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## G = C3×D5⋊C16order 480 = 25·3·5

### Direct product of C3 and D5⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C3×D5⋊C16
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C3×C5⋊2C8 — C3×C5⋊C16 — C3×D5⋊C16
 Lower central C5 — C3×D5⋊C16
 Upper central C1 — C24

Generators and relations for C3×D5⋊C16
G = < a,b,c,d | a3=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Smallest permutation representation of C3×D5⋊C16
On 240 points
Generators in S240
(1 61 34)(2 62 35)(3 63 36)(4 64 37)(5 49 38)(6 50 39)(7 51 40)(8 52 41)(9 53 42)(10 54 43)(11 55 44)(12 56 45)(13 57 46)(14 58 47)(15 59 48)(16 60 33)(17 82 113)(18 83 114)(19 84 115)(20 85 116)(21 86 117)(22 87 118)(23 88 119)(24 89 120)(25 90 121)(26 91 122)(27 92 123)(28 93 124)(29 94 125)(30 95 126)(31 96 127)(32 81 128)(65 218 168)(66 219 169)(67 220 170)(68 221 171)(69 222 172)(70 223 173)(71 224 174)(72 209 175)(73 210 176)(74 211 161)(75 212 162)(76 213 163)(77 214 164)(78 215 165)(79 216 166)(80 217 167)(97 230 183)(98 231 184)(99 232 185)(100 233 186)(101 234 187)(102 235 188)(103 236 189)(104 237 190)(105 238 191)(106 239 192)(107 240 177)(108 225 178)(109 226 179)(110 227 180)(111 228 181)(112 229 182)(129 194 146)(130 195 147)(131 196 148)(132 197 149)(133 198 150)(134 199 151)(135 200 152)(136 201 153)(137 202 154)(138 203 155)(139 204 156)(140 205 157)(141 206 158)(142 207 159)(143 208 160)(144 193 145)
(1 183 133 115 223)(2 116 184 224 134)(3 209 117 135 185)(4 136 210 186 118)(5 187 137 119 211)(6 120 188 212 138)(7 213 121 139 189)(8 140 214 190 122)(9 191 141 123 215)(10 124 192 216 142)(11 217 125 143 177)(12 144 218 178 126)(13 179 129 127 219)(14 128 180 220 130)(15 221 113 131 181)(16 132 222 182 114)(17 196 111 59 171)(18 60 197 172 112)(19 173 61 97 198)(20 98 174 199 62)(21 200 99 63 175)(22 64 201 176 100)(23 161 49 101 202)(24 102 162 203 50)(25 204 103 51 163)(26 52 205 164 104)(27 165 53 105 206)(28 106 166 207 54)(29 208 107 55 167)(30 56 193 168 108)(31 169 57 109 194)(32 110 170 195 58)(33 149 69 229 83)(34 230 150 84 70)(35 85 231 71 151)(36 72 86 152 232)(37 153 73 233 87)(38 234 154 88 74)(39 89 235 75 155)(40 76 90 156 236)(41 157 77 237 91)(42 238 158 92 78)(43 93 239 79 159)(44 80 94 160 240)(45 145 65 225 95)(46 226 146 96 66)(47 81 227 67 147)(48 68 82 148 228)
(1 223)(2 134)(3 185)(4 118)(5 211)(6 138)(7 189)(8 122)(9 215)(10 142)(11 177)(12 126)(13 219)(14 130)(15 181)(16 114)(18 60)(19 97)(20 174)(22 64)(23 101)(24 162)(26 52)(27 105)(28 166)(30 56)(31 109)(32 170)(33 83)(34 70)(35 151)(36 232)(37 87)(38 74)(39 155)(40 236)(41 91)(42 78)(43 159)(44 240)(45 95)(46 66)(47 147)(48 228)(49 161)(50 203)(51 103)(53 165)(54 207)(55 107)(57 169)(58 195)(59 111)(61 173)(62 199)(63 99)(67 81)(68 148)(71 85)(72 152)(75 89)(76 156)(79 93)(80 160)(84 230)(88 234)(92 238)(96 226)(100 201)(104 205)(108 193)(112 197)(115 183)(116 224)(119 187)(120 212)(123 191)(124 216)(127 179)(128 220)(131 221)(132 182)(135 209)(136 186)(139 213)(140 190)(143 217)(144 178)(145 225)(149 229)(153 233)(157 237)(163 204)(167 208)(171 196)(175 200)
(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)

G:=sub<Sym(240)| (1,61,34)(2,62,35)(3,63,36)(4,64,37)(5,49,38)(6,50,39)(7,51,40)(8,52,41)(9,53,42)(10,54,43)(11,55,44)(12,56,45)(13,57,46)(14,58,47)(15,59,48)(16,60,33)(17,82,113)(18,83,114)(19,84,115)(20,85,116)(21,86,117)(22,87,118)(23,88,119)(24,89,120)(25,90,121)(26,91,122)(27,92,123)(28,93,124)(29,94,125)(30,95,126)(31,96,127)(32,81,128)(65,218,168)(66,219,169)(67,220,170)(68,221,171)(69,222,172)(70,223,173)(71,224,174)(72,209,175)(73,210,176)(74,211,161)(75,212,162)(76,213,163)(77,214,164)(78,215,165)(79,216,166)(80,217,167)(97,230,183)(98,231,184)(99,232,185)(100,233,186)(101,234,187)(102,235,188)(103,236,189)(104,237,190)(105,238,191)(106,239,192)(107,240,177)(108,225,178)(109,226,179)(110,227,180)(111,228,181)(112,229,182)(129,194,146)(130,195,147)(131,196,148)(132,197,149)(133,198,150)(134,199,151)(135,200,152)(136,201,153)(137,202,154)(138,203,155)(139,204,156)(140,205,157)(141,206,158)(142,207,159)(143,208,160)(144,193,145), (1,183,133,115,223)(2,116,184,224,134)(3,209,117,135,185)(4,136,210,186,118)(5,187,137,119,211)(6,120,188,212,138)(7,213,121,139,189)(8,140,214,190,122)(9,191,141,123,215)(10,124,192,216,142)(11,217,125,143,177)(12,144,218,178,126)(13,179,129,127,219)(14,128,180,220,130)(15,221,113,131,181)(16,132,222,182,114)(17,196,111,59,171)(18,60,197,172,112)(19,173,61,97,198)(20,98,174,199,62)(21,200,99,63,175)(22,64,201,176,100)(23,161,49,101,202)(24,102,162,203,50)(25,204,103,51,163)(26,52,205,164,104)(27,165,53,105,206)(28,106,166,207,54)(29,208,107,55,167)(30,56,193,168,108)(31,169,57,109,194)(32,110,170,195,58)(33,149,69,229,83)(34,230,150,84,70)(35,85,231,71,151)(36,72,86,152,232)(37,153,73,233,87)(38,234,154,88,74)(39,89,235,75,155)(40,76,90,156,236)(41,157,77,237,91)(42,238,158,92,78)(43,93,239,79,159)(44,80,94,160,240)(45,145,65,225,95)(46,226,146,96,66)(47,81,227,67,147)(48,68,82,148,228), (1,223)(2,134)(3,185)(4,118)(5,211)(6,138)(7,189)(8,122)(9,215)(10,142)(11,177)(12,126)(13,219)(14,130)(15,181)(16,114)(18,60)(19,97)(20,174)(22,64)(23,101)(24,162)(26,52)(27,105)(28,166)(30,56)(31,109)(32,170)(33,83)(34,70)(35,151)(36,232)(37,87)(38,74)(39,155)(40,236)(41,91)(42,78)(43,159)(44,240)(45,95)(46,66)(47,147)(48,228)(49,161)(50,203)(51,103)(53,165)(54,207)(55,107)(57,169)(58,195)(59,111)(61,173)(62,199)(63,99)(67,81)(68,148)(71,85)(72,152)(75,89)(76,156)(79,93)(80,160)(84,230)(88,234)(92,238)(96,226)(100,201)(104,205)(108,193)(112,197)(115,183)(116,224)(119,187)(120,212)(123,191)(124,216)(127,179)(128,220)(131,221)(132,182)(135,209)(136,186)(139,213)(140,190)(143,217)(144,178)(145,225)(149,229)(153,233)(157,237)(163,204)(167,208)(171,196)(175,200), (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)>;

G:=Group( (1,61,34)(2,62,35)(3,63,36)(4,64,37)(5,49,38)(6,50,39)(7,51,40)(8,52,41)(9,53,42)(10,54,43)(11,55,44)(12,56,45)(13,57,46)(14,58,47)(15,59,48)(16,60,33)(17,82,113)(18,83,114)(19,84,115)(20,85,116)(21,86,117)(22,87,118)(23,88,119)(24,89,120)(25,90,121)(26,91,122)(27,92,123)(28,93,124)(29,94,125)(30,95,126)(31,96,127)(32,81,128)(65,218,168)(66,219,169)(67,220,170)(68,221,171)(69,222,172)(70,223,173)(71,224,174)(72,209,175)(73,210,176)(74,211,161)(75,212,162)(76,213,163)(77,214,164)(78,215,165)(79,216,166)(80,217,167)(97,230,183)(98,231,184)(99,232,185)(100,233,186)(101,234,187)(102,235,188)(103,236,189)(104,237,190)(105,238,191)(106,239,192)(107,240,177)(108,225,178)(109,226,179)(110,227,180)(111,228,181)(112,229,182)(129,194,146)(130,195,147)(131,196,148)(132,197,149)(133,198,150)(134,199,151)(135,200,152)(136,201,153)(137,202,154)(138,203,155)(139,204,156)(140,205,157)(141,206,158)(142,207,159)(143,208,160)(144,193,145), (1,183,133,115,223)(2,116,184,224,134)(3,209,117,135,185)(4,136,210,186,118)(5,187,137,119,211)(6,120,188,212,138)(7,213,121,139,189)(8,140,214,190,122)(9,191,141,123,215)(10,124,192,216,142)(11,217,125,143,177)(12,144,218,178,126)(13,179,129,127,219)(14,128,180,220,130)(15,221,113,131,181)(16,132,222,182,114)(17,196,111,59,171)(18,60,197,172,112)(19,173,61,97,198)(20,98,174,199,62)(21,200,99,63,175)(22,64,201,176,100)(23,161,49,101,202)(24,102,162,203,50)(25,204,103,51,163)(26,52,205,164,104)(27,165,53,105,206)(28,106,166,207,54)(29,208,107,55,167)(30,56,193,168,108)(31,169,57,109,194)(32,110,170,195,58)(33,149,69,229,83)(34,230,150,84,70)(35,85,231,71,151)(36,72,86,152,232)(37,153,73,233,87)(38,234,154,88,74)(39,89,235,75,155)(40,76,90,156,236)(41,157,77,237,91)(42,238,158,92,78)(43,93,239,79,159)(44,80,94,160,240)(45,145,65,225,95)(46,226,146,96,66)(47,81,227,67,147)(48,68,82,148,228), (1,223)(2,134)(3,185)(4,118)(5,211)(6,138)(7,189)(8,122)(9,215)(10,142)(11,177)(12,126)(13,219)(14,130)(15,181)(16,114)(18,60)(19,97)(20,174)(22,64)(23,101)(24,162)(26,52)(27,105)(28,166)(30,56)(31,109)(32,170)(33,83)(34,70)(35,151)(36,232)(37,87)(38,74)(39,155)(40,236)(41,91)(42,78)(43,159)(44,240)(45,95)(46,66)(47,147)(48,228)(49,161)(50,203)(51,103)(53,165)(54,207)(55,107)(57,169)(58,195)(59,111)(61,173)(62,199)(63,99)(67,81)(68,148)(71,85)(72,152)(75,89)(76,156)(79,93)(80,160)(84,230)(88,234)(92,238)(96,226)(100,201)(104,205)(108,193)(112,197)(115,183)(116,224)(119,187)(120,212)(123,191)(124,216)(127,179)(128,220)(131,221)(132,182)(135,209)(136,186)(139,213)(140,190)(143,217)(144,178)(145,225)(149,229)(153,233)(157,237)(163,204)(167,208)(171,196)(175,200), (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) );

