text:
# How whizard treat on-shell and off-shell particles

Asked by David Wang on 2020-09-07

Dear Whizard team,

What's the criteria of whizard in determining whether a particle is on-shell or off-shell.

For instance, with the default run_card, MadGraph treats particles as on-shell if their invariant mass is in the range M \pm 15*Gamma, with M the particle mass, Gamma the width, and 15 the default number that we have control over in the run_card.

Does Whizard have an analogous setting?

## Question information

Language:
English Edit question
Status:
Solved
For:
WHIZARD Edit question
Assignee:
Simon Braß Edit question
Solved by:
Juergen Reuter
Solved:
2020-09-17
Last query:
2020-09-17
2020-09-09
 Simon Braß (sbrass) said on 2020-09-07: #1

Dear David,

WHIZARD and Madgraph tackle the underlying problems regarding integration and event generation slightly different.
Therefore, it's quite difficult to give a "simple" answer.

However, given your question, I may guide you in following directions:

1) Phase-space generation? We do not have a width multiplicator, as we apply different phase-space mapping then Madgraph.
You can read on the options in our manual, "5.7 Cross sections: The main phase space setup wood".
However, the defaults should be fine.

2) Event generation with resonances? We have so-called resonance histories which allow to insert a resonance (history) into an event using a probabilistic approach (and more...).
The major point is to compute the full-matrix element (with no restrictions), and insert during event generation a (randomly) chosen intermediate decay channel (i.e. resonance history), which is then flagged in the event file accordingly.

Here, we allow to specify a width multiplicator, "resonance_on_shell_limit [= 4]".
I recommend you to read our manual, "10.6.4 Resonance insertion in the event record", which may answer everything else.

Cheers,
Simon

 David Wang (david-mhw) said on 2020-09-07: #2

Thanks! I think the second point you mentioned is related to our issue.
More specifically, the process is mu mu -> W Z/gamma -> W -> T bbar process where T is vector-like quark. We pick several collider energy to calculate the cross section. However, at some energy point, there is clearly a peak(order of magnitude higher) in the cross section values. This peak doesn't occur in the MadGraph simulation when applying the exact same cuts. So, I guess the resonance might not be physical.

 Juergen Reuter (j.r.reuter) said on 2020-09-08: #3

The second point that Simon mentioned is only relevant for parton showering, and you would see it then in detector-level observables like energy and multiplicity distributions of photons, neutral hadrons, etc. With peak in the cross section, you mean a peak in the curve for the total cross section as a function of the muon collider energy, correct? What is your process definition, what are the cuts? What are the input parameters? Was the integration stable? Could you paste here the SINDARIN file and the relevant parts of the whizard.log?

 Simon Braß (sbrass) said on 2020-09-09: #4

Hi David,

I think I did not went on in too much detail with my explanation:
1) You have to explicitly set on the resonance history, and then, I will only insert additional particle information into event record, without changing the sum of weights / cross section.
2) You are looking at total cross sections (for different beam energies), so, we are only talking about phase-space mapping and integration.
Without knowing your exact approach, I (and the other) can not provide further help at this point (Jürgen pointed this problem already out).

 David Wang (david-mhw) said on 2020-09-09: #5

Thanks! The following is part of my sin script and the whizard.log. This is a BSM process so beside scanning through the collider energy, we also scan through the possible particle mass.

This is a charged VBF process and the Feynman diagram should look like the following and the vertex are given by the TW card parameters.

mu mu -> W Z/gamma -> W -> T bbar

You can see that from the log file with the same sqrts(center of mass) energy 6 TeV, when MTP = 1TeV, the cross section is about 1e2, when MTP=2.5TeV, the cross section is about 1e3, then it drops to 1e-2 at MTP=5TeV(which I did not show but you get the idea). So, the weird resonance is occurred at MTP=2.5TeV.

The cuts are Pt and eta cuts on initial muons. One interesting fact is that if we do a cut on on-shell W,
and all M > 150 GeV ["vm~", "mu-"]
and all M > 150 GeV ["vm", "mu+"]
the resonance can be smoothed out quite a bit.
-----------------------------------------------------------------------------------------
# import the ufo model file
model = VLQ (ufo)

# define parameters
# TW card
xitpw = 1.0 xitpz = 0.0 xitph = 0.0
xibpw = 1.0 xibpz = 0.0 xibph = 0.0

# define the process
process myprocess = "mu-", "mu+" => ("tp", "b~", "vm~", "mu-") + ("tp~", "b", "vm", "mu+")
compile

cuts = all Pt > 30 GeV ["mu-","mu+"]
and all abs(Eta) < 2.5 ["mu+"]
and all abs(Eta) < 2.5 ["mu-"]

scan sqrts = (1 TeV, 3 TeV, 6 TeV, 10 TeV, 14 TeV)
{
# scan through different particle masses
scan MTP = ( 0.1 TeV, 0.2 TeV, 0.25 TeV, 0.35 TeV, 0.5 TeV, 0.8 TeV, 1 TeV, 1.2 TeV, 2.3 TeV, 2.5 TeV, 2.7 TeV, 4.8 TeV, 5 TeV, 5.2 TeV, 10 TeV, 25 TeV)
{
if MTP < sqrts then
seed = 12345

# iteration control
accuracy_goal = 1.4
integrate (myprocess){iterations = 6:20000: "gw", 3:50000}

# record mass points, cross section, error of cross section, collider energy
record lineshape (MTP/1e3, integral(myprocess), error(myprocess), sqrts/1e3)

