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Author: Subject: Does anyone know how to use one of these?
monolithic
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 Quote: Originally posted by draculic acid69 Ill try the boiling water when I've got an empty flask. What's the knob with rst written on it for? I'm thinking it means rheostat.but what's it there for if you can set the temperature with the number wheel thing? [Edited on 30-10-2019 by draculic acid69]

 Quote: V. Operation instruction Place the equipment on level worktable, equip test-tube stand, and insert plug of sensor into sensor socket. Put the container with liquid inside into the heating jacket, and dip the sensor into heating liquid. Switch on the power accords with the machine, then the equipment screen will show the current temperature of heating liquid. Press dial switch to set the temperature as heating needs. Make sure the power line and the power socket are connected to heating jacket well; the. When the displayed temperature reaches the setting point, the red indicator light will be lighten, then it gets into constant temperature state. Adjust the RST knob to the left side, where is the smaller value position; wait until the temperature does not rise anymore, for example, when the displayed temperature is lower than the setting temperature, then turn it to right side gradually.

http://webcache.googleusercontent.com/search?q=cache:http://...

I have no idea what it does. Probably a classic case of shitty Chinese reverse engineering where they rip something off without understanding what the fuck they're even doing.
G-Coupled
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The RST is some sort of fine calibration setting maybe?
draculic acid69
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So no one knows what the rst knob actually does?once the display temp is lower than the setting temperature you adjust the knob to the right bcoz it does what?
G-Coupled
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Maybe it will allow you to calibrate the display temperature to match the setting temperature?
draculic acid69
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Tried boiling a flask full of water.mantle still gets to 300~ish degrees'c even when dial is set to 100.it appears that the temperature probe isn't more than 2 degrees off when it comes to boiling water.dont have a higher boiling liquid available ATM. Did figure out my glass thermometer is a lil bit off.reads 108 in boiling water but the mantles 2-300'c spikes make it useless for anything other than boiling water.theres not one rxn I can do that wouldn't be ruined by this kind of overheating.the knob doesn't seem to make a difference which ever way it's turned.this thing as it is is no better than those 2000w hotplates from kmart.i might just rip the guts out and put a Pwm thyristor circuit in there like my other one which is working fine.
monolithic
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 Quote: Originally posted by draculic acid69 Tried boiling a flask full of water.mantle still gets to 300~ish degrees'c even when dial is set to 100.it appears that the temperature probe isn't more than 2 degrees off when it comes to boiling water.dont have a higher boiling liquid available ATM. Did figure out my glass thermometer is a lil bit off.reads 108 in boiling water but the mantles 2-300'c spikes make it useless for anything other than boiling water.theres not one rxn I can do that wouldn't be ruined by this kind of overheating.the knob doesn't seem to make a difference which ever way it's turned.this thing as it is is no better than those 2000w hotplates from kmart.i might just rip the guts out and put a Pwm thyristor circuit in there like my other one which is working fine.

Temperature controlled hot plates and mantles don't regulate the voltage to the heating element. It's expected that the mantle is going to hit 300 C or whatever its maximum output is. Put the probe in a flask of water, completely ignore the temperature of the heating element, and set the temperature of the mantle to something below boiling, like 70 C. Does it eventually stabilize at 70 C? Then it is working as it should. If the probe is off by 8 C then I don't think there's much you can do as there doesn't seem a way to adjust the calibration. You can buy PID controllers pretty cheap ($50 or less on Amazon or eBay) that offer a ton of configuration: support for different types of thermocouples/thermistors, support for K/F/C units of temperature, calibration adjustment, etc. It wouldn't be much work to wire one up to an SSR or mechanical relay, to control a hot plate. If you're looking for something to control the temperature output at the heating element, you need something like a soft heating mantle (Glas-Col or a knockoff) plugged into an AC transformer (Variac or a knockoff), set atop a magnetic stirrer or magnetic hot plate/stirrer with the heating circuit turned off. [Edited on 11-2-2019 by monolithic] G-Coupled Hazard to Others Posts: 208 Registered: 9-3-2017 Member Is Offline Mood: Slightly triturated Yeah, it looks like maybe the best thing to do with these Won Hung Lo hotplates might be to just use the mechanical guts of them, rip out/bypass the control circuitry and replace it with a working configuration of your own. IME, the vast majority of heaters use a kind of PWM to regulate the heat. By that, I mean that they use full power 'pulses' of current of varying lengths to the heating element to keep it within the desired temperature range. IDK if it'd improve things a lot if a controller was used which allowed the current to the heating element to be continuously variable instead. I guess it would remove the hysteresis that the 'full power PWM' heaters exhibit, where a 100C set temperature would result in the element would vary in temperature from like 96-103C or whatever. draculic acid69 International Hazard Posts: 628 Registered: 2-8-2018 Member Is Offline Quote: Originally posted by monolithic  Quote: Originally posted by draculic acid69 Tried boiling a flask full of water.mantle still gets to 300~ish degrees'c even when dial is set to 100.it appears that the temperature probe isn't more than 2 degrees off when it comes to boiling water.dont have a higher boiling liquid available ATM. Did figure out my glass thermometer is a lil bit off.reads 108 in boiling water but the mantles 2-300'c spikes make it useless for anything other than boiling water.theres not one rxn I can do that wouldn't be ruined by this kind of overheating.the knob doesn't seem to make a difference which ever way it's turned.this thing as it is is no better than those 2000w hotplates from kmart.i might just rip the guts out and put a Pwm thyristor circuit in there like my other one which is working fine. Temperature controlled hot plates and mantles don't regulate the voltage to the heating element. It's expected that the mantle is going to hit 300 C or whatever its maximum output is. Put the probe in a flask of water, completely ignore the temperature of the heating element, and set the temperature of the mantle to something below boiling, like 70 C. Does it eventually stabilize at 70 C? Then it is working as it should. If the probe is off by 8 C then I don't think there's much you can do as there doesn't seem a way to adjust the calibration. You can buy PID controllers pretty cheap ($50 or less on Amazon or eBay) that offer a ton of configuration: support for different types of thermocouples/thermistors, support for K/F/C units of temperature, calibration adjustment, etc. It wouldn't be much work to wire one up to an SSR or mechanical relay, to control a hot plate.

If you're looking for something to control the temperature output at the heating element, you need something like a soft heating mantle (Glas-Col or a knockoff) plugged into an AC transformer (Variac or a knockoff), set atop a magnetic stirrer or magnetic hot plate/stirrer with the heating circuit turned off.

[Edited on 11-2-2019 by monolithic]

Imagine your doing a 100ml size grignard and you set the temperature control to 60'c
and then within a minute it's boiling and then it's out of control spewing out of the flask bcoz this 'operating as it should' mantle heated up to over 250'c within a minute
(gets hot quick) other than boiling water that kind of temperature difference has the potential to ruin rxn's.

[Edited on 2-11-2019 by draculic acid69]
monolithic
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 Quote: Originally posted by draculic acid69 Imagine your doing a 100ml size grignard and you set the temperature control to 60'c and then within a minute it's boiling and then it's out of control spewing out of the flask bcoz this 'operating as it should' mantle heated up to over 250'c within a minute (gets hot quick) other than boiling water that kind of temperature difference has the potential to ruin rxn's. [Edited on 2-11-2019 by draculic acid69]

You should use a soft mantle, AC transformer, and a magnetic stirrer for situations like that. Even a hot water bath on a magnetic stirrer/hot plate would work well for something that has to run at 60 C, speaking from experience. Also, I doubt even your current setup would heat the reaction mixture to 250 C within a minute. The heating element would hit 250 C within a minute but it takes time to transfer that heat through the glass and to the reaction mixture.
SWIM
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Okay, I probably missed something in this thread, but did you try adjusting with the RST knob as directed?