G=PermutationGroup([[(1,61,34),(2,62,35),(3,63,36),(4,64,37),(5,49,38),(6,50,39),(7,51,40),(8,52,41),(9,53,42),(10,54,43),(11,55,44),(12,56,45),(13,57,46),(14,58,47),(15,59,48),(16,60,33),(17,82,113),(18,83,114),(19,84,115),(20,85,116),(21,86,117),(22,87,118),(23,88,119),(24,89,120),(25,90,121),(26,91,122),(27,92,123),(28,93,124),(29,94,125),(30,95,126),(31,96,127),(32,81,128),(65,218,168),(66,219,169),(67,220,170),(68,221,171),(69,222,172),(70,223,173),(71,224,174),(72,209,175),(73,210,176),(74,211,161),(75,212,162),(76,213,163),(77,214,164),(78,215,165),(79,216,166),(80,217,167),(97,230,183),(98,231,184),(99,232,185),(100,233,186),(101,234,187),(102,235,188),(103,236,189),(104,237,190),(105,238,191),(106,239,192),(107,240,177),(108,225,178),(109,226,179),(110,227,180),(111,228,181),(112,229,182),(129,194,146),(130,195,147),(131,196,148),(132,197,149),(133,198,150),(134,199,151),(135,200,152),(136,201,153),(137,202,154),(138,203,155),(139,204,156),(140,205,157),(141,206,158),(142,207,159),(143,208,160),(144,193,145)], [(1,183,133,115,223),(2,116,184,224,134),(3,209,117,135,185),(4,136,210,186,118),(5,187,137,119,211),(6,120,188,212,138),(7,213,121,139,189),(8,140,214,190,122),(9,191,141,123,215),(10,124,192,216,142),(11,217,125,143,177),(12,144,218,178,126),(13,179,129,127,219),(14,128,180,220,130),(15,221,113,131,181),(16,132,222,182,114),(17,196,111,59,171),(18,60,197,172,112),(19,173,61,97,198),(20,98,174,199,62),(21,200,99,63,175),(22,64,201,176,100),(23,161,49,101,202),(24,102,162,203,50),(25,204,103,51,163),(26,52,205,164,104),(27,165,53,105,206),(28,106,166,207,54),(29,208,107,55,167),(30,56,193,168,108),(31,169,57,109,194),(32,110,170,195,58),(33,149,69,229,83),(34,230,150,84,70),(35,85,231,71,151),(36,72,86,152,232),(37,153,73,233,87),(38,234,154,88,74),(39,89,235,75,155),(40,76,90,156,236),(41,157,77,237,91),(42,238,158,92,78),(43,93,239,79,159),(44,80,94,160,240),(45,145,65,225,95),(46,226,146,96,66),(47,81,227,67,147),(48,68,82,148,228)], [(1,223),(2,134),(3,185),(4,118),(5,211),(6,138),(7,189),(8,122),(9,215),(10,142),(11,177),(12,126),(13,219),(14,130),(15,181),(16,114),(18,60),(19,97),(20,174),(22,64),(23,101),(24,162),(26,52),(27,105),(28,166),(30,56),(31,109),(32,170),(33,83),(34,70),(35,151),(36,232),(37,87),(38,74),(39,155),(40,236),(41,91),(42,78),(43,159),(44,240),(45,95),(46,66),(47,147),(48,228),(49,161),(50,203),(51,103),(53,165),(54,207),(55,107),(57,169),(58,195),(59,111),(61,173),(62,199),(63,99),(67,81),(68,148),(71,85),(72,152),(75,89),(76,156),(79,93),(80,160),(84,230),(88,234),(92,238),(96,226),(100,201),(104,205),(108,193),(112,197),(115,183),(116,224),(119,187),(120,212),(123,191),(124,216),(127,179),(128,220),(131,221),(132,182),(135,209),(136,186),(139,213),(140,190),(143,217),(144,178),(145,225),(149,229),(153,233),(157,237),(163,204),(167,208),(171,196),(175,200)], [(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)]])

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H 10 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 16A ··· 16P 20A 20B 24A ··· 24H 24I ··· 24P 30A 30B 40A 40B 40C 40D 48A ··· 48AF 60A 60B 60C 60D 120A ··· 120H order 1 2 2 2 3 3 4 4 4 4 5 6 6 6 6 6 6 8 8 8 8 8 8 8 8 10 12 12 12 12 12 12 12 12 15 15 16 ··· 16 20 20 24 ··· 24 24 ··· 24 30 30 40 40 40 40 48 ··· 48 60 60 60 60 120 ··· 120 size 1 1 5 5 1 1 1 1 5 5 4 1 1 5 5 5 5 1 1 1 1 5 5 5 5 4 1 1 1 1 5 5 5 5 4 4 5 ··· 5 4 4 1 ··· 1 5 ··· 5 4 4 4 4 4 4 5 ··· 5 4 4 4 4 4 ··· 4

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 F5 C2×F5 C3×F5 D5⋊C8 C6×F5 D5⋊C16 C3×D5⋊C8 C3×D5⋊C16 kernel C3×D5⋊C16 C3×C5⋊C16 D5×C24 D5⋊C16 C120 D5×C12 C5⋊C16 C8×D5 C3×Dic5 C6×D5 C40 C4×D5 C3×D5 Dic5 D10 D5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 4 4 16 8 8 32 1 1 2 2 2 4 4 8

Matrix representation of C3×D5⋊C16 in GL5(𝔽241)

 15 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 240 1 0 0 0 240 0 1 0 0 240 0 0 1 0 240 0 0 0
,
 240 0 0 0 0 0 240 0 0 0 0 240 0 0 1 0 240 0 1 0 0 240 1 0 0
,
 1 0 0 0 0 0 44 197 172 0 0 216 197 0 44 0 44 0 197 216 0 0 172 197 44

G:=sub<GL(5,GF(241))| [15,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,240,240,240,240,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0],[240,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0],[1,0,0,0,0,0,44,216,44,0,0,197,197,0,172,0,172,0,197,197,0,0,44,216,44] >;

C3×D5⋊C16 in GAP, Magma, Sage, TeX

C_3\times D_5\rtimes C_{16}
% in TeX

G:=Group("C3xD5:C16");
// GroupNames label

G:=SmallGroup(480,269);
// by ID

G=gap.SmallGroup(480,269);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,176,80,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

Export

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text:
# Find the constant $a$ such that $Y_t$ is a martingale.

Let $$X_t$$ be the solution of SDE $$\text{d}X_t=3X_t\text dt+2X_t\text dB_t$$ and $$X_0=1$$ which $$B_t$$ denotes the Brownian motion with $$B_0=0$$. Let $$Y_t=e^{at}X_t$$, Find the constant $$a$$ such that $$Y_t$$ is a martingale.

$$dY_t=ae^{at}X_t+e^{at}dX_t=ae^{at}X_t+e^{at}(3X_t\text dt+2X_t\text dB_t)=ae^{at}X_t+3e^{at}X_t\text dt+2e^{at}X_t\text dB_t$$

That's all I can get, I fail to apply this to prove $$E[Y_{t+1}|\mathcal{F}_s]=Y_t$$ Any hints would be helpful.

You've made a mistake when applying Ito's lemma. You should have $$dY_t = ae^{at}X_t dt + e^{at} dX_t = (3+a)e^{at}X_t dt + 2e^{at} X_t dB_t = (3+a) Y_t dt + 2 Y_t dB_t$$ (notice the presence of $$dt$$!). For this to be a martingale, it must at least have no $$dt$$ component and hence (since the initial condition rules out $$Y = 0$$) we have $$a=-3$$. In this case, $$Y$$ is (up to a constant) the stochastic exponential of Brownian motion and hence is a martingale.

format: plain


text:
main-content

## Weitere Artikel dieser Ausgabe durch Wischen aufrufen

01.06.2015 | Ausgabe 3/2015 Open Access

# Coherent Detection of Turbo-Coded OFDM Signals Transmitted Through Frequency Selective Rayleigh Fading Channels with Receiver Diversity and Increased Throughput

Zeitschrift:
Wireless Personal Communications > Ausgabe 3/2015
Autor:
K. Vasudevan
Wichtige Hinweise
This work is supported by the India-UK Advanced Technology Center (IU-ATC) of Excellence in Next Generation Networks, Systems and Services under grant SR/RCUK-DST/IUATC Phase 2/2012-IIT K (C), sponsored by DST-EPSRC. This work was partly presented as an Invited Talk at the International Federation of Nonlinear Analysts (IFNA) World Congress, Greece, June 2012 and partly published in ISPCC, Sept. 2013, Shimla and ITST Nov. 2013, Finland.

## Abstract

In this work, we discuss techniques for coherently detecting turbo coded orthogonal frequency division multiplexed (OFDM) signals, transmitted through frequency selective Rayleigh (the magnitude of each channel tap is Rayleigh distributed) fading channels having a uniform power delay profile. The channel output is further distorted by a carrier frequency and phase offset, besides additive white Gaussian noise. A new frame structure for OFDM, consisting of a known preamble, cyclic prefix, data and known postamble is proposed, which has a higher throughput compared to the earlier work. A robust turbo decoder is proposed, which functions effectively over a wide range of signal-to-noise ratio (SNR). Simulation results show that it is possible to achieve a bit-error-rate (BER) of $$10^{-5}$$ at an SNR per bit as low as 8 dB and throughput of 82.84 %, using a single transmit and two receive antennas. We also demonstrate that the practical coherent receiver requires just about 1 dB more power compared to that of an ideal coherent receiver, to attain a BER of $$10^{-5}$$. The key contribution to the good performance of the practical coherent receiver is due to the use of a long preamble (512 QPSK symbols), which is perhaps not specified in any of the current wireless communication standards. We have also shown from computer simulations that, it is possible to obtain even better BER performance, using a better code. A simple and approximate Cramér–Rao bound on the variance of the frequency offset estimation error for coherent detection, is derived. The proposed algorithms are well suited for implementation on a DSP-platform.

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Documentation

Simulate GARCH Models

This example shows how to simulate from a GARCH process with and without specifying presample data. The sample unconditional variances of the Monte Carlo simulations approximate the theoretical GARCH unconditional variance.

Step 1. Specify a GARCH model.

Specify a GARCH(1,1) model ${\epsilon }_{t}={\sigma }_{t}{z}_{t},$ where the distribution of ${z}_{t}$ is Gaussian and

`${\sigma }_{t}^{2}=0.01+0.7{\sigma }_{t-1}^{2}+0.25{\epsilon }_{t-1}^{2}.$`

`Mdl = garch('Constant',0.01,'GARCH',0.7,'ARCH',0.25)`
```Mdl = garch with properties: Description: "GARCH(1,1) Conditional Variance Model (Gaussian Distribution)" Distribution: Name = "Gaussian" P: 1 Q: 1 Constant: 0.01 GARCH: {0.7} at lag [1] ARCH: {0.25} at lag [1] Offset: 0 ```

Step 2. Simulate from the model without using presample data.

Simulate five paths of length 100 from the GARCH(1,1) model, without specifying any presample innovations or conditional variances. Display the first conditional variance for each of the five sample paths. The model being simulated does not have a mean offset, so the response series is an innovation series.