# simulation
string $filename = sprintf "%s_%s_MTP%.3fTeV_s%.0fTeV" ($finalstate, $mode, MTP/1e3, sqrts/1e3) simulate (myprocess) {n_events = 10000 sample_format = lhef$sample = $filename} else endif } string$exten_name = sprintf "_to_s%.0fTeV.dat" (sqrts/1e3)
compile_analysis {$out_file =$title & $exten_name} } |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 19988 1.2789566E+02 2.11E+00 1.65 2.33* 6.28 2 19978 1.2738612E+02 5.49E-01 0.43 0.61* 21.08 | VAMP: accuracy goal reached; skipping iterations |-----------------------------------------------------------------------------| 2 39966 1.2741858E+02 5.32E-01 0.42 0.83 21.08 0.05 2 |-----------------------------------------------------------------------------| 3 49990 1.2735932E+02 3.52E-01 0.28 0.62 15.49 4 49990 1.2768718E+02 3.52E-01 0.28 0.62* 15.04 5 49990 1.2683915E+02 3.45E-01 0.27 0.61* 14.61 |-----------------------------------------------------------------------------| 5 149970 1.2728963E+02 2.02E-01 0.16 0.61 14.61 1.51 3 |=============================================================================| | Time estimate for generating 10000 events: 0d:00h:00m:12s | Starting integration for process 'myprocess' part 2 | Integrate: iterations = 6:20000:"gw", 3:50000 | Integrator: 16 chains, 19 channels, 8 dimensions | Integrator: Using VAMP channel equivalences | Integrator: 20000 initial calls, 20 bins, stratified = T | Integrator: VAMP |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 19988 1.2543916E+02 2.19E+00 1.74 2.46* 6.29 2 19979 1.2652631E+02 5.83E-01 0.46 0.65* 20.48 | VAMP: accuracy goal reached; skipping iterations |-----------------------------------------------------------------------------| 2 39967 1.2645421E+02 5.63E-01 0.45 0.89 20.48 0.23 2 |-----------------------------------------------------------------------------| 3 49989 1.2671555E+02 3.71E-01 0.29 0.65 13.83 4 49989 1.2666061E+02 3.69E-01 0.29 0.65* 13.03 5 49989 1.2749646E+02 3.79E-01 0.30 0.67 11.30 |-----------------------------------------------------------------------------| 5 149967 1.2694841E+02 2.15E-01 0.17 0.66 11.30 1.55 3 |=============================================================================| | Time estimate for generating 10000 events: 0d:00h:00m:15s | Integrate: sum of all components |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 0 2.5423804E+02 2.95E-01 0.12 0.00* 12.74 |=============================================================================| [user variable]$filename = "T_bbar_chargedVBF_MTP1.000TeV_s6TeV"
n_events = 10000
$sample = "T_bbar_chargedVBF_MTP1.000TeV_s6TeV" | Starting simulation for process 'myprocess' | Simulate: using integration grids from file 'myprocess.m1.vg' | Simulate: using integration grids from file 'myprocess.m2.vg' | RNG: Initializing TAO random-number generator | RNG: Setting seed for random-number generator to 12346 | OpenMP: Using 4 threads | Simulation: requested number of events = 10000 | corr. to luminosity [fb-1] = 3.9333E+01 | Events: writing to LHEF file 'T_bbar_chargedVBF_MTP1.000TeV_s6TeV.lhe' | Events: writing to raw file 'T_bbar_chargedVBF_MTP1.000TeV_s6TeV.evx' | Events: generating 10000 unweighted, unpolarized events ... | Events: event normalization mode '1' | ... event sample complete. | Events: actual unweighting efficiency = 12.92 % Warning: Encountered events with excess weight: 8 events ( 0.080 %) | Maximum excess weight = 9.533E-01 | Average excess weight = 2.806E-04 | Events: closing LHEF file 'T_bbar_chargedVBF_MTP1.000TeV_s6TeV.lhe' | Events: closing raw file 'T_bbar_chargedVBF_MTP1.000TeV_s6TeV.evx' | Restoring model 'VLQ' VLQ.MTP => 2.500000000000E+03 seed = 12345 accuracy_goal = 1.400000000000E+00 | RNG: Initializing TAO random-number generator | RNG: Setting seed for random-number generator to 12345 | Initializing integration for process myprocess: | Beam structure: [any particles] | Beam data (collision): | mu- (mass = 0.0000000E+00 GeV) | mu+ (mass = 0.0000000E+00 GeV) | sqrts = 6.000000000000E+03 GeV | Phase space: generating configuration ... | Phase space: ... success. | Phase space: writing configuration file 'myprocess.i1.phs' | Phase space: generating configuration ... | Phase space: ... success. | Phase space: writing configuration file 'myprocess.i2.phs' | ------------------------------------------------------------------------ | Process [scattering]: 'myprocess' | Library name = 'default_lib' | Process index = 1 | Process components: | 1: 'myprocess_i1': mu-, mu+ => tp, b~, vm~, mu- [omega] | 2: 'myprocess_i2': mu-, mu+ => tp~, b, vm, mu+ [omega] | ------------------------------------------------------------------------ | Phase space: 19 channels, 8 dimensions | Phase space: found 19 channels, collected in 16 groves. | Phase space: Using 19 equivalences between channels. | Phase space: wood | Phase space: 19 channels, 8 dimensions | Phase space: found 19 channels, collected in 16 groves. | Phase space: Using 19 equivalences between channels. | Phase space: wood | Applying user-defined cuts. | OpenMP: Using 4 threads | Starting integration for process 'myprocess' part 1 | Integrate: iterations = 6:20000:"gw", 3:50000 | Integrator: 16 chains, 19 channels, 8 dimensions | Integrator: Using VAMP channel equivalences | Integrator: 20000 initial calls, 20 bins, stratified = T | Integrator: VAMP |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 19988 1.5108061E+03 2.29E+01 1.51 2.14* 9.77 2 19979 1.4707417E+03 5.72E+00 0.39 0.55* 29.04 | VAMP: accuracy goal reached; skipping iterations |-----------------------------------------------------------------------------| 2 39967 1.4730991E+03 5.55E+00 0.38 0.75 29.04 2.89 2 |-----------------------------------------------------------------------------| 3 49990 1.4702467E+03 3.61E+00 0.25 0.55* 26.67 4 49990 1.4747081E+03 3.56E+00 0.24 0.54* 26.41 5 49990 1.4777945E+03 3.51E+00 0.24 0.53* 26.06 |-----------------------------------------------------------------------------| 5 149970 1.4743182E+03 2.05E+00 0.14 0.54 26.06 1.13 3 |=============================================================================| | Time estimate for generating 10000 events: 0d:00h:00m:07s | Starting integration for process 'myprocess' part 2 | Integrate: iterations = 6:20000:"gw", 3:50000 | Integrator: 16 chains, 19 channels, 8 dimensions | Integrator: Using VAMP channel equivalences | Integrator: 20000 initial calls, 20 bins, stratified = T | Integrator: VAMP |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 19988 1.4660652E+03 2.33E+01 1.59 2.25* 9.65 2 19979 1.4658522E+03 5.80E+00 0.40 0.56* 26.96 | VAMP: accuracy goal reached; skipping iterations |-----------------------------------------------------------------------------| 2 39967 1.4658646E+03 5.63E+00 0.38 0.77 26.96 0.00 2 |-----------------------------------------------------------------------------| 3 49994 1.4699485E+03 3.65E+00 0.25 0.56* 25.88 4 49994 1.4669227E+03 3.63E+00 0.25 0.55* 25.47 5 49994 1.4677451E+03 3.59E+00 0.24 0.55* 24.99 |-----------------------------------------------------------------------------| 5 149982 1.4681957E+03 2.09E+00 0.14 0.55 24.99 0.18 3 |=============================================================================| | Time estimate for generating 10000 events: 0d:00h:00m:07s | Integrate: sum of all components |=============================================================================| | It Calls Integral[fb] Error[fb] Err[%] Acc Eff[%] Chi2 N[It] | |=============================================================================| 1 0 2.9425139E+03 2.93E+00 0.10 0.00* 25.51 |=============================================================================| [user variable]$filename = "T_bbar_chargedVBF_MTP2.500TeV_s6TeV"