I'm not sure what it does either. But it seems to be something they definitely want you to do.

Maybe it calibrates the thermocouple.

Maybe it limits the current flow to the heating element.

Maybe its worth a try.

EDIT: Thanks for the information, G-coupled. (Below)

[Edited on 2-11-2019 by SWIM]

Do you want red P?
Eat lots of beets
G-Coupled
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They tried the RST knob, but it didn't appear to change anything.

I'd have a lookyloo inside and see where the RST knob is attached, and see if that might help figure out what it's supposed to do.
draculic acid69
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Tried the knob didn't do much.its wired directly to the board.
G-Coupled
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Yes, but attached to *what* - where do the traces go to from where it's attached?

Any chance of a photograph of all the gubbins inside?

[Edited on 3-11-2019 by G-Coupled]
Deathunter88
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Quote: Originally posted by draculic acid69
Quote: Originally posted by monolithic
 Quote: Originally posted by draculic acid69 Tried boiling a flask full of water.mantle still gets to 300~ish degrees'c even when dial is set to 100.it appears that the temperature probe isn't more than 2 degrees off when it comes to boiling water.dont have a higher boiling liquid available ATM. Did figure out my glass thermometer is a lil bit off.reads 108 in boiling water but the mantles 2-300'c spikes make it useless for anything other than boiling water.theres not one rxn I can do that wouldn't be ruined by this kind of overheating.the knob doesn't seem to make a difference which ever way it's turned.this thing as it is is no better than those 2000w hotplates from kmart.i might just rip the guts out and put a Pwm thyristor circuit in there like my other one which is working fine.

Temperature controlled hot plates and mantles don't regulate the voltage to the heating element. It's expected that the mantle is going to hit 300 C or whatever its maximum output is. Put the probe in a flask of water, completely ignore the temperature of the heating element, and set the temperature of the mantle to something below boiling, like 70 C. Does it eventually stabilize at 70 C? Then it is working as it should. If the probe is off by 8 C then I don't think there's much you can do as there doesn't seem a way to adjust the calibration. You can buy PID controllers pretty cheap ($50 or less on Amazon or eBay) that offer a ton of configuration: support for different types of thermocouples/thermistors, support for K/F/C units of temperature, calibration adjustment, etc. It wouldn't be much work to wire one up to an SSR or mechanical relay, to control a hot plate. If you're looking for something to control the temperature output at the heating element, you need something like a soft heating mantle (Glas-Col or a knockoff) plugged into an AC transformer (Variac or a knockoff), set atop a magnetic stirrer or magnetic hot plate/stirrer with the heating circuit turned off. [Edited on 11-2-2019 by monolithic] Imagine your doing a 100ml size grignard and you set the temperature control to 60'c and then within a minute it's boiling and then it's out of control spewing out of the flask bcoz this 'operating as it should' mantle heated up to over 250'c within a minute (gets hot quick) other than boiling water that kind of temperature difference has the potential to ruin rxn's. [Edited on 2-11-2019 by draculic acid69] monolithic is right, you can't expect everything to work the way you think it should in your brain. A heating mantle designed the way you think it should work would be useless, imagine wanting to boil some water and your mantle only heats to 100C. Your water is never going to boil. Your "hypothetical" situation is like saying, this masonry drill doesn't work well when I try to use it for dentistry, it just destroyed my patient's entire mouth. They should have finer speed control on it. It's not the problem with the equipment, but with the way you're using it. G-Coupled Hazard to Others Posts: 208 Registered: 9-3-2017 Member Is Offline Mood: Slightly triturated To be fair, until recently all kitchen microwaves worked like that until recently, using full power but with less duty cycles when the heat setting is turned down, but a couple of years ago Panasonic brought out a new type of microwave which used actual true variable power to heat things. [Edited on 3-11-2019 by G-Coupled] wg48temp9 National Hazard Posts: 323 Registered: 30-12-2018 Member Is Offline From reading the instructions the control knob varies the maximum heating power. Assuming it not faulty (stuck on full power) you should be able to verify that by checking the max temperature of the mantle say at 0% 50% and 100% rotation of the knob while the temperature thumb switches are set at max temperature or better yet check the power consumed with an in-lead power meter in place of measuring the mantle temperature. i am wg48 but not on my usual pc hence the temp handle. Thank goodness for Fleming and the fungi. draculic acid69 International Hazard Posts: 628 Registered: 2-8-2018 Member Is Offline Quote: Originally posted by Deathunter88 Quote: Originally posted by draculic acid69 Quote: Originally posted by monolithic  Quote: Originally posted by draculic acid69 Tried boiling a flask full of water.mantle still gets to 300~ish degrees'c even when dial is set to 100.it appears that the temperature probe isn't more than 2 degrees off when it comes to boiling water.dont have a higher boiling liquid available ATM. Did figure out my glass thermometer is a lil bit off.reads 108 in boiling water but the mantles 2-300'c spikes make it useless for anything other than boiling water.theres not one rxn I can do that wouldn't be ruined by this kind of overheating.the knob doesn't seem to make a difference which ever way it's turned.this thing as it is is no better than those 2000w hotplates from kmart.i might just rip the guts out and put a Pwm thyristor circuit in there like my other one which is working fine. Temperature controlled hot plates and mantles don't regulate the voltage to the heating element. It's expected that the mantle is going to hit 300 C or whatever its maximum output is. Put the probe in a flask of water, completely ignore the temperature of the heating element, and set the temperature of the mantle to something below boiling, like 70 C. Does it eventually stabilize at 70 C? Then it is working as it should. If the probe is off by 8 C then I don't think there's much you can do as there doesn't seem a way to adjust the calibration. You can buy PID controllers pretty cheap ($50 or less on Amazon or eBay) that offer a ton of configuration: support for different types of thermocouples/thermistors, support for K/F/C units of temperature, calibration adjustment, etc. It wouldn't be much work to wire one up to an SSR or mechanical relay, to control a hot plate.

If you're looking for something to control the temperature output at the heating element, you need something like a soft heating mantle (Glas-Col or a knockoff) plugged into an AC transformer (Variac or a knockoff), set atop a magnetic stirrer or magnetic hot plate/stirrer with the heating circuit turned off.

[Edited on 11-2-2019 by monolithic]

Imagine your doing a 100ml size grignard and you set the temperature control to 60'c
and then within a minute it's boiling and then it's out of control spewing out of the flask bcoz this 'operating as it should' mantle heated up to over 250'c within a minute
(gets hot quick) other than boiling water that kind of temperature difference has the potential to ruin rxn's.

[Edited on 2-11-2019 by draculic acid69]

monolithic is right, you can't expect everything to work the way you think it should in your brain. A heating mantle designed the way you think it should work would be useless, imagine wanting to boil some water and your mantle only heats to 100C. Your water is never going to boil.

Your "hypothetical" situation is like saying, this masonry drill doesn't work well when I try to use it for dentistry, it just destroyed my patient's entire mouth. They should have finer speed control on it. It's not the problem with the equipment, but with the way you're using it.