```rng default; % For reproducibility [Vn,Yn] = simulate(Mdl,100,'NumPaths',5); Vn(1,:) % Display variances```
```ans = 1×5 0.1645 0.3182 0.4051 0.1872 0.1551 ```
```figure subplot(2,1,1) plot(Vn) xlim([0,100]) title('Conditional Variances') subplot(2,1,2) plot(Yn) xlim([0,100]) title('Innovations')```

The starting conditional variances are different for each realization because no presample data was specified.

Step 3. Simulate from the model using presample data.

Simulate five paths of length 100 from the model, specifying the one required presample innovation and conditional variance. Display the first conditional variance for each of the five sample paths.

```rng default; [Vw,Yw] = simulate(Mdl,100,'NumPaths',5,... 'E0',0.05,'V0',0.001); Vw(1,:)```
```ans = 1×5 0.0113 0.0113 0.0113 0.0113 0.0113 ```
``` figure subplot(2,1,1) plot(Vw) xlim([0,100]) title('Conditional Variances') subplot(2,1,2) plot(Yw) xlim([0,100]) title('Innovations')```

All five sample paths have the same starting conditional variance, calculated using the presample data.

Note that even with the same starting variance, the realizations of the innovation series have different starting points. This is because each ${\epsilon }_{1}$ is a random draw from a Gaussian distribution with mean 0 and variance ${\sigma }_{1}=0.0113$.

Step 4. Look at the unconditional variances.

Simulate 10,000 sample paths of length 500 from the specified GARCH model. Plot the sample unconditional variances of the Monte Carlo simulations, and compare them to the theoretical unconditional variance,

`${\sigma }_{\epsilon }^{2}=\frac{\kappa }{\left(1-{\gamma }_{1}-{\alpha }_{1}\right)}=\frac{0.01}{\left(1-0.7-0.25\right)}=0.2.$`

```sig2 = 0.01/(1-0.7-0.25); rng default; [V,Y] = simulate(Mdl,500,'NumPaths',10000); figure plot(var(Y,0,2),'Color',[.7,.7,.7],'LineWidth',1.5) xlim([0,500]) hold on plot(1:500,ones(500,1)*sig2,'k--','LineWidth',2) legend('Simulated','Theoretical','Location','NorthWest') title('Unconditional Variance') hold off```

The simulated unconditional variances fluctuate around the theoretical unconditional variance.


text:
# Expressing any f(x,y) using only addition and unary functions?

Suppose we have a continuous function $f:R^2\rightarrow R$. I was told of the following remarkable theorem: $f$ can be expressed as the composition of continuous unary functions (that is, functions from $R\rightarrow R$) and addition.

Could anyone give me a reference (or name) for this theorem?

-

## 3 Answers

This is "due in successively more exact forms to Kolmogorov, Arnol'd and a succession of mathematicians ending with Kahane", to quote T.W. Korner on the subject.

I am informed that the proof I met is prepared using:

J.-P. Kahane Sur le treizieme probleme de Hilbert, le theoreme de superposition de Kolmogorov et les sommes algebriques d'arcs croissants in the conference proceedings Harmonic analysis, Iraklion 1978 Springer 1980

G. G. Lorenz, Approximation of functions Chelsea Publishing Co. 1986 (First Ed. 1966)

A. G. Vituskin On the representation of functions by superpositions and related topics in L'Enseignement Mathématique, 1977, Vol 23, pages 255-320

[This is all from these skeleton notes (no proofs) here (Links to pdf; See Chapter 1 and Chapter 11 for references)]

-
Thanks for the link! –  Andres Caicedo Sep 1 '10 at 15:11

I think this was proven by Kolmogorov. See the following reference: A. N. Kolmogorov, “On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition”, Dokl. Akad. Nauk SSSR 114 (1957), 953–956; English transl., Amer. Math. Soc. Transl. (2) 28 (1963), 55–59.

-

I find it interesting that this theorem is used as a theoretical basis for neural networks. I had never heard this before.

Věra Kůrková, Kolmogorov's theorem and multilayer neural networks 1992

-
There seems to be some debate on the relevance of Kolmogorov's theorem to learning theory. A rebuttal can be found in "Representation Properties of Networks: Kolmogorov's Theorem is Irrelevant" by Girosi and Poggio, 1989. A PDF copy is online here: cbcl.mit.edu/projects/cbcl/people/poggio/journals/… –  Bill Bradley Sep 7 '10 at 13:03
I was thinking along the same lines. It would guess that the functions postulated by Kolmogorov's theorem would be smooth only in simple cases. Thanks. –  dls Sep 8 '10 at 15:29


text:
# How to program an enemy to attack when someone gets near it?

Like in free-roam RPGs, like Skyrim, I want to make a super-rudimentary screen with just two objects: player and enemy. The enemy will walk around randomly through AI algorithms, and will only fight back gracefully when the player gets near them, and when the player runs away past a certain distance it will continue its AI walking cycle. How do I do this?

Any guidance would be nice.

• What have you tried already? What exactly are you having trouble with? It sounds like you know what you want to do, but haven't tried it yet, why not? – MichaelHouse Dec 10 '13 at 22:18

Here is a simple pseudo-code implementation that might give you some guidance, given the details you've described:

if ( length(playerPosition - enemyPosition) < enemyAttackRadius)
enemy.attackPlayer()

if (enemy.isAttacking() && length(playerPosition - enemyPosition) > enemyDisengageRadius)
enemy.stopAttacking()

• I see what you mean, but its still incorrect. (playerPosition - enemyPosition < enemyAttackRadius), how do you compare 2D or 3D coordinates with a scalar exactly? You would rather be using a distance function or the length of the resulting vector in that operation.. – Grimshaw Dec 10 '13 at 22:00
• I agree with you. (playerPosition - enemyPosition) must resolve to a scalar; the subtraction operator wasn't meant to be taken literally. In my mind it was a distance function, I suppose I should have been more specific. – Ryan Dec 10 '13 at 22:04

Here's one basic way to implement it.

if (distance from player to enemy < attack_distance)
{
enemy_target = player_position;
}
else if (distance from enemy to enemy destination is small)
{
enemy_destination = random_destination();
enemy_target = enemy_destination;
}
else
{
enemy_target = enemy_destination;
}

if (enemy can attack player)
{
attack();
}
else
{
move enemy towards enemy_target;
}

For this implementation, your game state consists of three positions - one each for the current position of the player and enemy, and one for the current enemy destination, which should be initialized to the enemy's position.

You can create an area/bounding box bound to the NPC, when player enters in the box, you start processing the data

The suggestions in the other answers are checking constantly checking the distance of the player and the NPC, while your game is small this is fine, but that can become slower and slower as you add NPCs.

So create an areabox class, and apply to all your NPC. Create events on the areabox, or thresholds inner boxes, on which you can attach other events, like: if is in this area : give up. This way you can also handle the NPC orientation cause I'm pretty sure you don't want it to attack if NPC is giving shoulders to your player.

Now depending what GL library you are using(OpenGl/DirectX), collision detection can be generalized and is way cheaper to run, also you don't need to do the math on the position all the time

//Pseudo code:
function OnAreaColisionEvent(){
tryToAttack = true; // or whatever
};

function OnGiveUpAreaCollisionEvent(){
stopChasing = true;
ReturnToPatrol();
}

function InAttackAreaEvent(){
if(closeEnough){
performDamage();
}
else{
runToPlayer();
}
}

UPDATE
follow this link for more detailed info on how to speedup collision detection Opengl Collision detection methods

• You do realize, you'd still have to check to see if the player is inside the bounding box on every update. Events are great, but they still need code checking when to fire them. – MichaelHouse Dec 10 '13 at 23:42
• performing checks on areabox is faster than checking on players, the volume takes more space, so if you process data on a player base search algorithm that can become real slow real fast, also compared to mesh collision which means now you need to check for every vertex or faces if there is a collision, with an areabox is only 6 faces math – Sherlock Dec 11 '13 at 16:38


text:
## Bit-Coded Value Help

Typically: "How do I... ", "How can I... " questions
UGV_robotics
Posts: 2
Joined: 12 Mar 2019, 00:03

### Bit-Coded Value Help

Hello I'm currently working with the Regular API function:

sim.auxiliaryConsoleOpen

and I'm having trouble understanding how to implement the different modes such as getting the lines to be wrapped and having the user being able to close the console window. I'm not sure how to put this into my current code:

Code: Select all

auxConsoleHandle=sim.auxiliaryConsoleOpen('Data',9,0x111,{1,2})
Currently the I get get the console window to open and have the title be named 'Data' with max number of lines being 9, but I do not understand how to implement these bit coded values that are stated in the description:

mode: bit-coded value. Bit0 (1) set indicates that the console window will automatically close at simulation end (when called from a simulation script, the console window will always automatically close at simulation end), bit1 (2) set indicates that lines will be wrapped, bit2 (4) set indicates that the user can close the console window, bit3 (8) is not used, bit4 (16) set indicates that the console will not automatically hide when the user switches to another scene (this only applies when called from an add-on script or the sandbox script).

If someone could provide an example of how to include these bit-coded values in my code, that would be greatly appreciated. Also I have no idea what I'm writing with the 0x111 so please ignore the confusion. Also the position {1,2} does not seem to do anything as the console window always opens in the center of my screen.


text:
# Approximating Rooted Steiner Network

Abstract : The Directed Steiner Tree (DST) problem is a cornerstone problem in network design, particularly, in the design of directed networks satisfying connectivity requirements. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an $O(k^\epsilon)$ hardness bound for the rooted $k$-connectivity problem in undirected graphs; this addresses a recent open question of Khanna.
Keywords :
Document type :
Conference papers

Cited literature [16 references]

https://hal.archives-ouvertes.fr/hal-00709994
Contributor : Guyslain Naves Connect in order to contact the contributor
Submitted on : Tuesday, June 19, 2012 - 5:55:08 PM
Last modification on : Thursday, June 18, 2020 - 12:32:04 PM
Long-term archiving on: : Thursday, September 20, 2012 - 2:41:43 AM

### File

bundit-august2.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00709994, version 1

### Citation

Joseph Cheriyan, Bundit Laekhanukit, Guyslain Naves, Adrian Vetta. Approximating Rooted Steiner Network. Symposium on Discrete Algorithms (SODA 2012), Jan 2012, Kyoto, Japan. ⟨hal-00709994⟩

Record views


text:
Product Collineations of Projective Planes

# Product Collineations of Projective Planes

Let $F$ be a field and $\mathbb{P}^2(F)$ be the projective plane over $F$. Consider two $3 \times 3$ invertible matrices $M$ and $N$. As we have seen, we can define collineations on $\mathbb{P}^2(F)$ as functions $\phi_M, \phi_N : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ for all $\mathbf{x} \in \mathbb{P}^2(F)$ by $\phi_M(\mathbf{x}) = \mathbf{x}M$ and $\phi_N(\mathbf{x}) = \mathbf{x}N$.

Now from linear algebra, recall that if $M$ and $N$ are invertible matrices, then their products $MN$ and $NM$ are invertible matrices. In particular, we have that:

(1)
\begin{align} \quad (MN)^{-1} = N^{-1}M^{-1} \end{align}

Since $(MN)(MN)^{-1} = MNN^{-1}M^{-1} = MIM^{-1} = MM^{-1} = I$ and $(MN)^{-1}(MN) = N^{-1}M^{-1}MN = N^{-1}IN = N^{-1}N = I$.

So we have verified that $MN$ is an invertible matrix… so what can be said about the collineation $\phi_{MN} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ defined for all $\mathbf{x} \in \mathbb{P}^2(F)$ by $\phi_{MN} (\mathbf{x}) = \mathbf{x}MN$?