 Juergen Reuter (j.r.reuter) said on 2020-09-09: #6

This just looks like pair-production of the VLQ which then decays to b, mu+, nu_mu etc. That would clearly peak if 2 * MTP ~ sqrts.
BTW, your first cut seems void, as Pt > 30 GeV [mu-, mu+] is the relative Pt of mu- with respect to mu+, not the Pt relative to the beam of either mu-- or mu+. The latter would be written Pt > 30 GeV [mu-:mu+] as a container. As there is never a mu- and a mu+ in the final state, this cut doesn't do anything.

 David Wang (david-mhw) said on 2020-09-17: #7

Thanks Juergen Reuter, that solved my question.


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# 11 Mr. R's Secret

Submitted by George Hunt-Wil... on Sat, 02/22/2014 - 13:52

## 11 Mr. R's Secret

This chapter is dedicated to the memory of the young radioman, who in 1952 made contact with intelligences from outer space via radiotelegraphy and radiotelephony.

Not wishing to disclose his identity, I called him simply: Mr. R (for radio'). In the years ahead of us, when history speaks of the turbulent Twentieth Century on Earth, and when all war, disease, greed and selfishness have become nebulous things of ancient legend, Mr. R will be remembered as a pioneer of the days when Earthman learned that he was not alone in the Universe.

Mr. R was the first radio operator to become a channel between the small planet we call home and the vastness of interstellar space. He became a bridge for men of goodwill throughout the Father's limitless realm.

Now his key is silent... for he has graduated to another dimension of time and space. He is now free of the restrictions of Earth, free to communicate forever with worlds without number.

We say, Good-bye, for just a little while, Mr. R... Light ... Love... Peace... as you travel the Great Path up to the stars!

After the highly successful radio experiments of 1952-53, our group decided to write a book on our experiences. We realized that the information we had received was of the greatest importance to all mankind. We asked Mr. R if he cared to write a book about his radio contacts with Outer Space, but he said that he was not interested in making his experiments known publicly, but he did not object if others wrote about them. At that time we had him sign an affidavit, subscribed and sworn to before Genevieve D. Scott, Notary Public, Winslow, Arizona, on the 7th day of March 1953. While he was willing to sign the affidavit, he also told us that he didn't want his name or call letters (since he was a licensed ham radio operator) used in any way whatsoever. This was a great disappointment to us, because we felt that without such information any book on the results of the experiments would fall far short of that which we desired for it. The book would lack the authority that we wanted and needed. However, he was a good friend and we told him we would never disclose his name. We are doing so now in this edition since Mr. R passed away on 23 April 1955, with what was said to be a heart condition and other contributing factors.'

The Frontispiece of this book contains the original affidavit revealing the names of Mr. R, licensed commercial and amateur radio operator, and his wife, Mrs. R, plus the name of the Arizona city in which they made their home. Mr. R was Lyman H. Streeter, radio operator for the Santa Fe Railroad. His amateur call letters were : W70JQ. He lived at 423 East Maple Street in Winslow, Arizona.

Lyman had good reasons for not wanting his identity known. First of all, he had never used his call letters during his communication with the space intelligences and he never logged any of his receptions. If these facts were made known, he knew he would lose his license, and this was the very last thing he wanted, for to him radio was his very life. Also, he knew that even if he had complied fully with the regulations of the Federal Communications Commission, his license would still be taken away, for no operator is permitted to make contact with an unlicensed operator and the UFOs aren't licensed with the FCC!

Lyman Streeter's first contact with intelligence from outer space took place the evening of 22 August 1952. He was a good radio operator holding both a commercial and a ham or amateur license. Lyman was very skeptical of the existence of flying saucers let alone the possibility of communicating with such objects by radio. However, he was willing to attempt contact.

Friday evening 22 August 1952, Lyman saw what he thought was a very small meteor display over Winslow. Then he observed what appeared to be a very bright light traveling at a high altitude in the sky directly above him. He turned on his receiver in his ham shack to 400 kc., and immediately, many strange signals were heard but not identified. Later on, the same evening, the Streeters and other witnesses heard strange, clear code signals coming to them as they sat in the main house. Lyman had his ham shack on the back of their property, and he had no transmitting or receiving equipment whatsoever in the main house.

At first, everyone thought the signals were coming from the radio shack in the back yard, but when they went to check, there was absolutely nothing to be heard there... in fact, the equipment wasn't even turned on. After they came back to the main house the mysterious code was heard again. It seemed to be coming from the very air itself. Since that memorable evening in 1952, many others have reported experiencing exactly the same thing.

About 2:00 a.m., 23 August 1952, code signals were again received. Lyman said it sounded as though two people were talking back and forth to each other, using code... but a code unfamiliar to him. It was definitely not standard International Morse Code. The code was coming over his receiver in loud, clear tones. Suddenly, he wrote down a word or two on his note pad; ZO and AFFA.

Later we learned that a superior of Lyman Streeter's in the radio work of the Santa Fe Railroad, a man high up in radio circles, had told him that he also had received strange signals at various times during his radio experiments and that he definitely believed such signals to be from space intelligences. However, he had been evasive when Lyman asked for further information and had seemed to want to discourage him in his own experiments. Why?

One day, this superior called Lyman and told him that Lowell Observatory on Mars Hill in Flagstaff, Arizona had observed UFOs on Friday 22 August 1952. Exactly the same day that Streeter received his first radio signals from space intelligences! This superior also said that a staff member at Lowell Observatory had reported to him earlier that on 22 August 1952, they would focus their large telescope for terrestrial observation over Winslow. But the man had not said what they were going to look for or why. Also, no mention had been made of the source of the information that caused this observation to be made in the first place.

Later, Lyman Streeter had to go to Mars Hill overlooking Flagstaff, where he was making a survey for the Santa Fe Railroad in connection with new radio equipment being installed on the top of Mt. Elden. In an area away from the big observatory, he saw a strange small building surrounded by a high wire fence. Inside the enclosed area he saw two gigantic dogs, obviously watchdogs. Navy men were moving various types of electronic equipment in and out of the little building, which looked new and recently painted. He wondered what such equipment was doing on Mars Hill. He was informed that if the new equipment belonging to the railroad being installed on Mt. Elden interfered with the government work on Mars Hill, then it would have to be placed elsewhere.