Not even close.i know how a usual mantle works it's just this one with a probe and setting the temperature using the dials is a lot different than what I'm used to.its just something with this much of a temperature spike above the set temperature makes it no more useful than a \$30
hotplate from Kmart. how often do you need to boil water in a flask that can't be done in a pot? I didn't buy it to boil water and I just can't trust it to maintain set rxn temperature with the spiking up to 250'c+ all of a sudden even when it's sitting in the right temp range the way it did several times.my thermometer only goes up to 250 so for all I know it could have hit 400'c and when your distillation or reflux is all clamped in its not like I can just lower the mantle out of the way till it cools down and with the temperature spikes all the rxn's I'm doing would be ruined by that.as for trying to trace where the temp probe goes on the board I can't really follow it.the 8 pins on the transformer confuse me.theres shit loads of resistors though.25 of them.
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 Sciencemadness Discussion Board » Fundamentals » Reagents and Apparatus Acquisition » Does anyone know how to use one of these? Select A Forum Fundamentals   » Chemistry in General   » Organic Chemistry   » Reagents and Apparatus Acquisition   » Beginnings   » Responsible Practices   » Miscellaneous   » The Wiki Special topics   » Technochemistry   » Energetic Materials   » Biochemistry   » Radiochemistry   » Computational Models and Techniques   » Prepublication Non-chemistry   » Forum Matters   » Legal and Societal Issues   » Detritus   » Test Forum


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## G = C7×D8.C4order 448 = 26·7

### Direct product of C7 and D8.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×D8.C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C7×C8.C4 — C7×D8.C4
 Lower central C1 — C2 — C4 — C8 — C7×D8.C4
 Upper central C1 — C28 — C2×C28 — C2×C56 — C7×D8.C4

Generators and relations for C7×D8.C4
G = < a,b,c,d | a7=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=b5c >

Smallest permutation representation of C7×D8.C4
On 224 points
Generators in S224
(1 51 43 35 27 19 11)(2 52 44 36 28 20 12)(3 53 45 37 29 21 13)(4 54 46 38 30 22 14)(5 55 47 39 31 23 15)(6 56 48 40 32 24 16)(7 49 41 33 25 17 9)(8 50 42 34 26 18 10)(57 105 97 89 81 73 65)(58 106 98 90 82 74 66)(59 107 99 91 83 75 67)(60 108 100 92 84 76 68)(61 109 101 93 85 77 69)(62 110 102 94 86 78 70)(63 111 103 95 87 79 71)(64 112 104 96 88 80 72)(113 161 153 145 137 129 121)(114 162 154 146 138 130 122)(115 163 155 147 139 131 123)(116 164 156 148 140 132 124)(117 165 157 149 141 133 125)(118 166 158 150 142 134 126)(119 167 159 151 143 135 127)(120 168 160 152 144 136 128)(169 217 209 201 193 185 177)(170 218 210 202 194 186 178)(171 219 211 203 195 187 179)(172 220 212 204 196 188 180)(173 221 213 205 197 189 181)(174 222 214 206 198 190 182)(175 223 215 207 199 191 183)(176 224 216 208 200 192 184)
(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)
(1 60)(2 59)(3 58)(4 57)(5 64)(6 63)(7 62)(8 61)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 72)(16 71)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 80)(24 79)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 88)(32 87)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 96)(40 95)(41 102)(42 101)(43 100)(44 99)(45 98)(46 97)(47 104)(48 103)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 112)(56 111)(113 173)(114 172)(115 171)(116 170)(117 169)(118 176)(119 175)(120 174)(121 181)(122 180)(123 179)(124 178)(125 177)(126 184)(127 183)(128 182)(129 189)(130 188)(131 187)(132 186)(133 185)(134 192)(135 191)(136 190)(137 197)(138 196)(139 195)(140 194)(141 193)(142 200)(143 199)(144 198)(145 205)(146 204)(147 203)(148 202)(149 201)(150 208)(151 207)(152 206)(153 213)(154 212)(155 211)(156 210)(157 209)(158 216)(159 215)(160 214)(161 221)(162 220)(163 219)(164 218)(165 217)(166 224)(167 223)(168 222)
(1 115 3 113 5 119 7 117)(2 114 4 120 6 118 8 116)(9 125 11 123 13 121 15 127)(10 124 12 122 14 128 16 126)(17 133 19 131 21 129 23 135)(18 132 20 130 22 136 24 134)(25 141 27 139 29 137 31 143)(26 140 28 138 30 144 32 142)(33 149 35 147 37 145 39 151)(34 148 36 146 38 152 40 150)(41 157 43 155 45 153 47 159)(42 156 44 154 46 160 48 158)(49 165 51 163 53 161 55 167)(50 164 52 162 54 168 56 166)(57 169 63 171 61 173 59 175)(58 176 64 170 62 172 60 174)(65 177 71 179 69 181 67 183)(66 184 72 178 70 180 68 182)(73 185 79 187 77 189 75 191)(74 192 80 186 78 188 76 190)(81 193 87 195 85 197 83 199)(82 200 88 194 86 196 84 198)(89 201 95 203 93 205 91 207)(90 208 96 202 94 204 92 206)(97 209 103 211 101 213 99 215)(98 216 104 210 102 212 100 214)(105 217 111 219 109 221 107 223)(106 224 112 218 110 220 108 222)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 8E 8F 14A ··· 14F 14G ··· 14L 14M ··· 14R 16A ··· 16H 28A ··· 28L 28M ··· 28R 28S ··· 28X 56A ··· 56X 56Y ··· 56AJ 112A ··· 112AV order 1 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 112 ··· 112 size 1 1 2 8 1 1 2 8 1 ··· 1 2 2 2 2 8 8 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 1 ··· 1 2 ··· 2 8 ··· 8 2 ··· 2 8 ··· 8 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C4 C4 C7 C14 C14 C14 C28 C28 D4 D4 D8 SD16 C7×D4 C7×D4 D8.C4 C7×D8 C7×SD16 C7×D8.C4 kernel C7×D8.C4 C7×C8.C4 C2×C112 C7×C4○D8 C7×D8 C7×Q16 D8.C4 C8.C4 C2×C16 C4○D8 D8 Q16 C56 C2×C28 C28 C2×C14 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 2 2 6 6 6 6 12 12 1 1 2 2 6 6 8 12 12 48

Matrix representation of C7×D8.C4 in GL2(𝔽113) generated by

 109 0 0 109
,
 18 0 0 44
,
 0 15 98 0
,
 0 35 65 0
G:=sub<GL(2,GF(113))| [109,0,0,109],[18,0,0,44],[0,98,15,0],[0,65,35,0] >;

C7×D8.C4 in GAP, Magma, Sage, TeX

C_7\times D_8.C_4
% in TeX

G:=Group("C7xD8.C4");
// GroupNames label

G:=SmallGroup(448,163);
// by ID

G=gap.SmallGroup(448,163);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,1970,360,172,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^5*c>;
// generators/relations

Export

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text:
# Documentation

## Ideal Airspeed Correction

### Introduction

This case study simulates indicated and true airspeed. It constitutes a fragment of a complete aerodynamics problem, including only measurement and calibration.

