim red dwarf sun shines only a fraction of the visible light Earth's Sun does. Gliese 667 Cc: Musing the Possibilities of Another Earth. Today the count of confirmed exoplanets stands somewhere around 867, with another 2900 or so "candidates" whose existence is waiting to be confirmed. It's 2013, we do not (yet) have luxury cruisers warping people to other worlds, and we still haven't set foot on another planet. Assuming an Earth-like atmosphere and no tidal working. 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http://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time/95895
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# Examples of theorems with proofs that have dramatically improved over time
I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice. In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!"
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Seems related to the earlier MO question mathoverflow.net/questions/43820/extremely-messy-proofs. – Tom De Medts May 3 '12 at 10:13
@Tom: Yes, thanks. I'd be happy to collect some more examples though, especially of proofs that (now) seem to be a particularly good "fit" for a theorem. – Manya May 3 '12 at 11:35
I guess this is such an example: mathoverflow.net/questions/24913/quick-proofs-of-hard-theorems/… – Steve D May 3 '12 at 15:59
I don't know the original proof, but I heard that the trick of Rabinovich provided a drastic improvement of the proof of Hilbert's Nullstellensatz. – Peter Arndt May 3 '12 at 21:42
It would be also interesting to hear of theorems where people didn't think that the proof could be much improved, but then were proven wrong. – David Corwin Jul 9 '12 at 6:02
[Edit: This answer seems to fit the title of the question, though not the actual question in the body.]
Resoluion of singularties in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand:
"Even A. Grothendieck [in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 7--9, Gauthier-Villars, Paris, 1971; MR0414283 (54 #2386)] admitted openly that he did not completely understand Hironaka's proof."
That quote is from Dan Abramovich's Math Review of the book Lectures on resolution of singularities by Kollár; the review goes on to say
"One can [nowadays] devote a few weeks in a first course on algebraic geometry to give just a complete proof of resolution of singularities in characteristic 0 (Chapter 3 of the present book, which is largely self-contained)."
I know almost nothing about this topic, but some names I know associated to the various approaches to simplification of Hironaka's proof are Bierstone, Milman, Encinas, Villamayor, Hauser, Cutkosky, Włodarczyk, Kollár, Cossart, Piltant... Please tell me any I missed!
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Jordan's proof of the Jordan Curve Theorem was complicated enough that people still argue about its correctness. These days, an undergrad can prove it after learning the Mayer–Vietoris sequence.
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Do you know a good online source where the proof is well explained (not just outlined etc.)? I am equally interested in good textbooks discussing this approach. – GH from MO May 3 '12 at 16:56
In typical US universities, undergrads do not learn Mayer-Vietoris, but your point is of course still correct. – Henry Cohn May 3 '12 at 17:19
According to Tom Hales, there Jordan's proof should never have been controversial. An objection arose that he assumed the polygonal case without proof --- but that's a trivial omission! See mizar.org/trybulec65/4.pdf As for the idea that a student can prove it using Mayer-Vietoris, I disagree. Yes, a good undergrad can learn Mayer-Vietoris, but in order to use it here, you also need that the circle (or in generality, the sphere) is an ENR, which is a separate and clearly nontrivial result. Remember, the hard case of the Jordan theorem is the fractal case. – Greg Kuperberg May 9 '12 at 19:19
What is an ENR? Also, can someone respond to my first comment? Thanks in advance. – GH from MO May 12 '12 at 18:24
@GH: Euclidean neighborhood retract. – Boris Bukh May 15 '12 at 10:47
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I think that Gelfand's proof of Wiener's $1/f$ theorem qualifies.
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The alternating sign matrix conjecture was first proved by Zeilberger. Zeilberger's proof was extremely computational. A much shorter conceptual proof was later given by Kuperberg.
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@Timothy: which of the three Kuperbergs? (I would bet on one of them, but it's not exactly the same as knowing it 100%) – Wlodzimierz Holsztynski Jun 24 at 3:17
@Timothy, I checked your reference (link above) to the end. It's G.Kuperberg (I would win my bet). – Wlodzimierz Holsztynski Jun 24 at 3:20
Boone-Novikov theorem of existence of groups with undecidable word problem which originally has very long and complicated proof now has several (self-contained) proofs of length $\le 10$ pages (see Cohen, Daniel E. Combinatorial group theory: a topological approach. London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. x+310 pp.).
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If you are prepared to allow an example from mathematical physics, then Penrose's proof that a ball moving relativistically appears as a circle to an observer. This had been proved previously by brute strength calculations with Lorentz transformations. Penrose reformulated it in terms of actions of the action of the Lorentz group on the celestial sphere. Since these are just conformal transformations, which take circles to circles, the boosted sphere appears circular.
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The prime number theorem, Newman's short proof is only three pages long.