 Theorem 1: Let $F$ be a field, $\mathbb{P}^2(F)$ be the projective plane over $F$, and $M$ and $N$ be $3 \times 3$ invertible matrices where $\phi_M, \phi_N : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ for all $\mathbf{x} \in \mathbb{P}^2(F)$ define the collineations $\phi_M(\mathbf{x}) = \mathbf{x}M$ and $\phi_N(\mathbf{x}) = \mathbf{x}M$. Then $\phi_{MN} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ is given by $\phi_{MN} = \phi_N \circ \phi_M$ for all $\mathbf{x} \in \mathbb{P}^2(F)$ and $\phi_{MN}^{-1} = \phi_M^{-1} \circ \phi_N^{-1}$.
• Proof: Let $\mathbf{x} \in F$. Then:
(2)
\begin{align} \quad (\phi_N \circ \phi_M)(\mathbf{x}) = \phi_N(\phi_M(\mathbf{x})) = \phi_M(\mathbf{x}M) = (\mathbf{x}M)N = \mathbf{x}MN = \phi_{MN} \end{align}
• Furthermore we have that:
(3)
\begin{align} \quad (\phi_M^{-1} \circ \phi_N^{-1})(\mathbf{x}) = \phi_M^{-1}(\phi_N^{-1}(\mathbf{x})) = \phi_M^{-1}(\mathbf{x}N^{-1}) = \mathbf{x}N^{-1}M^{-1} \end{align}
• So then:
(4)
\begin{align} \quad (\phi_M^{-1} \circ \phi_N^{-1})(\phi_{MN}(\mathbf{x})) = (\mathbf{x}MN)N^{-1}M^{-1} = \mathbf{x} \end{align}
• And also:
(5)
\begin{align} \quad \phi_{MN}((\phi_M^{-1} \circ \phi_N^{-1})(\mathbf{x})) = \phi_MN(\mathbf{x}N^{-1}M^{-1}) = (\mathbf{x}N^{-1}M^{-1})MN = \mathbf{x} \end{align}
• Therefore $\phi_{MN}^{-1} = \phi_M^{-1} \circ \phi_N^{-1}$. $\blacksquare$


text:
30.1.26 Cross

Function

$y=A_{2}+\frac{A_{1}-A_{2}}{1+(tx)^{m}}$

Brief Description

Cross model to describe pseudoplastic flow with asymptotic viscosities at zero and infinite shear rates.

Parameters

Number: 4

Names: A1, A2, t, m

Meanings: A1 = zero shear viscosity, A2 = infinite shear viscosity, t = time constant, m = power index

Lower Bounds: A1 > 0, A2 > 0, t > 0, m >= 0

Upper Bounds: none

Script Access

nlf_Cross(x,A1,A2,t,m)

Function File

fitfunc\Cross.fdf

Rheology


text:
?

Free Version
Difficult

# Wavelength of Photon Released from Hydrogen Atom

APPH12-Z33HJB

What is the wavelength of a photon released from a hydrogen atom when an electron drops from the $n=5$ state to the $n=2$ state?

A

2.856 nm

B

434 nm

C

465 nm

D

228 nm


text:
• July 14, 2020, 11:36:28 PM
• Welcome, Guest

Pages: 1 ... 10 11 12 [13] 14 15   Go Down

### AuthorTopic: Morane Saulnier G/H/L/LA  (Read 27831 times)

0 Members and 1 Guest are viewing this topic.

#### Mission_bug

• Modder
• member
• Offline
• Posts: 4765
##### Re: Morane Saulnier G/H/L/LA
« Reply #144 on: December 29, 2015, 05:38:26 AM »

Hello guys, this is the latest update I posted for the complete set of aircraft:

http://www.sas1946.com/main/index.php/topic,49180.msg542849.html#msg542849

Please ignore the first page link as that is out of date as is the one below posted by Istvan.

Gio asked me to post my latest additions as he was busy with other things so I was only able to post where I did, sorry if it is confusing but until everything is finalised please just add from those links.

The link Istvan gave comes after that but please only use the 3do folder as I suggest because that was a earlier version sent to Istvan for testing purposes before the I and P models were added.

The new skins look outstanding as always Istvan, looking forward to using them.

These aircraft use the Dreamk weapons packs so they must be installed, DBW 1916 uses a version of that which Istvan put together, if you use other installs then use the pack in Dreamks link.

When everything is complete hopefully we will be able to remove all links in the thread and move the most up to date into the first page.

Take care guys.

Wishing you all the very best, Pete.
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#### vpmedia

• Modder
• member
• Offline
• Posts: 6629
• www.vpmedia.hu/il2
##### Re: Morane Saulnier G/H/L/LA
« Reply #145 on: December 29, 2015, 07:11:42 AM »

Hi Pete! I finished updating the latest planes. Thanks for everything, very nice pack!

http://ulozto.net/xuJ64vs9/aircraft-moranesaulnier-h-and-l-model-family-7z

Basic psd template for Type L Pearly/Plate & Type N + skins for Type N & I:

http://ulozto.net/xQLsDwRs/moranesaulnier-skins-templates-7z
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#### Mission_bug

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• Posts: 4765
##### Re: Morane Saulnier G/H/L/LA
« Reply #146 on: December 29, 2015, 09:01:48 AM »

Thank you very much for the latest update Istvan, greatly appreciated.

We will now use that link as the most up to date for the pack so Gio or one of the admins can now add that in the first post.

For those using the pack posted in the link above for 4.12 with ModAct 5.30 these are the stationary files so if you are using my stationary plane mod for that install these class files can be removed and the entries for staionaries in the readme ignored, use those in the stationary mod.

Code: [Select]
0F2C78D8FCB213D401DA3BFA8A8149EA2AE56594A303115C2E253C687C09D4703C7071CC13AF59DE5E3D8032691F2008353B5C28A35BB6DA529ACF666F435A422248A6C81AEF456616091384B099D97AA70504681E8B211CAEA14D7AAA883C1ECF514A7C30F64068D03852C86B8D10B8DB49C7D675AD44C8E2688F4C80D3190EFBEE16789AB6F8E2FEA16C22DB9E81ECFF8C80A81E7B88B0
http://www.sas1946.com/main/index.php/topic,37950.msg541666.html#msg541666

If you want to use the pack as it comes and do not use my stationary mod then everything can be added as in the download.

Thank you very much to all those involved in this pack, your help and guidance is greatly appreciated.

If a cure is found for the existing problems they will be added later.

Take care guys.

Wishing you all the very best, Pete.
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#### SAS~Storebror

• Editor
• member
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• Posts: 19462
• Failure is not an option.
##### Re: Morane Saulnier G/H/L/LA
« Reply #147 on: December 29, 2015, 10:29:04 AM »

Why this insulting tone??
Sorry Istvan, didn't want to sound rude, I rather tried to express my confusion when I tried to install this mod and suddenly figured that I don't have sufficient knowledge how to get this working properly.
Imagine how confusing all of this must be to new users...

Now that the new pack is there, the required installation of your (fabulous I must say) DBW patch pack has been made clear and now that the instructions are clear, everything works fine and dandy for me too and I get precisely to the point mentioned by vampire when crashing planes don't get killed.
I can also confirm his log entries and the error that causes the LFs to become indestructable:
Code: [Select]
[17:13:58] java.lang.ArrayIndexOutOfBoundsException[17:13:58] at com.maddox.il2.objects.air.MoraneSaulnier_LF2.doWoundPilot(MoraneSaulnier_LF2.java:33)[17:13:58] at com.maddox.il2.fm.AircraftState.doSetPilotState(AircraftState.java:2255)[17:13:58] at com.maddox.il2.fm.AircraftState.setPilotState(AircraftState.java:2058)[17:13:58] at com.maddox.il2.fm.AircraftState.setPilotState(AircraftState.java:2049)[17:13:58] at com.maddox.il2.fm.AircraftState.hitPilot(AircraftState.java:2084)[17:13:58] at com.maddox.il2.objects.air.Aircraft.msgEndAction(Aircraft.java:687)[17:13:58] at com.maddox.rts.MsgEndAction.invokeListener(MsgEndAction.java:59)[17:13:58] at com.maddox.rts.Message._send(Message.java:1217)[17:13:58] at com.maddox.rts.Message.sendToObject(Message.java:1158)[17:13:58] at com.maddox.rts.Message.sendTo(Message.java:1134)[17:13:58] at com.maddox.rts.Message.trySend(Message.java:1115)[17:13:58] at com.maddox.rts.Time.loopMessages(Time.java:252)[17:13:58] at com.maddox.rts.RTSConf.loopMsgs(RTSConf.java:101)[17:13:58] at com.maddox.il2.game.MainWin3D.loopApp(MainWin3D.java:131)[17:13:58] at com.maddox.il2.game.Main.exec(Main.java:437)[17:13:58] at com.maddox.il2.game.GameWin3D.main(GameWin3D.java:235)
One part of the problem, at least in case of the LF2, is the hier.him and the meshes.
For each turret the internal java code logic requires the plane to have a regarding "Turret<x>A_D0" and a "Turret<x>B_D0" section in hier.him (with "x" being the 1-based index of the turret).
The LF2 only has "Turret1A_D0" but no "Turret1B_D0", so from IL2's internal logic it doesn't have any turret at all, therefore calling
Code: [Select]
this.FM.turret[0].setHealth(f);has to fail since "turret" is an empty array, even index "0" doesn't exist.

Best regards - Mike
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#### vpmedia

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##### Re: Morane Saulnier G/H/L/LA
« Reply #148 on: December 29, 2015, 11:43:49 AM »

Thanks for the advice. I'm really doing this pack without any java knowledge which makes things difficult sometimes but Pete (and Kant) has been a big help in this last two years. The current install format of DBW-1916 is indeed not ideal but I plan to make it easier, it will be one (split) 7z extraction and the whole thing is done.

Cheers
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#### Mission_bug

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##### Re: Morane Saulnier G/H/L/LA
« Reply #149 on: December 29, 2015, 12:26:00 PM »

Hello Mike, glad the confusion is now sorted out for you and hopefully everyone else.

Thank you very much for pointing out where the error was I will see if using the entries from one of the other similar aircraft can rectify the issue.

I have not actually touched the original aircraft hier files so it must have been there all along.

Take care.

Wishing you all the very best, Pete.
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#### Mission_bug

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##### Re: Morane Saulnier G/H/L/LA
« Reply #150 on: December 30, 2015, 02:05:17 AM »

Hello guys, I looked through each hier of the pack and all these have the entry Mike mentions:

Code: [Select]
GG2HILLFLF2NRGA1A2E1E3
This is that entry:

[Turret1A_D0]
Mesh Turret1A_D0_00
Parent cf_D0
Hidden
Separable
Attaching 0 1 0 0 0 -1 -1 0 0 0.0031 0.25402 0.32515

I removed it from each hier in the plane folder and started a QMB mission to see what happened, the result was mixed.

The angle at which the aircraft hits the ground seems to have some effect, some aircraft were completely destroyed and the text would dissapear, but others appeared to continue to burn so in effect remained as vampire_pilot mentions.

The two seat aircraft that do not have a turret essentially have two pilots cockpits in the java with no mention of Turret1A_D0 or 1B in that class file or any other for the type, although there is something about murder pilot and turret but that is most likely how all aircraft are to designate that the crew are in a position within the aircraft.

Maybe as Walter mentioned earlier the collision box might have a effect although the aircraft do break up, some though have parts that continue to burn through the mission.

Strange behaviour indeed but what is the cure.

Hopefully Mike will be able to advise further on this.

Take care.

Wishing you all the very best, Pete.