### Airspeed Correction Models

To view the airspeed correction models, enter the following at the MATLAB® command line:

```aeroblk_indicated aeroblk_calibrated ```

aeroblk_indicated Model

aeroblk_calibrated Model

### Measure Airspeed

To measure airspeed, most light aircraft designs implement pitot-static airspeed indicators based on Bernoulli's principle. Pitot-static airspeed indicators measure airspeed by an expandable capsule that expands and contracts with increasing and decreasing dynamic pressure. This is known as calibrated airspeed (CAS). It is what a pilot sees in the cockpit of an aircraft.

To compensate for measurement errors, it helps to distinguish three types of airspeed. These types are explained more completely in the following.

Airspeed Type

Description

Calibrated

Indicated airspeed corrected for calibration error

Equivalent

Calibrated airspeed corrected for compressibility error

True

Equivalent airspeed corrected for density error

#### Calibration Error

An airspeed sensor features a static vent to maintain its internal pressure equal to atmospheric pressure. Position and placement of the static vent with respect to the angle of attack and velocity of the aircraft determines the pressure inside the airspeed sensor and therefore the calibration error. Thus, a calibration error is specific to an aircraft's design.

An airspeed calibration table, which is usually included in the pilot operating handbook or other aircraft documentation, helps pilots convert the indicated airspeed to the calibrated airspeed.

#### Compressibility Error

The density of air is not constant, and the compressibility of air increases with altitude and airspeed, or when contained in a restricted volume. A pitot-static airspeed sensor contains a restricted volume of air. At high altitudes and high airspeeds, calibrated airspeed is always higher than equivalent airspeed. Equivalent airspeed can be derived by adjusting the calibrated airspeed for compressibility error.

#### Density Error

At high altitudes, airspeed indicators read lower than true airspeed because the air density is lower. True airspeed represents the compensation of equivalent airspeed for the density error, the difference in air density at altitude from the air density at sea level, in a standard atmosphere.

### Model Airspeed Correction

The `aeroblk_indicated` and `aeroblk_calibrated` models show how to take true airspeed and correct it to indicated airspeed for instrument display in a Cessna 150M Commuter light aircraft. The `aeroblk_indicated` model implements a conversion to indicated airspeed. The `aeroblk_calibrated` model implements a conversion to true airspeed.

Each model consists of two main components:

#### COESA Atmosphere Model Block

The COESA Atmosphere Model block is a mathematical representation of the U.S. 1976 COESA (Committee on Extension to the Standard Atmosphere) standard lower atmospheric values for absolute temperature, pressure, density, and speed of sound for input geopotential altitude. Below 32,000 meters (104,987 feet), the U.S. Standard Atmosphere is identical with the Standard Atmosphere of the ICAO (International Civil Aviation Organization).

The `aeroblk_indicated` and `aeroblk_calibrated` models use the COESA Atmosphere Model block to supply the speed of sound and air pressure inputs for the Ideal Airspeed Correction block in each model.

#### Ideal Airspeed Correction Block

The Ideal Airspeed Correction block compensates for airspeed measurement errors to convert airspeed from one type to another type. The following table contains the Ideal Airspeed Correction block's inputs and outputs.

Airspeed InputAirspeed Output
True Airspeed Equivalent airspeed
Calibrated airspeed
Equivalent Airspeed True airspeed
Calibrated airspeed
Calibrated Airspeed True airspeed
Equivalent airspeed

In the `aeroblk_indicated` model, the Ideal Airspeed Correction block transforms true to calibrated airspeed. In the `aeroblk_calibrated` model, the Ideal Airspeed Correction block transforms calibrated to true airspeed.

The following sections explain how the Ideal Airspeed Correction block mathematically represents airspeed transformations:

True Airspeed Implementation.  True airspeed (TAS) is implemented as an input and as a function of equivalent airspeed (EAS), expressible as

$TAS=\frac{EAS×a}{{a}_{0}\sqrt{\delta }}$

where

 α Speed of sound at altitude in m/s δ Relative pressure ratio at altitude a0 Speed of sound at mean sea level in m/s

Calibrated Airspeed Implementation.  Calibrated airspeed (CAS), derived using the compressible form of Bernoulli's equation and assuming isentropic conditions, can be expressed as

$CAS=\sqrt{\frac{2\gamma {P}_{0}}{\left(\gamma -1\right){\rho }_{0}}\left[{\left(\frac{q}{{P}_{0}}+1\right)}^{\left(\gamma -1\right)/\gamma }-1\right]}$

where

 ρ0 Air density at mean sea level in kg/m3 P0 Static pressure at mean sea level in N/m2 γ Ratio of specific heats q Dynamic pressure at mean sea level in N/m2

Equivalent Airspeed Implementation.  Equivalent airspeed (EAS) is the same as CAS, except static pressure at sea level is replaced by static pressure at altitude.

$EAS=\sqrt{\frac{2\gamma P}{\left(\gamma -1\right){\rho }_{0}}\left[{\left(\frac{q}{P}+1\right)}^{\left(\gamma -1\right)/\gamma }-1\right]}$

The symbols are defined as follows:

 ρ0 Air density at mean sea level in kg/m3 P Static pressure at altitude in N/m2 γ Ratio of specific heats q Dynamic pressure at mean sea level in N/m2

### Simulate Airspeed Correction

In the `aeroblk_indicated` model, the aircraft is defined to be traveling at a constant speed of 72 knots (true airspeed) and altitude of 500 feet. The flaps are set to 40 degrees. The COESA Atmosphere Model block takes the altitude as input and outputs the speed of sound and air pressure. Taking the speed of sound, air pressure, and airspeed as inputs, the Ideal Airspeed Correction block converts true airspeed to calibrated airspeed. Finally, the Calculate IAS subsystem uses the flap setting and calibrated airspeed to calculate indicated airspeed.

The model's Display block shows both indicated and calibrated airspeeds.

In the `aeroblk_calibrated` model, the aircraft is defined to be traveling at a constant speed of 70 knots (indicated airspeed) and altitude of 500 feet. The flaps are set to 10 degrees. The COESA Atmosphere Model block takes the altitude as input and outputs the speed of sound and air pressure. The `Calculate CAS` subsystem uses the flap setting and indicated airspeed to calculate the calibrated airspeed. Finally, using the speed of sound, air pressure, and true calibrated airspeed as inputs, the Ideal Airspeed Correction block converts calibrated airspeed back to true airspeed.

The model's Display block shows both calibrated and true airspeeds.


text:
Explosion pour l'équation de Schrödinger au régime du « log log »  [ Blow-up for the non linear Schrödinger equation in the “log log” regime ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 953, p. 33-54

In this talk we present some recent results by Merle and Raphael on analysis of blow-up solution for the ${L}^{2}$ critical non linear Schrödinger equation. In particular, we focus on their proof of the fact that initial data with negative energy (which had been known to blow up by the viriel identity) and with ${L}^{2}$ norm close to the ground states’s ${L}^{2}$ norm, do blow up in the “log log” regime and that this behaviour is stable.

On présente dans cet exposé des résultats récents de Merle et Raphael sur l’analyse des solutions explosives de l’équation de Schrödinger ${L}^{2}$ critique. On s’intéresse en particulier à leur preuve du fait que les solutions d’énergie négative (dont on savait qu’elles explosaient par l’argument du viriel) et dont la norme ${L}^{2}$ est proche de celle de l’état fondamental, explosent au régime du “log log”et que ce comportement est stable.

Classification:  35B30,  35B35,  35B65
Keywords: non linear Schrödinger equations, blow-up
@incollection{SB_2005-2006__48__33_0,
author = {Burq, Nicolas},
title = {Explosion pour l'\'equation de Schr\"odinger au r\'egime du \guillemotleft{} log log \guillemotright{}},
booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
author = {Collectif},
series = {Ast\'erisque},
publisher = {Soci\'et\'e math\'ematique de France},
number = {311},
year = {2007},
note = {talk:953},
pages = {33-54},
zbl = {1194.35400},
language = {fr},
url = {http://www.numdam.org/item/SB_2005-2006__48__33_0}
}

Burq, Nicolas. Explosion pour l'équation de Schrödinger au régime du « log log », in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 953, pp. 33-54. http://www.numdam.org/item/SB_2005-2006__48__33_0/

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text:
Contents

category theory