Several years later we learned that the government had placed electronic equipment at Lowell Observatory to assist in the tracking of two artificial satellites or space stations that were known to be circling the Earth at four hundred and at six hundred miles out in space. In August 1954, the magazine Aviation Week reported that the two objects were meteors, and insisted that Dr. Lincoln La Paz of the University of New Mexico had helped in the identification of them as natural rather than man-made objects. Dr. La Paz attacked the magazine's reference to him, but acknowledged the search for nearby satellites. He also said: The report is false in every particular insofar as reference to me is concerned.' Dr. La Paz and Dr. Clyde Tombaugh, discoverer of the planet Pluto, both worked on the Government project at Lowell Observatory, and both of these men are fully aware that the objects tracked by electronic equipment are not natural, and because of their actions in the heavens could only be artificial... constructed by other intelligent beings. When Sir Edward Appleton, famous British radio physicist, said 'These two unknown objects are discoveries of great astronomical Interest'... he didn't know the half of it. Or did he? It is very likely that Sir Edward knew what La Paz and Tombaugh knew, and the same thing that the United States Government knew when it had the Navy set up an experimental station on Mars Hill. Lyman Streeter stumbled into that secret project in 1952 when he was at Lowell Observatory. Other messages were received, over a period of several months, by Lyman Streeter in his ham shack at the back of the lot at 423 East Maple Street. He had been very skeptical at first. However, after he saw discs in the sky where his radio messages told him to look, discs over his own radio antenna, and after messages were received telling about things which no one but he could have known, and finally, messages coming over the receiver that were answers to questions that had never been transmitted to the intelligences in the usual manner, Streeter's attitude changed. They had either picked up verbal statements made in the ham shack by telepathy and/or by some kind of recording discs. Soon, our Mr. R had his proof. I remember his facial expression on many occasions; he appeared just too bewildered to even think.

Lyman usually transmitted on 40 meters and received on 405 kc. On one occasion he received via radiotelephone (voice) on 92 meters. However, we are certain that frequency has nothing whatsoever to do with it. Space intelligences have stated many times that they can make anything act as a receiver from radio equipment to the human brain.

At first, Streeter told us that he never had had any interest in UFOs before we contacted him to attempt communication with them. However, upon questioning, he later admitted that he had become very much interested in the possibility of Interstellar Communication in 1950 after he read an article on this subject in QST, one of the amateur ham radio magazines. He said that he had attempted some sort of contact at that time on a high frequency and a short wave length. However, the experiment was a failure and he gave it up.

There was always something strange overshadowing the life of Mr. R that I find impossible to describe in words. The only clue I might have would be a radio message from the UFOs we received during the evening of 28 September 1952. At 11:20 p.m. the following message came through the receiver:

Radioman has deep secret in his mind. We will not reveal. We are alarmed.'

Streeter turned to the rest of us in the ham shack and said, If they (the UFOs) had known about this before, they would never have picked me for your radioman.'

Immediately, code came through the receiver again : Be of peace!'

We waited for several minutes without talking until more code started.

Happy, happy! You, Radioman Kanet are installed in the records. Good! Attention! Surprised, my brother?'

What was meant by deep secret'? Evidently Lyman and his wife knew for they looked at each other in a strange way and she came over to sit beside him and hold his hand. Lyman seemed very concerned over this message. None of us was impolite enough to ask the Streeters what it all meant, so we said nothing. Anyway, the space intelligences didn't stay alarmed for long for they spoke of Streeter being installed in the records' and called him by another name: Radioman Kanet! This name was formerly left out of the message because we felt it was something that belonged to Streeter alone and could be of no interest to the public. However, it is well to mention it now for it might be a clue for which we are looking.

I remember how, one evening, Streeter told us, rather reluctantly, it seemed, that after attempting contact in 1950 he had appeared one day at work acting in a very strange manner. He went about his assigned radio tasks in the normal way, but his fellow workers noticed he wouldn't answer them when they spoke to him and behaved as if he were in a trance of some kind. His wife was called, and he was taken home. For eight days he was in this unusual zombie' condition. He said nothing to anyone during that period. Later, when he regained a state of normalcy, he admitted he couldn't remember a thing that had transpired during those eight days of amnesia.

After the contacts started on 22 August 1952, Streeter suddenly remembered what had happened several years previously (during the memory lapse). He told us that he apparently had left his earthly body (that would account for the zombie condition... the physical body had gone about its usual tasks at work under the direction of the animal mind, while the entity had been elsewhere) and awoke in a beautiful, large hall where many people were gathering. He was called before a tribunal and noticed that he was dressed in fine garments. He was called by a different name (Kanet?)--and told that he must work rapidly to complete his task upon the Earth planet. All he could remember from this eight day journey was the fact that he must work quickly.

His wife said that before the period of amnesia Lyman was just an ordinary radioman, but after his recovery he spent long hours studying electronics and would work for hours on end until he became a top-grade operator.

Besides working at the Santa Fe radio shop, he had his amateur equipment in his ham shack back of his home on Maple Street, and at the other end of this shack his workshop, where he repaired most of the radios, TV sets, recorders, etc., of the neighborhood, including the car radios of the local Police Department. (I remember that once the police interrupted one of our communications with the UFOs by driving into the yard one night and asking Lyman to fix their radio. He obliged, but kept trying to hear the code coming in, listen to the idle talk of the officers and fix the radio, all at the same time.)

Evidently, Lyman H. Streeter was a Wanderer'. His name had been Kanet, and he was born on Earth to assist the programme of the Space Confederation. His so-called amnesia experience must have been his awakening period. [See page 206 in the book, Other Tongues-Other Flesh (Neville Spearman) --Editor]

On 21 October 1952, something occurred that was to change the life of Mr. R. At 8:10 p.m. a small private plane crashed and burned at the airport at Winslow. This plane was on a mercy flight to a Phoenix hospital with a fourteen-months-old polio victim. All four passengers were instantly killed. One of the workers at the Winslow Timber Company (where Mrs. Streeter also was employed) was working late, and saw the plane take off and minutes later burst into flames. This man told the Civil Aeronautics Administration investigators that immediately after the crash, and before the ambulance and fire truck had arrived, an orange streak sped across the sky and apparently landed by the stricken plane! [UFOs appearing at the moment of birth or death are mentioned in the book : Road In The Sky, by George Hunt Williamson (Neville Spearman) -Editor.]