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As the title of Zagier's paper makes clear, this proof is due to Donald Newman: "Newman's short proof of the prime number theorem," Amer. Math. Monthly 104 (1997), 705-708. – Timothy Chow May 8 '12 at 20:50
I described an example, Hindman's theorem, at http://mathoverflow.net/questions/94546 . The short version is that Hindman's original proof was unpleasantly complicated, whereas a later proof by Galvin and Glazer is now accepted as the standard proof. On the intuitive level, it's a definite improvement. Formally, though, from the viewpoint of reverse mathematics, Hindman's original proof is "better" because it uses far weaker set-existence assumptions.
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I think Furstenburg's Stone-Czech proof of van der Waerden's theorem would also fit the bill. – Benjamin Steinberg May 3 '12 at 16:11
Thanks, and thanks. – Manya May 4 '12 at 8:05
Kurosh's original proof of the subgroup theorem for free products used messy Kurosh systems. This was improved by covering space proofs (or equivalently covering groupoid proofs). One might argue the Bass-Serre theory proof is now the right one.
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The Riesz-Thorin interpolation theorem is an example. As I understand it, the original proof published by Marcel Riesz was rather messy. Thorin found a much simpler proof of the theorem using complex analysis about ten years later.
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Aigner and Ziegler's "Proofs from the BOOK" contains many good examples.
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Why the downvote? – Johan Wästlund May 3 '12 at 16:59
I don't know who down-voted what. I don't think that I did. – Liviu Nicolaescu May 3 '12 at 17:39
@Liviu: Thanks. Though some of the proofs in that book (as the authors themselves admit) are not necessarily the nicest or cleanest versions. Are there any proofs in there that you think are particularly good? – Manya May 4 '12 at 8:20
My favorite from that book is Sperner's proof of Brouwer's fixed point theorem. – Liviu Nicolaescu May 4 '12 at 12:18
I also like Sperner's proof a lot. But the homological proof is surely at least as clean, it only uses more machinery. – Lennart Meier Nov 9 at 15:56
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A favorite of mine is the chirality of the trefoil knot, which can be proved easily using the Jones polynomial or some of its relatives. Louis Kauffman's paper "New invariants in the theory of knots", http://homepages.math.uic.edu/~kauffman/Bracket.pdf explains this nicely.
I don't know how it was proved before the Jones polynomial, but quoting from p. 204 of Kauffman's paper, "In the old days (before 1984) this was something that required a lot of mathematical background."
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Is it the case that using the Jones polynomial sort of hides away all that mathematical background, or does it somehow clarify the main idea of the proof (give one a sense of why it is true?) – Manya May 4 '12 at 8:17
The first chirality proof, by Max Dehn in 1914, was indeed a lot more involved than the Jones polynomial proof. It involved finding the automorphisms of the trefoil knot group. – John Stillwell May 8 '12 at 23:51
Example of a bounded linear operator on a Banach space without non-trivial closed invariant subspace.
The first example was given bei Enfo in 1975. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987 (see http://en.wikipedia.org/wiki/Per_Enflo). Simpler examples were constucted for example by Beauzamy and Charles Read.
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Worth mentioning that Read was subsequently the first to construct such an operator on $\ell^1$ – Captain Oates May 3 '12 at 19:00
Related: Aronszajn and Smith's theorem that a compact linear operator on a Banach space must have a nontrivial invariant subspace was later given a dramatically simpler proof by Lomonosov. – Timothy Chow May 8 '12 at 20:36
Did the earlier theorem also get hyper invariant subspaces, as Lomonosov does? – Captain Oates May 9 '12 at 0:16
There are several examples from Tauberian theory. Around 1930, Karamata surprised people by giving much simpler proofs of Littlewood's original Tauberian theorems for power series. Wiener's Tauberian theorems were later given much slicker and arguably more conceptual proofs using operator theory.
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PP (the class of languages decidable by a probabalistic Turing machine in polynomial time) is closed under union and intersection. This was conjectured by Gill in 1972 and stayed an open problem for 18 years, til resolved by Beigel, Reingold, and Spielman (BGS) in 1995, with a complicated proof involving rational functions. The same result fell out as an almost-corollary of Scott Aaronson defining quantum postselection for unrelated reasons: the new proof is less than a page. See:
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I disagree that the BRS proof is complicated. Given the rational function approximating sgn, the proof is just a paragraph. And the rational functions approximating sgn were mostly constructed already by Newman. In any case, BRS give their self-contained construction/proof in a couple dozen sentences. – Ryan O'Donnell Jun 23 at 23:01
Witten's proof of the positive energy theorem using spinors drastically simplified the original proof by Schoen and Yau.