# Contents

## Definition

A bifibration of categories is a functor

$E \to B$

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism $f : b_1 \to b_2$ in a bifibration has both a push-forward $f_* : E_{b_1} \to E_{b_2}$ as well as a pullback $f^* : E_{b_2} \to E_{b_1}$.

## Examples

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

## Relation to distributive laws

Fibrations and opfibrations on a category $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).

## Bifibration of model categories

In (CagneMelliès) the authors develop a notion of Quillen bifibration which combines the two concepts of Grothendieck bifibration and Quillen model structure, establishing conditions for when in a bifibration model structures on all the fibers and on the base combine to give one on the total category.

## Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into $2 Cat_{adj}$, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor $B^{coop} \to 2Cat$) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor $B\to 2Cat$) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.

## References

Last revised on June 27, 2018 at 09:51:37. See the history of this page for a list of all contributions to it.


text:
# I want to use a 12v dc actuator to make a can crusher.

#### bkhood1991

Joined Feb 9, 2016
29
Hi I am wanting to make a can crusher. I want to use a 12v dc actuator. I want it to simply crush the can and automatically retract back and keep going until you push a stop button. I was thinking maybe some photo eyes and latching relays or even a timer. But I need to know how you would wire this up so it will work right. And I need to convert the actuator 12v to 120v . Any suggestions about how to go about hooking this up. Maybe a step by step

#### ebeowulf17

Joined Aug 12, 2014
3,274
Might need to get more specific to get help. Do you have a specific actuator in mind? If so, more details (amperage, maybe dc resistance and/or inductance) will help. As for the control side, there are many regulars here who can probably help you out, but you need to figure out what you want. Does the optical sensor immediately trigger crushing? If so, is it going to crush someone's hands? Do you want delays before each crush? If it were my project, I'd want less automatic crushing and more safety, but it's your call.

#### Bernard

Joined Aug 7, 2008
5,556
Try: www.surpluscenter.com for linear actuators.12 V DC, 110 lb., 9 in. stroke about

#### bkhood1991

Joined Feb 9, 2016
29
I posted the link to the actuator I need . I just need to know what's next ? You have the actuator ? I need a convert it to 120v . Will I need relays for reversing polarity ?timer? Photo eye ? Ex

Joined Jul 18, 2013
20,898
That is .5"/sec and non-adjustable limit switches so if using these you would need to set the maximum travel for the L.S. when extended.
Max.

#### bkhood1991

Joined Feb 9, 2016
29
how would you wore it up if I wanted to have the actuator fully extend and retract back and keep doing that until I hit a stop button ? Would I wire the actuator to a latching relay to a timer then to another latching relay?

#### bkhood1991

Joined Feb 9, 2016
29
Feel free to call or text if easier .

MOD NOTE: Removed phone number.

Last edited by a moderator:

Joined Jul 18, 2013
20,898
So you want a cycle start P.B. etc then advance until the extend limit switch, then reverse until retract limits switch and repeat, with a cycle stop button included.
Max.

#### bkhood1991

Joined Feb 9, 2016
29
That sounds like it . Where would I wire my convertor into this ? Going from 12v to 120v . Wanting to be able to plug it into the wall

Joined Jul 18, 2013
20,898
First you need to obtain or build a 12vdc supply of a suitable power rating in order to run the motor .
Then you need the circuit required to run it, should be possible with relay logic.
Max.


text:
Note

Run interactively in the IBM Quantum lab.

# VQE on Aer simulator with noise¶

This notebook demonstrates using the Qiskit Aer qasm_simulator to run a simulation with noise, based on a given noise model. This can be useful to investigate behavior under different noise conditions. Aer not only allows you to define your own custom noise model, but also allows a noise model to be easily created based on the properties of a real quantum device. The latter is what this notebook will demonstrate since the goal is to show VQE with noise and not the more complex task of how to build custom noise models.

Qiskit Ignis provides a solution to mitigate the measurement error when running on a noisy simulation or a real quantum device. This solution is leveraged by the QuantumInstance module which allows any algorithm using it to automatically have measurement noise mitigation applied.

Further information on Qiskit Aer noise model can be found in the online Qiskit Aer documentation, also there is tutorial for building noise models.

Further information on measurement error mitigation in Qiskit Ignis can be found in the tutorial for measurement error mitigation.

[1]:

import numpy as np
import pylab

from qiskit import Aer
from qiskit.aqua import QuantumInstance, aqua_globals
from qiskit.aqua.algorithms import VQE, NumPyMinimumEigensolver
from qiskit.aqua.components.optimizers import SPSA
from qiskit.circuit.library import TwoLocal
from qiskit.aqua.operators import I, X, Z

Noisy simulation will be demonstrated here with VQE, finding the minimum (ground state) energy of an Hamiltonian, but the technique applies to any quantum algorithm from Qiskit Aqua.

So for VQE we need a qubit operator as input. Here, once again, we will take a set of paulis that were originally computed by Qiskit Chemistry, for an H2 molecule, so we can quickly create an Operator.

[2]:

H2_op = (-1.052373245772859 * I ^ I) + \
(0.39793742484318045 * I ^ Z) + \
(-0.39793742484318045 * Z ^ I) + \
(-0.01128010425623538 * Z ^ Z) + \
(0.18093119978423156 * X ^ X)

print(f'Number of qubits: {H2_op.num_qubits}')

Number of qubits: 2

/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/operators/operator_base.py:46: DeprecationWarning: The package qiskit.aqua.operators is deprecated. It was moved/refactored to qiskit.opflow (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_package('aqua.operators', 'qiskit.opflow', 'qiskit-terra')

As the above problem is still easily tractable classically we can use NumPyMinimumEigensolver to compute a reference value so we can compare later the results.

[3]:

npme = NumPyMinimumEigensolver()
result = npme.compute_minimum_eigenvalue(operator=H2_op)
ref_value = result.eigenvalue.real
print(f'Reference value: {ref_value:.5f}')

Reference value: -1.85728

/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/minimum_eigen_solvers/minimum_eigen_solver.py:36: DeprecationWarning: The package qiskit.aqua.algorithms.minimum_eigen_solvers is deprecated. It was moved/refactored to qiskit.algorithms.minimum_eigen_solvers (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_package('aqua.algorithms.minimum_eigen_solvers',
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/eigen_solvers/eigen_solver.py:36: DeprecationWarning: The package qiskit.aqua.algorithms.eigen_solvers is deprecated. It was moved/refactored to qiskit.algorithms.eigen_solvers (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_package('aqua.algorithms.eigen_solvers',

## Performance without noise¶

First we will run on the simulator without adding noise to see the result. I have created the backend and QuantumInstance, which holds the backend as well as various other run time configuration, which are defaulted here, so it easy to compare when we get to the next section where noise is added. There is no attempt to mitigate noise or anything in this notebook so the latter setup and running of VQE is identical.

[4]:

seed = 170
iterations = 125
aqua_globals.random_seed = seed
backend = Aer.get_backend('qasm_simulator')
qi = QuantumInstance(backend=backend, seed_simulator=seed, seed_transpiler=seed)

counts = []
values = []
def store_intermediate_result(eval_count, parameters, mean, std):
counts.append(eval_count)
values.append(mean)

var_form = TwoLocal(rotation_blocks='ry', entanglement_blocks='cz')
spsa = SPSA(maxiter=iterations)
vqe = VQE(var_form=var_form, optimizer=spsa, callback=store_intermediate_result, quantum_instance=qi)
result = vqe.compute_minimum_eigenvalue(operator=H2_op)
print(f'VQE on Aer qasm simulator (no noise): {result.eigenvalue.real:.5f}')
print(f'Delta from reference energy value is {(result.eigenvalue.real - ref_value):.5f}')

<ipython-input-1-e6494e819248>:3: DeprecationWarning: The variable qiskit.aqua.aqua_globals is deprecated. It was moved/refactored to qiskit.utils.algorithm_globals (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
aqua_globals.random_seed = seed