Several days later, a man appeared at the Streeter home and introduced himself as 'Mr. Clark.' He asked Mrs. Streeter for Lyman and was told that he was outside in the radio shack. The man went out and introduced himself to Lyman, saying that although he was not on an official visit, he was with the CAA (Civil Aeronautics Administration) and had just completed his investigations of the crash of 21 October. He showed Lyman his credentials. Then, on a friendlier note, he said:

'I'm a ham operator myself.'

Then he gave his call letters. (Streeter later checked up and found that there was a Clark with the call letters the man had given, but his address was not the same as that which had been given him.)

The visitor then sat down, and said bluntly:

'Streeter, what do you think of the flying, saucers?'

Lyman sensed something strange underneath all the apparent friendliness, but he answered honestly:

I have my own personal opinion, if that's what you mean. I think they come from Outer Space.'

And then, very quickly, the man looked Lyman in the eyes and said:

Streeter realized that this man, Clark, or whoever he was, knew whether he had had radio communication or not, so he said simply:

I have conducted certain radio experiments under established scientific procedures. Yes, I have had contact with the flying saucers!'

Whereupon Clark asked, Would you mind showing me what you received from them? It's no secret is it?'

Streeter reached for his notes in a drawer, as he answered. No, it's no secret. I'll be glad to read you what I have, but I must tell you that because the messages were sent so fast, I missed many of the words, so the messages are incomplete.'

Lyman started to read one message after another. Soon he noticed that whenever he came to a blank space, Clark would fill it in! Finally, Streeter said:

Look here, you must know more about the messages I received than I do!'

That's right,' Clark answered. We monitored everything you transmitted and received.'

Lyman was startled, to say the least! He said: What do you mean by we, the Government?'

Clark hesitated, put his cigarette out, looked intently at Lyman for another moment, and then said:

Of course, who else?'

Streeter could hardly believe his ears! Confirmation of his experiments at last! But any ideas along this line were soon to be changed rather drastically. Lyman quickly answered:

You mean to tell me, Mr. Clark, you admit that flying saucers really exist, are coming from Outer Space, and that I have personally had radio communication with these objects?'

Clarke didn't hesitate in his answers:

Of course I admit it, but you won't be able to prove it to anyone. You see, no one knows I've been here but you and your wife, and besides, I would never admit we talked about such things. I have been in Winslow strictly for the CAA investigation of the recent plane crash. Remember?'

Weakly, Lyman said :

Just what does your visit to me mean?'

This is the story,' said Clark. About the same time you were receiving your coded messages from the extra-terrestrials, fifteen other ham operators throughout the United States received the same kind of information. We have contacted them all and every one of them is willing to co-operate with his Government. Are you?'

Lyman, still unable to believe what was happening answered:

What do you mean by co-operation, Mr. Clark?'

Then, the CAA man pulled his ace-in-the-hole.

Look, Streeter, we have you dead to rights. You never used your call letters, you never logged your information, and even if you had done all of that, we still can take away your license because you were in communication with unlicensed operators! The FCC (Federal Communications Commission) frowns on such activity, you know! Now, you don't want to lose your radio privileges--after all, you are commercially licensed besides holding your amateur ticket, you're making, a good living from radio. If your license is taken away, what would you do?'

This was the day Mr. R had been dreading. Many times he had told us that there was the possibility of getting in trouble with the FCC because of the unusual nature of the radio experiments. Clark then started waving the stars and stripes by saying:

Lyman, your Government is doing all it can to enlighten the people in connection with the coming of the flying saucers. But the time is not ripe yet. A vast educational programme has been planned, and gradually people will come to realize that all space is inhabited, but now the effect would be disastrous if people were to know the truth. Join the other fifteen operators and cooperate with us. We understand your friend Williamson is ready to have a book published called The Saucers Speak! and that it deals with the radio contacts. You know about this don't you?'

Yes,' said Lyman, but my name and my city are not mentioned. No one will be able to locate me from the account in the book. My friends have promised to leave me out of it.'

'That's all right,' said Clark, but the Government doesn't want the radio contact story to be released at this time. You'd better get in contact with Williamson and tell him that the book just can't come out at all, and be firm about it!'

What can I tell him? I've already given him permission to do the story as long as my identity is withheld. He will think it's strange for me to want to stop the book now that it's ready to be distributed!'

That is your problem, Streeter,' said Clark. But the story just can't come out now. Clark went patriotic again, and added: Your Government will release such information in due time, but more evidence must be accumulated first. Tell your friend that your job is in danger, and your license. Tell him anything, but stop that book! We can't contact him, for we have his source of information--namely you, as radio operator. Besides, if we talk to him it will only give him proof that the entire affair was authentic. After all, he may still have a few doubts in the back of his mind. Who knows?'

Streeter was almost speechless, as he said:

What is expected of me?'

You must co-operate to the limit with us, but I must warn you that we will not be able to back you up or support your story in any way whatsoever if you ever tell your contact story or mention my being here today, or our conversation!'

In other words,' said Lyman, it will be one-sided co-operation?'

In the interests of national security,' said Clark. 'Yes, it will be strictly one-sided!'

Mr. R had no choice. Although his visitor didn't state it in so many words, it was plain to see that if he didn't co-operate, the FCC regulations would be brought into the picture and he would lose his radio privileges, and, of course, his job with the Santa Fe Railroad! His only answer could have been that which it was:

What is the first step?'

We want you to increase your power here for transmitting,' said Clark. How soon can you do this and be ready for some experiments?'

I can't afford to change my equipment now,' said Lyman, it would be too expensive!'

We will send you the necessary parts, Streeter,' Clark said as he rose to go. 'But not a word to anyone, and you'd better not give any more information to Williamson.'

Clark had been sent, not as an official from the FCC, for the Government felt that would be too severe a shock for Mr. R and that he might not give out any information at all. So, they had sent a CAA investigator who was also a ham, knowing full well that there is a comradeship between all radio amateurs, and this was to serve as the tie that binds and so establish communication between Clark and Streeter.

Mr. R received his needed equipment within a few weeks and started to experiment at once, but never again did he receive anything from the space intelligences after his decision to co-operate with the powers that be! Later, he was visited by two officials of the FCC itself who told him to increase his efforts at contact. This he did, thereby affecting many of the neighborhood electronic gadgets, so he stopped. He was attempting to beam all messages straight up in a powerful way that would not be needed in ordinary ham procedures in contacting other amateurs throughout the world. Why he was told to increase his power, I do not know, because while science says that high frequency and high wattage is necessary for any kind of interstellar communication, we know now that power is not important, but that the type wave used is important.