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#### SAS~Storebror

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##### Re: Morane Saulnier G/H/L/LA
« Reply #151 on: December 30, 2015, 05:11:46 AM »

I can take a look at this next year, right now I don't feel much like taking more tasks on my desk.
Just to clarify the internal structure from IL-2's point of view:
Turrets are "created" when the aircraft mesh is getting loaded.
The game just looks for "Turret<x>A_D0" and "Turret<x>B_D0" inside the hier.him, with "x" going from 1 to 9.
As soon as these two are found, a Turret is created, and as soon as these two are not found, the game stops searching.
So if for instance your hier.him has "Turret1A_D0", "Turret1B_D0", "Turret2A_D0" and "Turret2B_D0", then this makes two turrets.
If in contrast it has "Turret1A_D0", "Turret1B_D0", "Turret2A_D0" and "Turret2C_D0", then this makes one turret.
And if it has "Turret1A_D0", "Turret1C_D0", "Turret2A_D0" and "Turret2B_D0", then this makes no turrets at all.

The number of pilots solely depends on the defined number of pilots in the fmd file, the hier.him has to have their regarding Pilot meshes, otherwise loading the plane fails.

The problem of these planes is that when they crash, the game tries to damage the turrets, then kill the pilots (and thereby the gunner) and finally destroy the aircraft.
Since at the first step the code already bails out with an exception, step 2 (kill pilots) and 3 (destroy aircraft) never gets reached, that's why the planes keep floating.
When you hit the ground very steep, the order might change because a vital part of the plane (like the tank) might get damaged to a degree where the game destroys the aircraft automatically, already before the turret or the pilot(s) get hurt.
They will get hurt later (and throw the same error), but then the plane is already gone.

Best regards - Mike
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#### Mission_bug

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##### Re: Morane Saulnier G/H/L/LA
« Reply #152 on: December 30, 2015, 09:16:45 AM »

Hello Mike, thank you very much for explaining exactly what happens, greatly appreciated.

Ok, so using what you say I added the entries you say to use for no turret to the hier
of three of the offending aircraft:

Code: [Select]
[Turret1A_D0]Mesh Turret1A_D0Parent cf_D0HiddenSeparableAttaching 0 1 0 0 0 -1 -1 0 0 0.0031 0.25402 0.32515[Turret1B_D0]Mesh Turret1A_D0Parent cf_D0HiddenSeparableAttaching 0 1 0 0 0 -1 -1 0 0 0.0031 0.25402 0.32515[Turret2A_D0]Mesh Turret2A_D0Parent cf_D0HiddenSeparableAttaching 0 1 0 0 0 -1 -1 0 0 0.0031 0.25402 0.32515[Turret2B_D0]Mesh Turret2A_D0Parent cf_D0HiddenSeparableAttaching 0 1 0 0 0 -1 -1 0 0 0.0031 0.25402 0.32515

In this instance L, LF and LF2.

Also added to the plane folder were four meshes named according to the entries added to the hier:

Code: [Select]
Turret1A_D0Turret1B_D0Turret2A_D0Turret2B_D0
A number of missions were flown using AI, my aircraft the pursuer in Nieuports and kills were confirmed:

The previous issue of parts burning and bouncing was gone, only a fire where the destroyed aircraft crashed.

Previously you could scroll through what was the flight and still have your view take you to where the offending burning
was, now no more, also gone was the text that remained showing the aircraft type.

So, effectively what you imply in the reply is that even though the aircraft might not have a turret you still need to
add those entries if there are two crew, is that correct?

Just for completness sake I removed the meshes and just left the entries but that loaded my aircraft with a disabled engine
so they were returned.

Can we now conclude the issue reported by vampire_pilot has now been overcome or was my testing just lucky in those instances?

Take care.

Wishing you all the very best, Pete.
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#### Whiskey_Sierra_972

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##### Re: Morane Saulnier G/H/L/LA
« Reply #153 on: December 30, 2015, 09:45:32 AM »

You can send the modified flies to vampire pilot and let him reply for successful debugging....
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#### Mission_bug

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##### Re: Morane Saulnier G/H/L/LA
« Reply #154 on: December 31, 2015, 06:03:42 AM »

You can send the modified flies to vampire pilot and let him reply for successful debugging....

Hello Walter, the information was posted above so anyone who wished to experiment could.

I spent most of yesterday off and on trying to sort the issue and I think I have it dealt with, however, the tests showed another issue of extra pilots bailing out of a single seater so I had to trace that issue.

The FMs were altered so that the aircraft now have a dedicated FM for the single seaters where previously they used the same as the two seaters, it turned out that was not the issue of the extra bailout, that was a hook in the CF_D0 where the original two seat aircraft had been converted to a single.

There is a further issue in that the cockpit folders have no damage model but that will have to wait for another time, most of these aircraft will I suppose be used as cannon fodder so as long as the external damage model works that will be sufficient for the moment.

Today will be taken up watching the Nieuports shooting everything up again just to make sure all is well with the bug vampire_pilot pointed out, once I am satisfied a new update will be uploaded.

Take care.

Wishing you all the very best, Pete.

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#### Whiskey_Sierra_972

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##### Re: Morane Saulnier G/H/L/LA
« Reply #155 on: December 31, 2015, 06:26:45 AM »

Great news mate!


text:
# Epislon & Delta for Open / Not Open Sets

1. May 21, 2012

### rroy81

Hello All,

I am finding the hardest time in understanding how to work δ & ε Open Set Problems?

Can someone please explain this approach to me?

2. May 21, 2012

### Vargo

Hello!

That's kind of vague. Is there a particular problem you are working on? Or is there a Theorem you are trying to understand? Some examples would be easier to respond to.

3. May 21, 2012

### rroy81

So the problem given is:
Is the set S = {P=(x,y): |y| > 0} open or not?

I want to know how do I tackle or decipher this?...I have spent a good 2 days researching and reviewing the text book example and I am simply not getting it.

4. May 21, 2012

### lugita15

What is the definition of open set you're using? There are several equivalent definitions.

5. May 21, 2012

### rroy81

So the definition or what I understand a "Open Set" is;
Any arbitrary point P within a set must be < than the δ or radius in order to satisfy as an "Open Set".

I am simply confused and lost on how one would write a proof and hence I need a better explanation on understanding this types of "Prove"

6. May 22, 2012

### thrill3rnit3

I suggest going back and re-reading (and understand?) the formal definition. What you have said above is not quite accurate. The structure of the proof (ie. what you have to do) comes really from the definition itself.

7. May 22, 2012

### Stephen Tashi

As thrill3rint3 indicates, that is not a coherent statement. It's doesn't explain what the $\delta$ radius is. Is that a translation from a non-English textbook?

My guess about the definition in your text materials is that it says something equivalent to:
A set S is open if and only if for each point x in S, there exists an open interval O of radius $\delta > 0$ such that O contains x and O is a subset of S. (The definition in you book might want x to be the center of O.)

The way you start such a proof is to say "let x be a point in the set S = {w: |w| > 0}". Then you show you can find an open interval O that meets these condition. You can divide the proof into 2 cases, if you wish. For example, if you take the case that x < 0 then you can say "Consider the case that x < 0. Since x < 0, -x/2 > 0. Let $\delta = -x/2$." Then you must show that an open interval O with center x and radius $\delta$ is a subset of S.

Proofs involving showing one set is a subset of another often are done by showing that if we pick an arbitrary point in the first set then it must be in the second set. So you may began this part of the proof by saying "Let p be a point in 0 = {y:| y - x | < $\delta$ }." Then you prove p is in S by showing |p| > 0.

If it's any consolation, writing the proof in such detail is not easy. You should consider whether your textbook has proved any theorems that could be used. Without seeing your text, I don't know what it has proven. For example, has the textbook proven that a half open interval such as {q: q > C} is an open set?

8. May 22, 2012

### HallsofIvy

Staff Emeritus
Equivalent to what Stephen Tashi said, a set, A, is open if and only if, for any point p in A, there exist $\delta> 0$ such that, for any point q, if, $d(p, q)< \delta$ the q is in A also.

Many textbooks start by defining an "interior point" of a set, A, as any point, p, for which the above is true, then define A to be "open" if all of its points are interior points.

All this, by the way, is only true for metric topologies in which d(p, q), the distance between points p and q, is defined. There are more general definitions.

9. May 22, 2012

### rroy81

Ok guys, my apologies. I did more research on this problem set and this is exactly what is given.

Is the set S = {P=(x,y): |y| > 0} open or not?
{P=(x,y): |y| > 0} = {P=(x,y): (y > 0) or (y <0)} = {P=(x,y):y ≠ 0}

I need to show whether this is a open set or not?

Hope this iis more clearer.

10. May 22, 2012

### rroy81

firstly: THANK YOU all for taking the time and trying to point me in the right direction.

Now, I am sitting here pondering on this problems "Open Set" problems.

I believe the question I should be asking is whether someone can explain to me Open Sets using δ & ε? with examples and solution

Than maybe I can wrap my head around this.

TIA

11. May 22, 2012

### SteveL27

Can you draw or conceptualize a picture of the set in question? What points are not in S? What does S look like? Can you describe it in words? Hint: It's "Everything but _______" where the blank is a very familiar set.

Now if x is a point in S, can you draw a little circle around it that's entirely within S?

12. May 22, 2012

### rroy81

SteveL27, I have attached a word doc with picture.

#### Attached Files:

• ###### Doc1.doc
File size:
182.5 KB
Views:
65
13. May 22, 2012

### SteveL27

Yes, that's the correct intuition.

Can you turn this into a proof? Is that where you're stuck?

14. May 22, 2012

### thrill3rnit3

Think about it this way, for intuition, which you can then formalize to create your "proof".

You want to make sure that for all points in that given set S, you can create a "disk", or "neighborhood" [ with radius δ > 0 ], around each one of them, in such a way that all the points inside this disk is contained in S. That is essentially what it means to be an "open" set.

So, for this kind of proof, it suffices to show that there exist such a "neighborhood" for an arbitrary point P in S. Pick such a point. Now, ask yourself: Where can you run into trouble when creating the "neighborhood"? Is it possible for you to circumvent that problem for any choice of point P? In other words, given a point (x,y), what must delta be so that you don't run into the problem?

Last edited: May 22, 2012
15. May 22, 2012

### rroy81

SteveL27, yes this is exactly where I am stuck. I do believe I understand the question however, I am having a very difficult time put words on paper that explains it.

thrill3rnit3, I will attempt what I think..here goes..
I believe that "let P be any point in S such that (P-P0)/2 < δ"..again for the life of me I can't express in or formulate it into a statement.

16. May 22, 2012

### thrill3rnit3

What is P0? You need to make sure every detail in your proof is accounted for.

17. May 22, 2012

### rroy81

P0 really is meant to say P subscript 0. Didn't know how to do this in here. Couldn't Copy & paste from word either.

18. May 22, 2012

### SteveL27

If you can describe S in words, it will be very helpful.

S consists of all the points in the plane except for ___________.

Now if P is a point of S, can we find a little circle around P that does not contain any points of ____________?

19. May 22, 2012

### thrill3rnit3

Hm, I think it would be nice to have an systematic approach to this proof.

What do we want to show? We want to show that for any point p (x,y) there exists a radius δ > 0 such that, for any point q, if the distance from p to q is less than δ,

ie. if l p - q l < δ where l l represents the [ Euclidean? ] metric,

then q belongs to S.

The "there exists a radius δ > 0" is a very important part. It means that you have to come up with a delta, which depends on the chosen point p, so that the above holds.

Don't try to write down the "proof" yet. Make sure you understand what is going on. Can you come up with a delta such that the conditions are satisfied?