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The isosceles triangle theorem (pons asinorum), that the angles opposite the equal sides of an isosceles triangle are equal, was originally proved by Euclid by constructing several auxiliary lines. Pappus' proof uses no auxiliary lines, but only side-angle-side by "flipping" the triangle over to its mirror image.
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It occurs to me that Morse theory is a good example. At the time of Morse, algebraic topology (even the notion of CW complex or cell complex) is barely developed, which made his combinatorial arguments extremely difficult to read.
Well, nowadays people can simply learn these topics by referring to the definite account of Milnor or Bott.
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Faltings' theorem (aka Mordell conjecture) can be taken as such an example. Different methods have been used so far with various difficulties.
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Widom's formula for calculating determinants of banded Toeplitz matrices. The original paper is hard to understand and uses quite intricate techniques.
Now, a quite simple proof can be found in Böttchers "Spectral Properties of Banded Toeplitz Matrices". Actually, it also follows quite directly from the formula on Hall-Littlewood polynomials here: http://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood_polynomials
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Kottman proved that in any infinite-dimensional Banach space one can find a sequence $(x_n)_{n=1}^\infty$ of unit vectors with
$$\|x_n-x_m\|>1$$ whenever $n\neq m$. The original proof is quite messy, but there is a yet another proof, attributed to Starbird, which can be found in Diestel's book Sequences and series in Banach spaces. It uses essentially linear algebra and the Hahn-Banach theorem only.
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The global (or homology) version of Cauchy’s theorem was given an elementary proof by John Dixon. I believe this is mentioned in Rudin's Real and Complex Analysis. A proof is available online at http://www.math.uiuc.edu/~r-ash/CV/CV3.pdf. This states "The elementary proof to be presented below is due to John Dixon, and appeared in Proc. Amer. Math. Soc. 29 (1971), pp. 625-626, but the theorem as stated is originally due to E.Artin."
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Szemeredi's theorem and its special case roth's theorem have been given quite conceptual proofs by Hilel Furstenberg using ergodic methods which I think is quite natural while the initial proofs were extremely complicated.
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Tverberg Theorem (1965): Let $x_1,x_2,\dots, x_m$ be points in $R^d$, $m \ge (r-1)(d+1)+1$. Then there is a partition $S_1,S_2,\dots, S_r$ of $\{1,2,\dots,m\}$ such that $\cap _{j=1}^rconv (x_i: i \in S_j) \ne \emptyset$.
Tverberg's theorem was conjectured by Birch who also proved the planar case. The case $r=2$ is a 1920 theorem of Radon which follows easily from linear algebra consideration.
(The first thing to note is that Tverberg's theorem is sharp. If you have only $(r-1)(d+1)$ points in $R^d$ in a "generic" position then for every partition into $r$ parts even the affine spans of the points in the parts will not have a point in common.)
The first proof of this theorem appeared in 1965. It was rather complicated and was based on the idea to first prove the theorem for points in some special position and then show that when you continuously change the location of the points the theorem remains true. A common dream was to find an extension of the proof of Radon's theorem, a proof which is based on the two types of numbers - positive and negative. Somehow we need three, four, or $r$ types of numbers. In 1981 Helge Tverberg found yet another proof of his theorem. This proof was inspired by Barany's proof of the colored Caratheodory theorem (mentioned below) and it was still rather complicated. It once took me 6-7 hours in class to present it.
What could be the probability of hearing two new simple proofs of Tverberg'stheorem on the same day? While visiting the Mittag-Leffler Institute in 1992, I met Helge one day around lunch and asked him if he has found a new proof. To my surprise, he told me about a new proof that he found with Sinisa Vrecica. This is a proof that can be presented in class in 2 hours! It appeared (look here) along with a far-reaching conjecture (still unproved). Later in the afternoon I met Karanbir Sarkaria and he told me about a proof he found to Tverberg's theorem which was absolutely startling. This is a proof you can present in a one hour lecture; it also somehow goes along with the dream of having $r$ "types" of numbers replacing the role of positive and negative real numbers. Another very simple proof of Tverberg's theorem was found by Jean-Pierre Roudneff in 1999.
For further details see these blog posts (I,II).
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Dear @Gil Kalai: I fixed the hyperlinks in this answer. Also, it should be noted that all the links other than the blog posts require an AMS subscription. – Ricardo Andrade Nov 10 at 10:44
See Ostrowski's proof of Luroth theorem in Schinzel's book "Polynomials with special regard to reducibility"
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ah ic. I think MB proved something more general. For every subset $S\subseteq\Bbb N$, there is a finite subset $S_f\subsetneq S$ such that if a quadratic form represents $S_f$, then it represents $S$. Is there a link to the proof? How constructive is the set $S_f$? – J.A Nov 8 at 19:30
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2013-12-18 15:29:40
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