Streeter called me in Prescott and asked me to stop the book. Since he had already agreed to my writing the book, I thought he was only worried over the risk of his possible identification, and since I felt quite sure that Lyman was completely safe in my presentation, the book could still come out without any harm to him. Later I was to learn that he was concerned over something else.

I had realized for many weeks that Lyman wasn't acting normally. He avoided us, and while he didn't really seem unfriendly, at the same time he was quite cool, as though he knew something that he couldn't tell us. Later we learned that this was true, due to the fact that he had been visited by Clark... who gave his full name as William (Bill) Clark.

We were quite busy with other UFO happenings, however, about this time. Mr. and Mrs. Bailey and my wife and I were on the desert near Desert Centre, California, on zo November 1952, when George Adamski made his memorable contact with a Venusian. Many people have thought that because of our radio contacts we must have arranged that meeting on the desert, but that is not so, for due to Streeter's new commitments, we were not in radio contact at that time. No arrangements were made beforehand by anyone; we simply went out on the desert and what transpired was completely unexpected by all present.

Many other things of interest happened during December 1952, January and February 1953 . UFO sightings increased and nature went on a rampage. On 21 December 1952, Lyman Streeter and five other residents of Arizona, observed a large, cigar-shaped object over Winslow. They watched it from 5:00 p.m. until dark. Two smaller UFOs were seen to enter the larger craft and a few minutes later one left the mother-ship. All of this was observed through binoculars.

On 3 February 1953, my wife and I were coming home from downtown Prescott, Arizona, when we observed two objects, brighter than the planet Venus, which was in a different position in the sky, come within a few feet of the ground. These craft were close enough that the general bell-like outline and the light on top could be easily observed. There was absolutely no sound. I called the US Weather Bureau and was told that there were no planes or balloons in my area. Later the same night I observed another object with an amber light pass very low over my house.

On 15 February 1953 we decided to visit Streeter in Winslow, although at that time we still did not know about his talk with Clark and we didn't understand his coolness towards us. There were several things we felt needed clearing up in regard to the book, which would shortly be released. We hoped the UFOs would have some ideas or advice for us. We told Lyman that we didn't know whether all the facts should be given to the public or not. He had little to say, for he had already told us that he didn't want the report to come out because of the danger to him and his position. He had the radio receiver turned on for he always kept it that way when he was in the ham shack. He transmitted nothing in connection with our request. In fact, he transmitted nothing at all! Suddenly a radio code signal just seemed to slide in on 405 kc. At first, Streeter couldn't make any sense out of the dot and dash system used. Finally, one word stood out: Centuras. This was followed by a very understandable message:

'OK. This time it's for you to decide. We cannot.'

The message ended at 12:05 a.m. 16 February 1953. The UFO intelligences would not decide for us. We had to do that! So, in this book I haven't given the account exactly as it happened. We thought perhaps it would have been better to omit the telepathic section of our experiences, for after all it is not untruthful to tell only part of a story! After this communication we felt it best to tell the telepathic experiences as well as the subsequent radio contacts.

Later, Streeter called me again and told me to stop the presses and prevent the publisher from coming out with the book. I thought this a strange request, for already I had assured him that his identity was hidden. But he said:

I told you I wanted to be protected, and you didn't live up to your promise because now I hear the book is to be released!'

I told him that I had never agreed to stopping the book, but merely to protect him, and added:

But, Lyman, it will be impossible for anyone to know who you are or where you live. You are completely safe.'

When he still insisted that I stop the book, by force if need be, I became very suspicious, and started an investigation which led to my discovery of the Clark episode. Previously, he had agreed to my writing the book provided that he be left out of it, but suddenly when the book was practically in the hands of the public, he did a quick turn-about. Something was wrong somewhere! I wanted to know what, and I found out. First, one visitor in the form of Clark and then two more men from the FCC. There is little doubt in my mind that Lyman H. Streeter had been visited by the notorious three men in black!'

A. David Middelton, who has a wide and varied background in electronics and the communication field decided to investigate Mr. R. Somehow he learned his identity, and personally interviewed Streeter. Middelton is a Senior Member of the Institute of Radio Engineers and was formerly a director of the American Radio Relay League. He holds an Extra Class Amateur License and maintains W 5 CA. Middelton was a project engineer at Fort Monmouth during World War II and also a civilian radar field engineer attached to the U S Navy. Moving to ARRL headquarters, he was Assistant Editor of QST, and later corresponding Editor of CQ magazine. Currently he is working for the United States Government in electronics.

In January 1955, Middelton (W5CA) discussed the radio contacts with Lyman Streeter in person in Winslow. Later, I provided additional details. After full consideration of all data,' said Middelton, it is my opinion that these contacts were made by Mr. R as described.'

About the same time, early in 1955, Middelton received reports of radio contact established between UFOs and a VE3 radio operator in Canada. He stated at the time that he 'believed the contacts were authentic although technical details were lacking.'

The question Middelton asked, and rightly so, was:

Why have bonafide amateur UFO contact data been conspicuous by their absence in ham discussions on and off the air? Could it be that the amateurs involved have been afraid to report such contacts (Q S O s)? Afraid of what, or whom? Or, maybe there just have not been UFO amateur contacts other than the W7 and VE3 experiences!'

Because of Clark's statement about the fifteen other amateur radio operators, we know there were more than W7 and VE3. That leaves us with: Afraid of what, or whom?'

After examining the facts in regard to Lyman Streeter's contacts, Middelton wrote the following letter:

 Federal Communications Commission Amateur Division, Washington, DC 9 April 1955 Dear O M s, This is a formal and serious inquiry. In view of information reaching me from seriously interested sources, the situation might arise wherein an amateur operator, duly licensed and operating within the amateur bands of this country, might be called by a station purporting to be from Outer Space or from Unidentified Flying Objects. Such stations also operating in our bands. The calls signs used by these U F Os are not any more unusual than many of the strange and weird ones being assigned by some foreign countries, I understand. In view of this fact, it would be difficult to ascertain if the station was, in fact, just another DX country, or one not on this planet, as far as the call sign is concerned. Will you please inform me as to the feelings and desires of the Amateur Division of the F C C on this matter, if any. Also, will you please inform me of regulations covering this matter? I realize that this is a strange inquiry but it is definitely in order, and made in a sincere effort to ascertain the FCC views on this matter. Also, what about amateurs transmitting within our authorized bands but receiving on frequencies outside the bands? This, too, is of concern. This involves Q S Os wherein the amateur did not make the call up. I would appreciate having your reply to this matter, with any degree of secrecy you wish to place on it. Respectfully yours, (Signed) A. DAVID MIDDELTON, W5C A