Look at Stephen Tashi's post [ #7 ]. He basically outlines what you need to do. Let us know if there is a part that you are having trouble with.


text:
Chemistry » Energy and Chemical Reactions » Rates Of Reaction And Factors Affecting Rate

What is a reaction rate?

What is a reaction rate?

In a chemical reaction, the substances that are undergoing the reaction are called the reactants, while the substances that form as a result of the reaction are called the products. The reaction rate describes how quickly or slowly the reaction takes place. So how do we know whether a reaction is slow or fast? One way of knowing is to look either at how quickly the reactants are used during the reaction or at how quickly the products form. For example, iron and sulfur react according to the following equation:

$$\text{Fe}(\text{s}) + \text{S}(\text{s})$$ $$\rightarrow$$ $$\text{FeS}(\text{s})$$

In this reaction, we can observe the speed of the reaction by measuring how long it takes before there is no iron or sulfur left in the reaction vessel. In other words, the reactants have been used. Alternatively, one could see how quickly the iron sulfide (the product) forms. Since iron sulfide looks very different from either of its reactants, this is easy to do.

In another example:

$$2\text{Mg}(\text{s}) + \text{O}_2(\text{g})$$ $$\rightarrow$$ $$2\text{MgO}(\text{s})$$

In this case, the reaction rate depends on the speed at which the reactants (oxygen gas and solid magnesium) are used, or the speed at which the product (magnesium oxide) is formed.

Definition: Reaction rate

The average rate of a reaction describes how quickly reactants are used or how quickly products are formed during a chemical reaction.

The average rate of a reaction is expressed as the number of moles of reactant used, divided by the total reaction time, or as the number of moles of product formed, divided by the total reaction time.

Average reaction rate for:

• the use of a reactant = $$\dfrac{\text{moles reactant used}}{\text{reaction time (s)}}$$

• the formation of a product = $$\dfrac{\text{moles product formed}}{\text{reaction time (s)}}$$

Using the magnesium reaction shown earlier:

• Average reaction rate of Mg being used = $$\dfrac{\text{moles Mg used}}{\text{reaction time (s)}}$$

• Average reaction rate of $$\text{O}_2$$ being used = $$\dfrac{\text{moles O}_2\text{ used}}{\text{reaction time (s)}}$$

• Average reaction rate of MgO being formed = $$\dfrac{\text{moles MgO formed}}{\text{reaction time (s)}}$$

Example: Reaction Rates

Question

The following reaction takes place:

$$4\text{Li}(\text{s}) + \text{O}_{2}(\text{g})$$ $$\to$$ $$2\text{Li}_{2}\text{O}(\text{s})$$

After two minutes, $$\text{4}$$ $$\text{g}$$ of lithium has been used. Calculate the rate of the reaction.

Step 1: Calculate the number of moles of lithium that are used in the reaction

$$\text{n} = \dfrac{\text{m}}{\text{M}}$$

$$\phantom{\rule{6.pt}{0ex}}=\dfrac{\text{4}\text{ g}}{\text{6.94}\text{ g.mol^{-1}}}$$

$$\phantom{\rule{6.pt}{0ex}}=$$ $$\text{0.58}$$ $$\text{mol}$$

Step 2: Calculate the time (in seconds) for the reaction

t = $$\text{2}$$ $$\text{minutes}$$ = $$\text{2}$$ $$\times$$ $$\text{60}$$ $$\text{s}$$ = $$\text{120}$$ $$\text{seconds}$$

Step 3: Calculate the rate of the reaction

Rate of reaction of $$\text{Li}$$ used $$= \dfrac{\text{moles of lithium used}}{\text{time}}$$

$$\phantom{\rule{73.pt}{0ex}} = \dfrac{\text{0.58}\text{ mol}}{\text{120}\text{ s}}$$

$$\phantom{\rule{73.pt}{0ex}} =$$ $$\text{0.005}$$ $$\text{mol.s^{-1}}$$

The rate of the reaction is $$\text{0.005}$$ $$\text{mol.s^{-1}}$$


text:
$$\lim_{n\to\infty} \left({\frac{2n+1}{3n+2}}\right)^{n} = ?$$

Simple method gives an indeterminate expression. Any idea how to think out this case?

-

Probably a bit cumbersome but straightforward:

$$\lim_{n \to \infty} \bigg(\frac{2n +1}{3n+2} \bigg)^n=\lim_{n \to \infty}\bigg( \frac{2}{3} \cdot \bigg(\frac{n+\frac{1}{2}}{n+\frac{2}{3}} \bigg)\bigg)^n = \lim_{n \to \infty}\bigg( \frac{2}{3} \cdot \bigg(\frac{n+\frac{2}{3}-\frac{1}{6}}{n+\frac{2}{3}}\bigg)\bigg)^n =\lim_{n \to \infty}\bigg( \frac{2}{3} \cdot \bigg(1-\frac{1}{6n+4} \bigg)\bigg)^n= \lim_{n \to \infty}\bigg( \frac{2}{3} \bigg)^n \lim_{n \to \infty} \bigg(1-\frac{1}{6n+4} \bigg)^{(6n +4-4)\cdot \frac{1}{6}}=0 \cdot e^{-\frac{1}{6}} \cdot 1 = 0$$

-

HINT: $$0<\frac{2n+1}{3n+2}<\frac23$$ for $n\ge 1$.

-
+1 So painfully simple –  DonAntonio Dec 5 '12 at 1:11
Probably sounds quit cheesy, but officially your solution is much better than mine) –  Alex Dec 7 '12 at 0:02
@Alex: Apparently not for the OP! Yours does have a certain baroque impressiveness. :-) –  Brian M. Scott Dec 7 '12 at 0:06

Brian's answer is right on the money. Another way to see it is noting the algebraic equality

$$\left(\frac{2n+1}{3n+2}\right)^n=e^{n\log\frac{2n+1}{3n+2}}$$

But

$$n\log\frac{2n+1}{3n+2}\xrightarrow [n\to\infty]{}-\infty$$

since $\,\log\frac{2}{3}<0\,$ , so the limit's like

$$\lim_{m\to-\infty}e^m=0$$

But I'd choose Brian's answer as it is way simpler...:)

-

if f(n) is less then a constant k which is less then 1, for large enough n, and all proceeding n, then as n tends to infinity $f(n)^n$=0, so the limit of your expression is zero.

-
Maybe could change to less than $k$ which is less than $1$, to make sure the criterion is not used on, for example, $(1-1/n)^n$. –  André Nicolas Dec 5 '12 at 1:04


text:
Probability of $k$ of a kind (or greater) on $n$ dice with $m$ sides, with duplicates allowed

How would I calculate the probability of at least one matching set of $k$ (or more) dice, from a roll of $n$ dice with $m$ sides?

I can do this for $k$ > $n/2$, I believe, by summing a formula for an exact match of $k$ dice (e.g., exactly 7 of a kind on 10 dice with 6 sides), from $k$ to $n$.

If I had a general formula, given $k \leq n/2$), for the largest match being exactly $k$ dice (with duplicates allowed): then I could continue the sum-of-probabilities approach for all $k$. However, the two existing questions I could find on this are either asking for no duplicate $k$-of-a-kind matches, or are asking about a specific case where $k = (n/2)-1$. I'm not sure how to generalize either of those to a case where duplicates (but not larger matches) are allowed.

A possibly useful sanity check that I've been using: when $n > (k-1)m$, the probability of getting a match of at least $k$ (the main probability I want to calculate) is $1$; and when $n > km$, the probability of the largest match being exactly $k$ (a useful but non-essential intermediate probability, for me) is $0$.


text:
# Journal article

## Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements

Abstract:

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge–Ampere equation  for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge–Ampere  equation to this mesh generation problem; this is a mixed finite eleme...

Publication status:
Published
Peer review status:
Peer reviewed

### Access Document

Files:
• (Accepted manuscript, pdf, 1.9MB)
Publisher copy:
10.1137/16M1109515

### Authors

More by this author
Institution:
University of Oxford
Division:
MPLS
Department:
Physics
Sub department:
Atmos Ocean & Planet Physics
Role:
Author
Publisher:
Society for Industrial and Applied Mathematics Publisher's website
Journal:
SIAM Journal on Scientific Computing Journal website
Volume:
40
Issue:
2
Pages:
A1121–A1148
Publication date:
2018-04-24
Acceptance date:
2018-01-16
DOI:
EISSN:
1095-7197
ISSN:
1064-8275
Keywords:
Pubs id:
pubs:821168
UUID:
uuid:8cd83e94-7701-4031-bb5c-d31ae0c604c9
Local pid:
pubs:821168
Source identifiers:
821168
Deposit date:
2018-01-24


text:
# Simulating an orbit, primary is not at focus

I've been toying around with some -very- simple orbital simulators, mostly using preexisting physics libraries (I took a layman's stab at doing it with vectors too). The thing that is confusing me is that my orbits do not behave as in reality - the primary is always at the center of the ellipse rather than at one of the foci. I get the same result regardless of the engine or library I use.

I've simply been putting in a primary and an orbiter, with the primary at the center of the layout. I plug in the formula $$F = G\frac{m_1m_2}{r_1^2}$$ with the force directed towards the primary. I've tried adjusting the time step, but I get the same result. I'm simply confused as to what would cause this.

Update

You're right, I was multiplying, I just didn't know it (something about the way the Construct2 interpreted my commands). I had it configured to apply the force which was essentially G(arbitrary) / distance * distance (didn't like it when I tried ^2). When I put distance squared in a variable and then divided by it things worked.

Now I had tried this before in Panda3d with the same problem, so I'll have to go back and look at that.

-

## 1 Answer

A few basic checks:

• What size time steps are you taking? Way too big will lead to wild errors. Way too small and changes in velocity and position will be incorrect due to roundoff errors (finite precision).
• Is the primary mass much larger than the orbiter? In real life, the Moon and Earth orbit around a common inertial center. Since the Earth is much heavier, we can declare it stationary at the risk of small errors. I don't think this explains your odd results, however. But it could be a helpful clue to know who is not guilty. (What was the famous Sherlock quote?)
• What initial velocity are you giving the orbiter?
• Do you have a choice of integration method? I can't imagine the wrong choice giving the result you get, but still, clues help...
• You really are telling it inverse square, right? And it's inverse, right? If you multiply not divide by r squared, that will explain centered ellipses.
-
I've been using the frame rate as the time step for now. I've also tried fixing the time-step to an arbitrary number, in fractions of a second, with the same result. At the moment, the force is applied every frame (this is a really arcade-y, proof of concept). The primary is a fixed point. The force is exerted on the orbiter in the direction of the primary. I've picked some arbitrary numbers to plug into the formula. For instance, masses are left out, g is divided by the distance squared. –  T. Mcleod Mar 28 '13 at 17:11
The initial velocity is actually zero, I have it so the orbiter can be controlled (so I can experiment more). I can get into an orbit easily. As for integration method, I really am not well enough educated. At the moment I am using different physics engines to do the hard work for me. –  T. Mcleod Mar 28 '13 at 17:21


text:
## LaTeX forum ⇒ Math & Science ⇒ squared values in SIunitx for resolutions

Information and discussion about LaTeX's math and science related features (e.g. formulas, graphs).
fifo
Posts: 3
Joined: Fri Apr 24, 2020 12:09 pm

### squared values in SIunitx for resolutions

Dear all,

spent half the day to find a solution for this, but since now I didn't. In my thesis I write a lot about image and volume resolutions. Those resolutions are all isometric, so I'd like to shorten the notation to a squared or cubed value. Unfortunately SIunitx does not allow the modifiers \squared or \cubed in the value part of \SI. I attached this MWE, where I implemented my desired visualization manually in math mode:

\documentclass{scrbook} \usepackage[T1]{fontenc}\usepackage[utf8]{inputenc} \usepackage{siunitx}\DeclareSIUnit\pixel{px}  \begin{document} This is about my new favourite image resolution of \SI{100x100}{\pixel}.I would like to write it in a shorter notation, because of its isometric nature:$100^2\,\text{px}$. How do I implement that with SIunitx? \end{document}

Does anybody know a way to implement this with SIunitx directly?