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/quantum_instance.py:135: DeprecationWarning: The class qiskit.aqua.QuantumInstance is deprecated. It was moved/refactored to qiskit.utils.QuantumInstance (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_class('aqua.QuantumInstance',
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/components/optimizers/optimizer.py:49: DeprecationWarning: The package qiskit.aqua.components.optimizers is deprecated. It was moved/refactored to qiskit.algorithms.optimizers (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_package('aqua.components.optimizers',
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/vq_algorithm.py:70: DeprecationWarning: The class qiskit.aqua.algorithms.VQAlgorithm is deprecated. It was moved/refactored to qiskit.algorithms.VariationalAlgorithm (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_class('aqua.algorithms.VQAlgorithm',

VQE on Aer qasm simulator (no noise): -1.85464
Delta from reference energy value is 0.00264

We captured the energy values above during the convergence so we can see what went on in the graph below.

[5]:

pylab.rcParams['figure.figsize'] = (12, 4)
pylab.plot(counts, values)
pylab.xlabel('Eval count')
pylab.ylabel('Energy')
pylab.title('Convergence with no noise')

[5]:

Text(0.5, 1.0, 'Convergence with no noise')

## Performance with noise¶

Now we will add noise. Here we will create a noise model for Aer from an actual device. You can create custom noise models with Aer but that goes beyond the scope of this notebook. Links to further information on Aer noise model, for those that may be interested in doing this, were given above.

First we need to get an actual device backend and from its configuration and properties we can setup a coupling map and a noise model to match the device. While we could leave the simulator with the default all to all map, this shows how to set the coupling map too. Note: We can also use this coupling map as the entanglement map for the variational form if we choose.

Note: simulation with noise will take longer than without noise.

Terra Mock Backends:

We will use real noise data for an IBM Quantum device using the date stored in Qiskit Terra. Specifically, in this tutorial, the device is ibmq_vigo.

[6]:

import os
from qiskit.providers.aer import QasmSimulator
from qiskit.providers.aer.noise import NoiseModel
from qiskit.test.mock import FakeVigo
device_backend = FakeVigo()

backend = Aer.get_backend('qasm_simulator')
counts1 = []
values1 = []
noise_model = None
os.environ['QISKIT_IN_PARALLEL'] = 'TRUE'
device = QasmSimulator.from_backend(device_backend)
coupling_map = device.configuration().coupling_map
noise_model = NoiseModel.from_backend(device)
basis_gates = noise_model.basis_gates

print(noise_model)
print()

aqua_globals.random_seed = seed
qi = QuantumInstance(backend=backend, seed_simulator=seed, seed_transpiler=seed,
coupling_map=coupling_map, noise_model=noise_model,)

def store_intermediate_result1(eval_count, parameters, mean, std):
counts1.append(eval_count)
values1.append(mean)

var_form = TwoLocal(rotation_blocks='ry', entanglement_blocks='cz')
spsa = SPSA(maxiter=iterations)
vqe = VQE(var_form=var_form, optimizer=spsa, callback=store_intermediate_result1, quantum_instance=qi)
result1 = vqe.compute_minimum_eigenvalue(operator=H2_op)
print(f'VQE on Aer qasm simulator (with noise): {result1.eigenvalue.real:.5f}')
print(f'Delta from reference energy value is {(result1.eigenvalue.real - ref_value):.5f}')

NoiseModel:
Basis gates: ['cx', 'id', 'kraus', 'roerror', 'rz', 'save_amplitudes', 'save_amplitudes_sq', 'save_density_matrix', 'save_expval', 'save_expval_var', 'save_probabilities', 'save_probabilities_dict', 'save_stabilizer', 'save_state', 'save_statevector', 'save_statevector_dict', 'set_density_matrix', 'set_stabilizer', 'set_statevector', 'snapshot', 'sx', 'x']
Instructions with noise: ['measure', 'id', 'cx', 'sx', 'x']
Qubits with noise: [0, 1, 2, 3, 4]
Specific qubit errors: [('id', [0]), ('id', [1]), ('id', [2]), ('id', [3]), ('id', [4]), ('sx', [0]), ('sx', [1]), ('sx', [2]), ('sx', [3]), ('sx', [4]), ('x', [0]), ('x', [1]), ('x', [2]), ('x', [3]), ('x', [4]), ('cx', [3, 4]), ('cx', [4, 3]), ('cx', [3, 1]), ('cx', [1, 3]), ('cx', [1, 2]), ('cx', [2, 1]), ('cx', [0, 1]), ('cx', [1, 0]), ('measure', [0]), ('measure', [1]), ('measure', [2]), ('measure', [3]), ('measure', [4])]

VQE on Aer qasm simulator (with noise): -1.80215
Delta from reference energy value is 0.05512

[7]:

if counts1 or values1:
pylab.rcParams['figure.figsize'] = (12, 4)
pylab.plot(counts1, values1)
pylab.xlabel('Eval count')
pylab.ylabel('Energy')
pylab.title('Convergence with noise')

## Performance with noise and measurement error mitigation¶

Now we will add method for measurement error mitigation, which increases the fidelity of measurement. Here we choose CompleteMeasFitter to mitigate the measurement error. The calibration matrix will be auto-refresh every 30 minute (default value).

Note: simulation with noise will take longer than without noise.

[8]:

from qiskit.ignis.mitigation.measurement import CompleteMeasFitter

counts2 = []
values2 = []
if noise_model is not None:
aqua_globals.random_seed = seed
qi = QuantumInstance(backend=backend, seed_simulator=seed, seed_transpiler=seed,
coupling_map=coupling_map, noise_model=noise_model,
measurement_error_mitigation_cls=CompleteMeasFitter,
cals_matrix_refresh_period=30)

def store_intermediate_result2(eval_count, parameters, mean, std):
counts2.append(eval_count)
values2.append(mean)

var_form = TwoLocal(rotation_blocks='ry', entanglement_blocks='cz')
spsa = SPSA(maxiter=iterations)
vqe = VQE(var_form=var_form, optimizer=spsa, callback=store_intermediate_result2, quantum_instance=qi)
result2 = vqe.compute_minimum_eigenvalue(operator=H2_op)
print(f'VQE on Aer qasm simulator (with noise and measurement error mitigation): {result2.eigenvalue.real:.5f}')
print(f'Delta from reference energy value is {(result2.eigenvalue.real - ref_value):.5f}')

VQE on Aer qasm simulator (with noise and measurement error mitigation): -1.86432
Delta from reference energy value is -0.00704

[9]:

if counts2 or values2:
pylab.rcParams['figure.figsize'] = (12, 4)
pylab.plot(counts2, values2)
pylab.xlabel('Eval count')
pylab.ylabel('Energy')
pylab.title('Convergence with noise, measurement error mitigation enabled')

Lets bring the results together here for a summary.

We produced a reference value using a classical algorithm and then proceeded to run VQE on a qasm simulator. While the simulation is ideal (no noise) there is so called shot-noise due to sampling - increasing the number of shots reduces this as more and more samples are taken, but shots was left at the default of 1024 and we see a small effect in the outcome.

Then we added noise using a model taken off a real device and can see the result is affected. Finally we added measurement noise mitigation which adjusts the results in an attempt to alleviate the affect of noise in the classical equipment measuring the qubits.

[10]:

print(f'Reference value: {ref_value:.5f}')
print(f'VQE on Aer qasm simulator (no noise): {result.eigenvalue.real:.5f}')
print(f'VQE on Aer qasm simulator (with noise): {result1.eigenvalue.real:.5f}')
print(f'VQE on Aer qasm simulator (with noise and measurement error mitigation): {result2.eigenvalue.real:.5f}')

Reference value: -1.85728
VQE on Aer qasm simulator (no noise): -1.85464
VQE on Aer qasm simulator (with noise): -1.80215
VQE on Aer qasm simulator (with noise and measurement error mitigation): -1.86432

[11]:

import qiskit.tools.jupyter
%qiskit_version_table

### Version Information

Qiskit SoftwareVersion
Qiskit0.25.4
Terra0.17.2
Aer0.8.2
Ignis0.6.0
Aqua0.9.1
IBM Q Provider0.12.3
System information
Python3.8.10 (default, May 4 2021, 07:16:51) [GCC 9.3.0]
OSLinux
CPUs2
Memory (Gb)6.791343688964844
Wed May 05 19:44:56 2021 UTC

### This code is a part of Qiskit

[ ]:


text:
Types of conic sections
Table of conics, Cyclopaedia, 1728

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a circular conical surface) with a plane. Cyclopaedia or A Universal Dictionary of Arts and Sciences ( folio, 2 vols Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

## Types of conics

The three types of conics are the hyperbola, ellipse, and parabola. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the The circle and the ellipse arise when the intersection of cone and plane is a closed curve. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves, though often one is ignored.

### Degenerate cases

There are multiple degenerate cases, in which the plane passes through the apex of the cone. The intersection in these cases can be a straight line (when the plane is tangential to the surface of the cone); a point (when the angle between the plane and the axis of the cone is larger than tangential); or a pair of intersecting lines (when the angle is smaller). In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume

Where the cone is a cylinder (the vertex is at infinity) cylindric sections are obtained. Although these yield mostly ellipses (or circles)[1] as normal, a degenerate case of two parallel lines can also be produced.

### Eccentricity

Ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with fixed focus F and directrix.

The four defining conditions above can be combined into one condition that depends on a fixed point F (the focus), a line L (the directrix) not containing F and a nonnegative real number e (the eccentricity). In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is a / e, where $a \$ is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae The distance from the center to a focus is $ae \$.

In the case of a circle, the eccentricity e = 0, and one can imagine the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity.

The eccentricity of a conic section is thus a measure of how far it deviates from being circular.

For a given $a \$, the closer $e \$ is to 1, the smaller is the semi-minor axis. In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and

## Cartesian coordinates

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. The equation will be of the form

$Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0\;$ with $A \$, $B \$, $C \$ not all zero.

then:

• if $B^2 - 4AC < 0 \$, the equation represents an ellipse (unless the conic is degenerate, for example $x^2 + y^2 + 10 = 0 \$);
• if $A = C \$ and $B = 0 \$, the equation represents a circle;
• if $B^2 - 4AC = 0 \$, the equation represents a parabola;
• if $B^2 - 4AC > 0 \$, the equation represents a hyperbola;
• if we also have $A + C = 0 \$, the equation represents a rectangular hyperbola. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

Note that A and B are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the previous sections.

Through change of coordinates these equations can be put in standard forms:

• Circle: $x^2+y^2=r^2\,$
• Ellipse: ${x^2\over a^2}+{y^2\over b^2}=1 \$, ${x^2\over b^2}+{y^2\over a^2}=1 \$
• Parabola: $y^2=4ax\, \$, $x^2=4ay\, \$
• Hyperbola: ${x^2\over a^2}-{y^2\over b^2}=1 \$, ${x^2\over a^2}-{y^2\over b^2}=-1 \$
• Rectangular Hyperbola: $xy=c^2 \$

Such forms will be symmetrical about the x-axis and for the circle, ellipse and hyperbola symmetrical about the y-axis.
The rectangular hyperbola however is only symmetrical about the lines $y = x\$ and $y = -x\$. Therefore its inverse function is exactly the same as its original function.

These standard forms can be written as parametric equations,

• Circle: $(a\cos\theta,a\sin\theta)\,$,
• Ellipse: $(a\cos\theta,b\sin\theta)\,$,
• Parabola: $(a t^2,2 a t)\,$,
• Hyperbola: $(a\sec\theta,b\tan\theta)\,$ or $(\pm a\cosh u,b \sinh u)\,$. In Mathematics, parametric equations are a method of defining a curve
• Rectangular Hyperbola: $(ct,{c \over t})\,$

## Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as:

A1x2 + A2y2 + A3z2 + 2B1xy + 2B2xz + 2B3yz = 0. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

Or in matrix notation

$\begin{bmatrix}x & y & z\end{bmatrix} . \begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix} . \begin{bmatrix}x\\y\\z\end{bmatrix} = 0.$

The matrix $M=\begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix}$ is called the matrix of the conic section. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

$\Delta = \det(M) = \det\left(\begin{bmatrix}A_1 & B_1 & B_2\\B_1 & A_2 & B_3\\B_2&B_3&A_3\end{bmatrix}\right)$ is called the determinant of the conic section. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n If Δ = 0 then the conic section is said to be degenerate, this means that the conic section is in fact a union of two straight lines. A conic section that intersects itself is always degenerate, however not all degenerate conic sections intersect themselves, if they do not they are straight lines.

For example, the conic section $\begin{bmatrix}x & y & z\end{bmatrix} . \begin{bmatrix}1 & 0 & 0\\0 & -1 & 0\\0&0&0\end{bmatrix} . \begin{bmatrix}x\\y\\z\end{bmatrix} = 0$ reduces to the union of two lines:

$\{ x^2 - y^2 = 0\} = \{(x+y)(x-y)=0\} = \{x+y=0\} \cup \{x-y=0\}$.

Similarly, a conic section sometimes reduces to a (single) line:

$\{x^2+2xy+y^2 = 0\} = \{(x+y)^2=0\}=\{x+y=0\} \cup \{x+y=0\} = \{x+y=0\}$.

$\delta = \det\left(\begin{bmatrix}A_1 & B_1\\B_1 & A_2\end{bmatrix}\right)$ is called the discriminant of the conic section. In Algebra, the discriminant of a Polynomial with real or complex Coefficients is a certain expression in the coefficients of the If δ = 0 then the conic section is a parabola, if δ<0, it is an hyperbola and if δ>0, it is an ellipse. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a A conic section is a circle if δ>0 and A1 = A2, it is an rectangular hyperbola if δ<0 and A1 = -A2. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the It can be proven that in the complex projective plane CP2 two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection points and there is always 1 intersection point (possibilities: 4 distinct intersection points, 2 singular intersection points and 1 double intersection points, 2 double intersection points, 1 singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. In Euclidean geometry, the Intersection of a line and a line can be the Empty set, a point, or a line If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating[2].

Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is said to be tangent and it is called the tangent line. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Because every straight line intersects a conic section twice, each conic section has two points at infinity (the intersection points with the line at infinity). Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" "Ideal line" redirects here For the ideal line in racing see Racing line. If these points are real, the conic section must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic section has one double point at infinity it is a parabola. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated it is neither a parabola nor an ellipse nor a hyperbola.

## Polar coordinates

In polar coordinates, a conic section with one focus at the origin and, if any, the other on the x-axis, is given by the equation

$r = { l \over {1 + e \cos \theta} }$,

where e is the eccentricity and l is the semi-latus rectum (see below). In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by As above, for e = 0, we have a circle, for 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

## Parameters

Various parameters can be associated with a conic section.

conic sectionequationeccentricity (e)linear eccentricity (c)semi-latus rectum (l)focal parameter (p)
circle$x^2+y^2=r^2 \,$00$r \,$$\infty$
ellipse$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$$\frac{\sqrt{a^2-b^2}}{a}$$\sqrt{a^2-b^2}$$\frac{b^2}{a}$$\frac{b^2}{\sqrt{a^2-b^2}}$
parabola$y^2=4ax \,$, $x^2=4ay \,$1a$2a \,$2a
hyperbola$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$$\frac{\sqrt{a^2+b^2}}{a}$$\sqrt{a^2+b^2}$$\frac{b^2}{a}$$\frac{b^2}{\sqrt{a^2+b^2}}$
conic parameters in the case of an ellipse

For every conic section, there exist a fixed point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions e is called the eccentricity of the conic section. In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci). In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle

The latus rectum (2l) is the chord parallel to the directrix and passing through the focus (or one of the two foci). In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

The semi-latus rectum (l) is half the latus rectum.

The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix. Distance is a numerical description of how far apart objects are

The relation p = l / e holds.

## Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. In Differential calculus, an inflection point, or point of inflection (or inflexion) is a point on a Curve at which the Curvature This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence. Laminar flow, sometimes known as streamline flow occurs when a fluid flows in parallel layers with no disruption between the layers In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes

## Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Gravitation is a natural Phenomenon by which objects with Mass attract one another If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by

In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose A projective transformation is a transformation used in Projective geometry: it is the composition of a pair of Perspective projections It describes what

For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions

## Intersecting two conics

The solutions to a two second degree equations system in two variables may be seen as the coordinates of the intersections of two generic conic sections. In particular two conics may possess none, two, four possibly coincident intersection points. The best method to locate these solutions is to exploits the homogeneous matrix representation of conic sections, i. In Mathematics, the matrix representation of conic sections is one way of studying a Conic section, its axis, vertices, foci, e. a 3x3 symmetric matrix which depends on six parameters. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T}

The procedure to locate the intersection points follows these steps:

• given the two conics C1 and C2 consider the pencil of conics given by their linear combination λC1 + μC2
• identify the homogeneous parameters (λ,μ) which corresponds to the degenerate conic of the pencil. This can be done by imposing that detC1 + μC2) = 0, which turns out to be the solution to a third degree equation.
• given the degenerate cone C0, identify the two, possibly coincident, lines constituting it
• intersects each identified line with one of the two original conic; this step can be done efficiently using the dual conic representation of C0
• the points of intersection will represent the solution to the initial equation system

## Dandelin spheres

See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix. In Geometry, a nondegenerate Conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus Each


text:
## Cryptology ePrint Archive: Report 2021/1121

Constant-Time Arithmetic for Safer Cryptography

Lúcás Críostóir Meier and Simone Colombo and Marin Thiercelin and Bryan Ford

Abstract: The humble integers, $\mathbb{Z}$, are the backbone of many cryptosystems. When bridging the gap from theoretical systems to real-world implementations, programmers often look towards general purpose libraries to implement the arbitrary-precision arithmetic required. Alas, these libraries are often conceived without cryptography in mind, leaving applications potentially vulnerable to timing attacks. To address this, we present saferith, a library providing safer arbitrary-precision arithmetic for cryptography, through constant-time operations. The main challenge was in designing an API to provide this functionality alongside these stronger constant-time guarantees. We benchmarked the performance of our library against Go's big.Int library, and found an acceptable slowdown of only 2.56x for modular exponentiation, the most expensive operation. Our library was also used to implement a variety cryptosystems and applications, in collaboration with industrial partners ProtonMail and Taurus. Porting implementations to use our library is relatively easy: it took the first author under 8 hours to port Go's implementation of P-384.

Category / Keywords: applications / constant-time, elliptic curve cryptosystem, DSA, implementation, timing attack, RSA

Date: received 2 Sep 2021, last revised 2 Sep 2021

Contact author: lucascriostoir meier at epfl ch

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1121

[ Cryptology ePrint archive ]


text:
# 3-opt Information

https://en.wikipedia.org/wiki/3-opt

In optimization, 3-opt is a simple local search algorithm for solving the travelling salesperson problem and related network optimization problems. Compared to the simpler 2-opt algorithm, it is slower but can generate higher-quality solutions.

3-opt analysis involves deleting 3 connections (or edges) in a network (or tour), to create 3 sub-tours. Then the 7 different ways of reconnecting the network are analysed to find the optimum one. This process is then repeated for a different set of 3 connections, until all possible combinations have been tried in a network. A single execution of 3-opt has a time complexity of ${\displaystyle O(n^{3})}$. [1] Iterated 3-opt has a higher time complexity.

This is the mechanism by which the 3-opt swap manipulates a given route:

def reverse_segment_if_better(tour, i, j, k):
"""If reversing tour[i:j] would make the tour shorter, then do it."""
# Given tour [...A-B...C-D...E-F...]
A, B, C, D, E, F = tour[i-1], tour[i], tour[j-1], tour[j], tour[k-1], tour[k % len(tour)]
d0 = distance(A, B) + distance(C, D) + distance(E, F)
d1 = distance(A, C) + distance(B, D) + distance(E, F)
d2 = distance(A, B) + distance(C, E) + distance(D, F)
d3 = distance(A, D) + distance(E, B) + distance(C, F)
d4 = distance(F, B) + distance(C, D) + distance(E, A)

if d0 > d1:
tour[i:j] = reversed(tour[i:j])
return -d0 + d1
elif d0 > d2:
tour[j:k] = reversed(tour[j:k])
return -d0 + d2
elif d0 > d4:
tour[i:k] = reversed(tour[i:k])
return -d0 + d4
elif d0 > d3:
tmp = tour[j:k] + tour[i:j]
tour[i:k] = tmp
return -d0 + d3
return 0

The principle is pretty simple. You compute the original distance ${\displaystyle d_{0}}$ and you compute the cost of each modification. If you find a better cost, apply the modification and return ${\displaystyle \delta }$ (relative cost). This is the complete 3-opt swap making use of the above mechanism:

def three_opt(tour):
"""Iterative improvement based on 3 exchange."""
while True:
delta = 0
for (a, b, c) in all_segments(len(tour)):
delta += reverse_segment_if_better(tour, a, b, c)
if delta >= 0:
break

def all_segments(n: int):
"""Generate all segments combinations"""
return ((i, j, k)
for i in range(n)
for j in range(i + 2, n)
for k in range(j + 2, n + (i > 0)))

For the given tour, you generate all segments combinations and for each combinations, you try to improve the tour by reversing segments. While you find a better result, you restart the process, otherwise finish.

1. ^ Blazinskas, Andrius; Misevicius, Alfonsas (2011). "Combining 2-OPT, 3-OPT and 4-OPT with K-SWAP-KICK perturbations for the traveling salesman problem". S2CID  15324387. {{ cite journal}}: Cite journal requires |journal= ( help)


text:
# Creating and Using Packages

To organise a class into a package we use the package keyword. The package corresponds with the directory name relative to the parent folder.

## Example

Here is an example of the Main class defined in the package edu.self package :

package edu.self;

public class Main {

public static void main(String[] args) {
}
}

and here is the folder structure on the operating system :

### Directory Structure

1. src is the parent directory
2. edu is the parent package
3. self is the sub package defined in the edu package.

Packages can be nested as in the above case by using a dot, (.). The directory structure will have to correspond.

## Using Packages Members

To use a class or interface defined in the package they will need to be imported. The members are imported using the import keyword. Members in the java.lang package do not need to be imported, they are available automatically.

The import statement should come after the package statement and before any class definitions.

### Importing a Class

After adding the Socket.IO library to our project we will have access to the classes and interfaces of the library.

To use the Socket class, defined in the package, package io.socket.client, we can import it as follows :

import io.socket.client.Socket;

Now we have access to the Socket class in our code.

Instead of imported we can also refer to the member by its fully qualified name as follows :

io.socket.client.Socket socket;

### Importing Entire Package

Or we can import the whole package by using the asterisk, *.

import io.socket.client.*;

### Apparent Hierarchies of Packages

Packages appear to be hierarchical but they are not. Package naming is used to show relationships but not show inclusion.

Importing io.socket.* imports all the types in the io.socket package, but does not import io.socket.emmiter or io.socket.client packages.

### Name Ambiguities

If a member in one package shares its name with a member in another package and both packages are imported, you must refer to each member by its qualified name.

In our chat application, they are two Socket classes, one is defined in the io.socket.engineio.client and another in the io.socket.client. To use both in our class, the Socket class will need to be fully qualified as follows :

io.socket.client.Socket clientSocket;
io.socket.engineio.client.Socket engineSocket;

## Static Import

Static methods and fields can be imported so that they can be used without their class or interface prefix by using the import static keyword.

We can import the static abs method in the java.lang.Math class as follows :

import static java.lang.Math.abs;

Now we can use the abs method anywhere without prefixing it with the Math class as follows :

import static java.lang.Math.abs;

public class Main {
public static void main(String[] args) {
int a = abs(-10); //now we can use without typing Math.abs()
}
}