Middelton wanted to ascertain FCC views on the matter, and that is exactly what the FCC didn't want him to do. His high reputation in this field, and the fact that his technical articles have been widely acclaimed in radio circles and reprinted throughout the world, didn't do very much to convince the FCC they could answer him with any degree of secrecy' at all! He received a letter from them as follows:

 Federal Communications Commission Washington 25, DC 25 April 1955 File: 7400 A. David Middelton Tijeras, New Mexico Dear Sir, This is in reference to your letter of 9 April 1955, requesting comments concerning regulations which would apply to possible communications with unidentified objects. In regard to amateur radio stations, Section 12.101, of Part 12 of the Commission's Rules Governing Amateur Radio Service specifies the points of communications permitted for amateur stations licensed by the Commission. Within the limitations of Section 12.101, amateurs may communicate with stations which transmit on frequencies outside the amateur frequency bands. Very truly yours, (Signed) MARY JANE MORRISSecretary

Between the time Middelton wrote to the FCC on 9 April, and the time he received an answer from Washington on 25 April. Lyman H. Streeter (Mr. R) had passed away on 23 April.

Needless to say, the extreme mental anxiety and pressure placed upon Lyman Streeter didn't exactly place him in a state of exuberant health. He was a very young man to die, but I feel certain that 'Kanet,' the radioman, had done his job and had completed the mission he came here to do. Who is to say? Surely not the FCC!

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## Exercise 4: Visualising data using ggplot

Alternative (optional) solutions to Exercise 4 for those who use (or are interested in using) the ggplot approach to plotting data. See Chapter 5 of the Introduction to R book for more information about how to use ggplot.

# make ggplot2, gridExtra and GGally packages available
# Note: you may need to install these packages first
# Use : install.packages('ggplot2', dep = TRUE)
# Use : install.packages('gridExtra', dep = TRUE)
# Use : install.packages('GGally', dep = TRUE)

library(ggplot2)
library(gridExtra)
library(GGally)

1. As in previous exercises, either create a new R script or continue with your previous R script in your RStudio Project. Again, make sure you include any metadata you feel is appropriate (title, description of task, date of creation etc) and don’t forget to comment out your metadata with a # at the beginning of the line.

1. Download the data file ‘squid1.xlsx’ from the Data link and save it to the data directory you created during exercise 1. Open this file in Microsoft Excel (or even better use an open source equivalent - LibreOffice is a good free alternative) and save it as a tab delimited file type. Name the file ‘squid1.txt’ and save it to the data directory.

1. These data were originally collected as part of a study published in Aquatic Living Resources1 in 2005. The aim of the study was to investigate the seasonal patterns of investment in somatic and reproductive tissues in the long finned squid Loligo forbesi caught in Scottish waters. Squid were caught monthly from December 1989 - July 1991 (month and year variables). After capture, each squid was given a unique specimen code, weighed (weight) and the sex determined (sex - only female squid are included here). The size of individuals was also measured as the dorsal mantle length (DML) and the mantle weight measured without internal organs (eviscerate.weight). The gonads were weighed (ovary.weight) along with the accessory reproductive organ (the nidamental gland, nid.weight, nid.length). Each individual was also assigned a categorical measure of maturity (maturity.stage, ranging from 1 to 5 with 1 = immature, 5 = mature). The digestive gland weight (dig.weight) was also recorded to assess nutritional status of the individual. If you’re not familiar with squid morphology and are interested in finding out more see here.

1. Import the ‘squid1.txt’ file into R using the read.table() function and assign it to a variable named squid. Use the str() function to display the structure of the dataset and the summary() function to summarise the dataset. How many observations are in this dataset? How many variables? The year, month and maturity.stage variables were coded as integers in the original dataset. Here we would like to code them as factors. Create a new variable for each of these variables in the squid dataframe and recode them as factors. Use the str() function again to check the coding of these new variables.

squid <- read.table('workshop/data/squid1.txt', header =TRUE,
stringsAsFactors = TRUE)