Attachments
MWE-I.tex
Minimal Working Example

### Who is online

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text:
# Linker errors when Compiling in Dev-C++

The errors i’m getting are orruring when compiling a GLUT app. Here’s the errors, has anybody seen them before??

c:\documents and settings\viper\desktop\hw3\hw3.o(.text+0x17):hw3.c: undefined reference to __glutInitWithExit@12' c:\documents and settings\viper\desktop\hw3\hw3.o(.text+0x3b):hw3.c: undefined reference to __glutCreateWindowWithExit@8’
c:\documents and settings\viper\desktop\hw3\hw3.o(.text+0x67):hw3.c: undefined reference to `__glutCreateMenuWithExit@8’

It is entirely possible that I’m losing it, but I seem to remember running into a similar problem because the code I was compiling used a variant from the standard glut… It was under an MS env though.

Did you include glut32.lib on the library includes? And are they the right version?

I have the exact same problem. I’ve put the glut/glu/gl headers in the include directory, I’ve put all the libraries in the lib directory, specifically added each one to the project, and put the 3 dlls in the windows system directory. none of this has gotten rid of the linker errors. Any ideas?

This topic was automatically closed 183 days after the last reply. New replies are no longer allowed.


text:
## The Annals of Statistics

### On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint

#### Abstract

We identify the asymptotic behavior of the estimators proposed by Rojo and Samaniego and Mukerjee of a distribution $F$ assumed to be uniformly stochastically smaller than a known baseline distribution $G$.We show that these estimators are $\sqrt{n}$-convergent to a limit distribution with mean squared error smaller than or equal to the mean squared error of the empirical survival function. By examining the mean squared error of the limit distribution, we are able to identify the optimal estimator within Mukerjee’s class under a variety of different assumptions on $F$ and $G$. Similar theoretical results are developed for the two-sample problem where$F$ and $G$ are both unknown. The asymptotic distribution theory is applied to obtain confidence bands for the survival function $\bar{F}$ based on published data from an accelerated life testing experiment.

#### Article information

Source
Ann. Statist., Volume 28, Number 1 (2000), 116-150.

Dates
First available in Project Euclid: 14 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1016120367

Digital Object Identifier
doi:10.1214/aos/1016120367

Mathematical Reviews number (MathSciNet)
MR1762906

Zentralblatt MATH identifier
1106.62332

Subjects
Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62E20

#### Citation

Arcones, Miguel A.; Samaniego, Francisco J. On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann. Statist. 28 (2000), no. 1, 116--150. doi:10.1214/aos/1016120367. https://projecteuclid.org/euclid.aos/1016120367

#### References

• Bagai, I. and Kochar, S. C. (1986). On tail-ordering and comparison of failure rates. Comm. Statist. A Theory Methods 15 1377-1388.
• Barlow, R. E., Toland, R. H. and Freeman, T. (1988). A Bayesian analysis of the stress-rupture life of Kevlar/Epoxy spherical pressure vessels. In Accelerated Life Testing and Experts' Opinions (C. A. Clarotti and D. V. Lindley, eds.) 203-236. North-Holland, Amsterdam.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. J. Appl. Probab. 31 180-192.
• Breslow, N. and Crowley, J. (1974). A large sample study of the life table and product limit estimates under random censorship. Ann. Statist. 2 473-453.
• Cap´era a, P. (1988). Tail ordering and asymptotic efficiency of rank tests. Ann. Statist. 16 470-478.
• Dykstra, R., Kochar, S. and Robertson, T. (1991). Statistical inference for uniform stochastic ordering in several populations. Ann. Statist. 19 870-888.
• Gill, R. D. (1981). Testing with replacement and the product limit estimator. Ann. Statist. 9 853-860.
• Keilson, J. and Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10 101-198.
• Mukerjee, H. (1996). Estimation of survival functions under uniform stochastic ordering. J. Amer. Statist. Assoc. 91 1684-1689.
• Rojo, J. (1998). Estimating of the quantile function of an IFRA distribution. Scand. J. Statist. 25 293-310.
• Rojo, J. and Samaniego, F. J. (1991). On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard. Statist. Probab. Lett. 11 267-271.
• Rojo, J. and Samaniego, F. J. (1993). On estimating a survival curve subject to a uniform stochastic ordering constraint. J. Amer. Statist. Assoc. 88 566-572.
• Ross, S. M. (1983). Stochastic Processes. Wiley, New York.
• Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston.
• Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
• Yanagimoto, T. and Sibuya, M. (1972). Stochastically larger component of a random vector. Ann. Inst. Statist. Math. 24 259-269.
• Whitt, W. (1980). Uniform conditional stochastic order. J. Appl. Probab. 17 112-123.


text:
Dataplot Vol 2 Vol 1

CORRELATION

Name:
CORRELATION (LET)
Type:
Let Subcommand
Purpose:
Compute the correlation coefficient between two variables.
Description:
The correlation coefficient is a measure of the linear relationship between two variables. It is computed as:

$$S_{xx} = \sum_{i=1}^{N}{(X_{i}-\bar{X})^2}$$

$$S_{yy} = \sum_{i=1}^{N}{(Y_{i}-\bar{Y})^2}$$

$$S_{xy} = \sum_{i=1}^{N}{(X_{i}-\bar{X}) (Y_{i} - \bar{Y})}$$

$$r = \frac{S_{xy}}{\sqrt{S_{xx}} \sqrt{S_{yy}}}$$

A perfect linear relationship yields a correlation coefficient of +1 (or -1 for a negative relationship) and no linear relationship yields a correlation coefficient of 0.

It may be of interest to determine if the correlation is significantly different than 0. The CDF value for this test is

CDF = FCDF(VAL,1,N-2)

where FCDF is the F cumulative distribution function with 1 and N - 2 degrees of freedom (N is the number of observations) and

$$\mbox{VAL} = \left| \frac{(N-2) r^2}{1 - r^2} \right|$$

with r denoting the computed correlation. The pvalue is 1 - CDF.

Syntax 1:
LET <par> = CORRELATION <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
<par> is a parameter where the computed correlation is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Syntax 2:
LET <par> = CORRELATION ABSOLUTE VALUE <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
<par> is a parameter where the absolute value of the computed correlation is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the absolute value of the correlation coefficient. This is typically used in screening applications where there is an interest in identifying high magnitude correlations regardless of the direction of the correlation.

Syntax 3:
LET <par> = CORRELATION PVALUE <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
<par> is a parameter where the computed correlation pvalue is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the pvalue (described above) of the correlation.

Syntax 4:
LET <par> = CORRELATION CDF <y1> <y2>
<SUBSET/EXCEPT/FOR qualification>
where <y1> is the first response variable;
<y2> is the second response variable;
<par> is a parameter where the computed correlation cdf is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the cdf (described above) of the correlation.

Examples:
LET A = CORRELATION Y1 Y2
LET A = CORRELATION Y1 Y2 SUBSET TAG > 2
Note:
The two variables must have the same number of elements.
Note:
Dataplot statistics can be used in a number of commands. For details, enter

Default:
None
Synonyms:
None
Related Commands:
 CORRELATION MATRIX = Generate a correlation matrix. RANK CORRELATION = Compute the rank correlation of two variables. KENDALLS TAU = Compute the Kendall tau correlation of two variables. WINSORIZED CORRELATION = Compute the Winsorized correlation of two variables. BIWEIGHT MIDCORRELATION = Compute the biweight mid-correlation of two variables. PERCENTAGE BEND CORRELATION = Compute the percentage bend correlation of two variables. COVARIANCE = Compute the covariance of two variables. PARTIAL CORRELATION = Compute the partial correlation of three variables. PARTIAL CORRELATION MATRIX = Generate the partial correlation matrix. CORRELATION STAT PLOT = Generate a correlation vs. subset plot.
Reference:
Consult any introductory statistics text.

Peavy, Bremer, Varner, Hogben (1986), "OMNITAB 80: An Interpretive System for Statistical and Numerical Data Analysis," NBS Special Publication 701.

Applications:
Linear Regression
Implementation Date:
Pre-1987
2012/06: CORRELATION PVALUE and CORRELATION CDF added
Program 1:

SKIP 25
LET CORR = CORRELATION Y X
LET PVAL = CORRELATION PVALUE Y X
LET CDF = CORRELATION CDF Y X
SET WRITE DECIMALS 3

The following output is generated.
 PARAMETERS AND CONSTANTS--

CORR    --          0.946
PVAL    --          0.000
CDF     --          1.000

Program 2:

SKIP 25
READ IRIS.DAT Y1 Y2 Y3 Y4 TAG
.
TITLE CASE ASIS
TITLE OFFSET 2
LABEL CASE ASIS
TIC MARK OFFSET UNITS DATA
Y1LABEL |Correlation|
YLIMITS 0 1
MAJOR YTIC MARK NUMBER 6
MINOR YTIC MARK NUMBER 1
Y1TIC MARK LABEL DECIMAL 1
Y1LABEL DISPLACEMENT 20
X1LABEL Species
XLIMITS 1 3
MAJOR XTIC MARK NUMBER 3
MINOR XTIC MARK NUMBER 0
XTIC MARK OFFSET 0.3 0.3
X1LABEL DISPLACEMENT 14
CHARACTER X BLANK
LINES BLANK SOLID
.
MULTIPLOT CORNER COORDINATES 5 5 95 95
MULTIPLOT SCALE FACTOR 2
MULTIPLOT 2 3
.
TITLE Sepal Length vs Sepal Width
CORRELATION ABSOLUTE VALUE PLOT Y1 Y2 TAG
.
TITLE Sepal Length vs Petal Length
CORRELATION ABSOLUTE VALUE PLOT Y1 Y3 TAG
.
TITLE Sepal Length vs Petal Width
CORRELATION ABSOLUTE VALUE PLOT Y1 Y4 TAG
.
TITLE Sepal Width vs Petal Length
CORRELATION ABSOLUTE VALUE PLOT Y2 Y3 TAG
.
TITLE Sepal Width vs Petal Width
CORRELATION ABSOLUTE VALUE PLOT Y2 Y4 TAG
.
TITLE Petal Length vs Petal Width
CORRELATION ABSOLUTE VALUE PLOT Y3 Y4 TAG
.
END OF MULTIPLOT

NIST is an agency of the U.S. Commerce Department.

Date created: 01/23/2013
Last updated: 09/12/2018


text:
Maximum theoretical range estimate

## Syntax

```maxrng = radareqrng(lambda,SNR,Pt,Tau) maxrng = radareqrng(...,Name,Value) ```

## Description

`maxrng = radareqrng(lambda,SNR,Pt,Tau)` estimates the theoretical maximum detectable range `maxrng` for a radar operating with a wavelength of `lambda` meters with a pulse duration of `Tau` seconds. The signal-to-noise ratio is `SNR` decibels, and the peak transmit power is `Pt` watts.

`maxrng = radareqrng(...,Name,Value)` estimates the theoretical maximum detectable range with additional options specified by one or more `Name,Value` pair arguments.

## Input Arguments

 `lambda` Wavelength of radar operating frequency (in meters). The wavelength is the ratio of the wave propagation speed to frequency. For electromagnetic waves, the speed of propagation is the speed of light. Denoting the speed of light by c and the frequency (in hertz) of the wave by f, the equation for wavelength is: `$\lambda =\frac{c}{f}$` `Pt` Transmitter peak power in watts. `SNR` The minimum output signal-to-noise ratio at the receiver in decibels. `Tau` Single pulse duration in seconds.

### Name-Value Pair Arguments

 `'Gain'` Transmitter and receiver gain in decibels (dB). When the transmitter and receiver are colocated (monostatic radar), `Gain` is a real-valued scalar. The transmit and receive gains are equal. When the transmitter and receiver are not colocated (bistatic radar), `Gain` is a 1-by-2 row vector with real-valued elements. The first element is the transmitter gain, and the second element is the receiver gain. Default: 20 `'Loss'` System loss in decibels (dB). `Loss` represents a general loss factor that comprises losses incurred in the system components and in the propagation to and from the target. Default: 0 `'RCS'` Radar cross section in square meters. The target RCS is nonfluctuating. Default: 1 `'Ts'` System noise temperature in Kelvin. The system noise temperature is the product of the system temperature and the noise figure. Default: 290 Kelvin `'unitstr'` Units of the estimated maximum theoretical range. `unitstr` takes one of the following values `'km'` kilometers`'m'` meters`'mi'` miles`'nmi'` nautical miles (U.S.) Default: `'m'`

## Output Arguments

 `maxrng` The estimated theoretical maximum detectable range. The units of `maxrng` depends on the value of `unitstr`. By default `maxrng` is in meters. For bistatic radars, `maxrng` is the geometric mean of the range from the transmitter to the target and the receiver to the target.

## Examples

collapse all

Estimate the theoretical maximum detectable range for a monostatic radar operating at 10 GHz using a pulse duration of 10 μs. Assume the output SNR of the receiver is 6 dB.

```lambda = physconst('LightSpeed')/10e9; SNR = 6; tau = 10e-6; Pt = 1e6; maxrng = radareqrng(lambda,SNR,Pt,tau)```
```maxrng = 4.1057e+04 ```

Estimate the theoretical maximum detectable range for a monostatic radar operating at 10 GHz using a pulse duration of 10 μs. The target RCS is 0.1 m². Assume the output SNR of the receiver is 6 dB. The transmitter-receiver gain is 40 dB. Assume a loss factor of 3 dB.

```lambda = physconst('LightSpeed')/10e9; SNR = 6; tau = 10e-6; Pt = 1e6; RCS = 0.1; Gain = 40; Loss = 3; maxrng2 = radareqrng(lambda,SNR,Pt,tau,'Gain',Gain,... 'RCS',RCS,'Loss',Loss)```
```maxrng2 = 1.9426e+05 ```

collapse all

### Point Target Radar Range Equation

The point target radar range equation estimates the power at the input to the receiver for a target of a given radar cross section at a specified range. The model is deterministic and assumes isotropic radiators. The equation for the power at the input to the receiver is

`${P}_{r}=\frac{{P}_{t}{G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{\left(4\pi \right)}^{3}{R}_{t}^{2}{R}_{r}^{2}L}$`

where the terms in the equation are:

• Pt — Peak transmit power in watts

• Gt — Transmitter gain in decibels

• Gr — Receiver gain in decibels. If the radar is monostatic, the transmitter and receiver gains are identical.

• λ — Radar operating frequency wavelength in meters

• σ — Target's nonfluctuating radar cross section in square meters

• L — General loss factor in decibels that accounts for both system and propagation loss

• Rt — Range from the transmitter to the target

• Rr — Range from the receiver to the target. If the radar is monostatic, the transmitter and receiver ranges are identical.

Terms expressed in decibels, such as the loss and gain factors, enter the equation in the form 10x/10 where x denotes the variable. For example, the default loss factor of 0 dB results in a loss term of 100/10=1.

The equation for the power at the input to the receiver represents the signal term in the signal-to-noise ratio. To model the noise term, assume the thermal noise in the receiver has a white noise power spectral density (PSD) given by:

`$P\left(f\right)=kT$`

where k is the Boltzmann constant and T is the effective noise temperature. The receiver acts as a filter to shape the white noise PSD. Assume that the magnitude squared receiver frequency response approximates a rectangular filter with bandwidth equal to the reciprocal of the pulse duration, 1/τ. The total noise power at the output of the receiver is:

`$N=\frac{kT{F}_{n}}{\tau }$`

where Fn is the receiver noise factor.

The product of the effective noise temperature and the receiver noise factor is referred to as the system temperature. This value is denoted by Ts, so that Ts=TFn .

`$\frac{{P}_{r}}{N}=\frac{{P}_{t}\tau \text{}\text{ }{G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{\left(4\pi \right)}^{3}k{T}_{s}{R}_{t}^{2}{R}_{r}^{2}L}$`

You can derive this expression using the following equations:

### Theoretical Maximum Detectable Range

For monostatic radars, the range from the target to the transmitter and receiver is identical. Denoting this range by R, you can express this relationship as ${R}^{4}={R}_{t}^{2}{R}_{r}^{2}$.

Solving for R

`$R={\left(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{P}_{r}{\left(4\pi \right)}^{3}k{T}_{s}L}\right)}^{1/4}$`

For bistatic radars, the theoretical maximum detectable range is the geometric mean of the ranges from the target to the transmitter and receiver:

`$\sqrt{{R}_{t}{R}_{r}}={\left(\frac{N{P}_{t}\tau {G}_{t}{G}_{r}{\lambda }^{2}\sigma }{{P}_{r}{\left(4\pi \right)}^{3}k{T}_{s}L}\right)}^{1/4}$`

## References

[1] Richards, M. A. Fundamentals of Radar Signal Processing. New York: McGraw-Hill, 2005.

[2] Skolnik, M. Introduction to Radar Systems. New York: McGraw-Hill, 1980.

[3] Willis, N. J. Bistatic Radar. Raleigh, NC: SciTech Publishing, 2005.


text:
# nLab A-infinity-algebra

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

An $A_\infty$-algebra is a monoid internal to a homotopical category such that the associativity law holds not as an equation, but only up to higher coherent homotopy.

## Definition

###### Definition

An $A_\infty$-algebra is an algebra over an operad over an A-∞ operad.

## Realizations

### In chain complexes

Let here $\mathcal{E}$ be the category of chain complexes $\mathcal{Ch}_\bullet$. Notice that often in the literature this choice of $\mathcal{E}$ is regarded as default and silently assumed.

An $A_\infty$-algebra in chain complexes is concretely the following data.

A chain $A_\infty$-algebra is the structure of a degree 1 coderivation

$D : T^c V \to T^c V$

on the reduced tensor coalgebra $T^c V = \oplus_{n\geq 1} V^{\otimes n}$ (with the standard noncocommutative coproduct, see differential graded Hopf algebra) over a graded vector space $V$ such that

$D^2 = 0 \,.$

Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows one to decompose $D$ as a sum:

$D = D_1 + D_2 + D_3 + \cdots$

with each $D_k$ specified entirely by its action

$D_k : V^{\otimes k} \to V \,.$

which is a map of degree $2-k$ (or can be alternatively understood as a map $D_k : (V[1])^{\otimes k}\to V[1]$ of degree $1$).

Then:

• $D_1 : V\to V$ is the differential with $D_1^2 = 0$;

• $D_2 : V^{\otimes 2} \to V$ is the product in the algebra;

• $D_3 : V^{\otimes 3} \to V$ is the associator which measures the failure of $D_2$ to be associative;

• $D_4 : V^{\otimes 4} \to V$ is the pentagonator (or so) which measures the failure of $D_3$ to satisfy the pentagon identity;

• and so on.

One can also allow $D_0$, in which case one talks about weak $A_\infty$-algebras, which are less understood.

There is a resolution of the operad $\mathrm{Ass}$ of associative algebras (as operad on the category of chain complexes) which is called the $A_\infty$-operad; the algebras over the $A_\infty$-operad are precisely the $A_\infty$-algebras.

A morphism of $A_\infty$-algebras $f : A\to B$ is a collection $\lbrace f_n\rbrace_{n\geq 1}$ of maps

$f_n : (A[1])^{\otimes n}\to B[1]$

of degree $0$ satisfying

$\sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j}) = \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots f_{i_r}).$

For example, $f_1\circ D_1 = D_1\circ f_1$.

#### Rectification

###### Theorem

If $A$ is a dg-algebra, regarded as a strictly associative $A_\infty$-algebra, its chain cohomology $H^\bullet(A)$, regarded as a chain complex with trivial differentials, naturally carries the structure of an $A_\infty$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_\infty$-algebra.

More details are at Kadeishvili's theorem.

###### Remark

This theorem provides a large supply of examples of $A_\infty$-algebras: there is a natural $A_\infty$-algebra structure on all cohomologies such as

etc.

### In Topological space

An $A_\infty$-algebra in Top is also called an A-∞ space .

#### Examples

Every loop space is canonically an A-∞ space. (See there for details.)

#### Rectification

###### Theorem

Every $A_\infty$-space is weakly homotopy equivalent to a topological monoid.

This is a classical result by (Stasheff, BoardmanVogt). It follows also as a special case of the more general result on rectification in a model structure on algebras over an operad (see there).

### In spectra

A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

algebraic deformation quantization

dimensionclassical field theoryLagrangian BV quantum field theoryfactorization algebra of observables
general $n$P-n algebraBD-n algebra?E-n algebra
$n = 0$Poisson 0-algebraBD-0 algebra? = BD algebraE-0 algebra? = pointed space
$n = 1$P-1 algebra = Poisson algebraBD-1 algebra?E-1 algebra? = A-∞ algebra

## References

A survey of $A_\infty$-algebras in chain complexes is in

• Bernhard Keller, A brief introduction to $A_\infty$-algebras (pdf)

Classical articles on $A_\infty$-algebra in topological spaces are

• Jim Stasheff, Homotopy associativity of H-spaces I , Trans. Amer. Math. Soc. 108 (1963), p. 275-292.

A brief survey is in section 1.8 of

• Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

The 1986 thesis of Alain Prouté explores the possibility of obtaining analogues of minimal models for $A_\infty$ algebras. It was published in TAC much later.

Last revised on February 17, 2017 at 07:20:31. See the history of this page for a list of all contributions to it.


text:
Enable text based alternatives for graph display and drawing entry

Try Another Version of This Question

Evaluate the expression  (-4+i) (-4-2i)  and write the result in the form a+ b i.


text:
# Building the Reliability Stack for Machine Learning

Eric Wong, Massachusetts Institute of Technology (MIT)
Host: Johns Hopkins Department of Computer Science

Currently, machine learning (ML) systems have impressive performance but can behave in unexpected ways. These systems latch onto unintuitive patterns and are easily compromised, a source of grave concern for deployed ML in settings such as healthcare, security, and autonomous driving. In this talk, I will discuss how we can redesign the core ML pipeline to create reliable systems. First, I will show how to train provably robust models, which enables formal robustness guarantees for complex deep networks. Next, I will demonstrate how to make ML models more debuggable. This amplifies our ability to diagnose failure modes, such as hidden biases or spurious correlations. To conclude, I will discuss how we can build upon this reliability stack’’ to enable broader robustness requirements, and develop new primitives that make ML debuggable by design.

## Speaker Biography

Eric Wong is a postdoctoral researcher in the Computer Science and Artificial Intelligence Laboratory at Massachusetts Institute of Technology. His research focuses on the foundations for reliable systems: methods that allow us to diagnose, create, and verify robust systems. He is a 2020 Siebel Scholar and received an honorable mention for his thesis on the robustness of deep networks to adversarial examples at Carnegie Mellon University.