str(squid)
# 'data.frame': 519 obs. of  13 variables:
#  $sample.no : int 105128901 105128901 105128901 105128901 ... #$ specimen         : int  1002 1003 1005 1007 1008 1009 1011 1013 ...
#  $year : int 1989 1989 1989 1989 1989 1989 1989 1989 ... #$ month            : int  12 12 12 12 12 12 12 12 12 12 ...
#  $weight : num 152 106 138 141 126 ... #$ sex              : int  2 2 2 2 2 2 2 2 2 2 ...
#  $maturity.stage : int 3 1 2 2 3 1 2 3 3 4 ... #$ DML              : int  174 153 169 175 169 116 135 192 170 205 ...
#  $eviscerate.weight: num 87.5 62.6 79.4 83.1 72.2 ... #$ dig.weight       : num  4.648 3.138 0.307 4.123 3.605 ...
#  $nid.length : num 39.4 24.1 39 41.4 39.8 20 14 55 44 53 ... #$ nid.weight       : num  2.46 0.319 1.169 1.631 2.03 ...
#  $ovary.weight : num 1.68 0.103 0.289 0.252 0.86 ... summary(squid) # convert variables to factors squid$Fmaturity <- factor(squid$maturity.stage) squid$Fmonth <- factor(squid$month) squid$Fyear <- factor(squid$year) str(squid) # 'data.frame': 519 obs. of 16 variables: #$ sample.no        : int  105128901 105128901 105128901 ...
#  $specimen : int 1002 1003 1005 1007 1008 1009 ... #$ year             : int  1989 1989 1989 1989 1989 1989 ...
#  $month : int 12 12 12 12 12 12 12 12 12 12 ... #$ weight           : num  152 106 138 141 126 ...
#  $sex : int 2 2 2 2 2 2 2 2 2 2 ... #$ maturity.stage   : int  3 1 2 2 3 1 2 3 3 4 ...
#  $DML : int 174 153 169 175 169 116 135 ... #$ eviscerate.weight: num  87.5 62.6 79.4 83.1 72.2 ...
#  $dig.weight : num 4.648 3.138 0.307 4.123 3.605 ... #$ nid.length       : num  39.4 24.1 39 41.4 39.8 20 14 ...
#  $nid.weight : num 2.46 0.319 1.169 1.631 2.03 ... #$ ovary.weight     : num  1.68 0.103 0.289 0.252 0.86 ...
#  $Fmaturity : Factor w/ 5 levels "1","2","3","4" "5"... #$ Fmonth           : Factor w/ 12 levels "1","2","3","4" ...
#  $Fyear : Factor w/ 3 levels "1989","1990",..1 ... 1. How many observations are there per month and year combination (hint: remember the table() or xtabs() functions?)? Don’t forget to use the factor recoded versions of these variables. Do you have data for each month in each year? Which years have the most observations? (optional) Use a combination of the xtabs() and ftable() functions to create a flattened table of the number of observations for each year, maturity stage and month (aka a contingency table). table(squid$Fmonth, squid$Fyear) # 1989 1990 1991 # 1 0 3 37 # 2 0 0 30 # 3 0 40 29 # 4 0 10 33 # 5 0 1 30 # 6 0 0 14 # 7 0 42 1 # 8 0 29 0 # 9 0 82 0 # 10 0 19 0 # 11 0 76 0 # 12 12 31 0 ftable(xtabs(~ Fyear + Fmaturity + Fmonth, data = squid)) # Fmonth 1 2 3 4 5 6 7 8 9 10 11 12 # Fyear Fmaturity # 1989 1 0 0 0 0 0 0 0 0 0 0 0 2 # 2 0 0 0 0 0 0 0 0 0 0 0 3 # 3 0 0 0 0 0 0 0 0 0 0 0 5 # 4 0 0 0 0 0 0 0 0 0 0 0 2 # 5 0 0 0 0 0 0 0 0 0 0 0 0 # 1990 1 0 0 0 0 0 0 8 0 1 1 1 2 # 2 0 0 0 0 0 0 22 21 76 17 31 4 # 3 0 0 0 0 0 0 0 5 5 1 31 6 # 4 2 0 15 7 0 0 4 3 0 0 10 13 # 5 1 0 25 3 1 0 8 0 0 0 3 6 # 1991 1 0 0 0 2 0 4 0 0 0 0 0 0 # 2 1 1 0 1 0 6 0 0 0 0 0 0 # 3 2 0 0 1 1 0 0 0 0 0 0 0 # 4 16 8 6 13 6 1 1 0 0 0 0 0 # 5 18 21 23 16 23 3 0 0 0 0 0 0 1. The humble cleveland dotplot is a great way of identifying if you have potential outliers in continuous variables. Create dotplots (using the dotchart() function) for the following variables; DML, weight, nid.length and ovary.weight. Do these variables contain any unusually large or small observations? Don’t forget, if you prefer to create a single figure with all 4 plots you can always split your plotting device into 2 rows and 2 columns. Use the pdf() function to save a pdf version of your plot(s) in your output directory you created in Exercise 1. p1 <- ggplot(data = squid) + geom_point(aes(x = DML, y = as.numeric(rownames(squid)))) + ylab("") p1 p2 <- ggplot(data = squid) + geom_point(aes(x = weight, y = as.numeric(rownames(squid)))) + ylab("") p2 p3 <- ggplot(data = squid) + geom_point(aes(x = nid.length, y = as.numeric(rownames(squid)))) + ylab("") p3 p4 <- ggplot(data = squid) + geom_point(aes(x = ovary.weight, y = as.numeric(rownames(squid)))) + ylab("") p4 # arrange all 4 plots in a single # graphics device pall <- grid.arrange(p1, p2, p3, p4, nrow = 2) pall # save the plot ggsave('workshop/figures/gg_dotplot.pdf', plot = pall, device = 'pdf') 1. It looks like the variable nid.length contains an unusually large value. Actually, this value is biologically implausible and clearly an error. The researchers were asked to go back and check their field notebooks and sure enough they discover a typo. It looks like a tired researcher accidentally inserted a zero by mistake when transcribing these data. We can clearly see this value is over 400 so we can use the which() function to identify which observation this is which(squid$nid.length > 400). It looks like this is the 11th observation of the squid$nid.length variable. Use your skill with the square brackets [ ] to first confirm the this is the correct value (you should get 430.2) and then change this value to 43.2. Now redo the dotchart to visualise your correction. Caution: You can only do this because you have confirmed that this is an transcribing error. You should not remove or change values in your data just because you feel like it or they look ‘unusual’. This is scientific fraud! Also, the advantage of making this change in your R script rather than in Excel is that you now have a permanent record of the change you made and can state a clear reason for the change. which(squid$nid.length > 400)
# [1] 11

squid$nid.length[11] # [1] 430.2 squid$nid.length[11] <- 43.2
squid$nid.length[11] # [1] 43.2 ggplot(data = squid) + geom_point(aes(x = nid.length, y = as.numeric(rownames(squid)))) + ylab("") 1. When exploring your data it is often useful to visualise the distribution of continuous variables. Create histograms for the variables; DML, weight, eviscerate.weight and ovary.weight. Again, its up to you if you want to plot all 4 plots separately or in the same figure. Export your plot(s) as pdf file(s). One potential problem with histograms is that the distribution of data can look quite different depending on the number of ‘breaks’ used. The hist() function does it’s best to create appropriate ‘breaks’ for your plots (it uses the Sturges algorithm for those that want to know) but experiment with changing the number of breaks for the DML variable to see how the shape of the distribution changes (see the course manual for further details of how to change the breaks). ggplot(data = squid) + geom_histogram(aes(DML)) ggplot(data = squid) + geom_histogram(aes(weight)) ggplot(data = squid) + geom_histogram(aes(eviscerate.weight)) ggplot(data = squid) + geom_histogram(aes(ovary.weight)) # experimenting with different breaks bp1 <- ggplot(data = squid) + geom_histogram(aes(DML), bins = 20) bp1 bp2 <- ggplot(data = squid) + geom_histogram(aes(DML), bins = 10) bp2 bp3 <- ggplot(data = squid) + geom_histogram(aes(DML), bins = 5) bp3 bp4 <- ggplot(data = squid) + geom_histogram(aes(DML), bins = 2) bp4 bpall <- grid.arrange(bp1, bp2, bp3, bp4, nrow = 2) # save the plot to file ggsave('workshop/figures/gg_hist.pdf', plot = bpall, device = 'pdf') 1. Scatterplots are great for visualising relationships between two continuous variables. Plot the relationship between DML on the x axis and weight on the y axis. How would you describe this relationship? Is it linear? One approach to linearising relationships is to apply a transformation on one or both variables. Try transforming the weight variable with either a natural log (log()) or square root (sqrt()) transformation. I suggest you create new variables in the squid dataframe for your transformed variables and use these variables when creating your new plots (ask if you’re not sure how to do this). Which transformation best linearises this relationship? Again, save your plots as a pdf file (or try saving in your output directory as jpeg or png format using the jpeg() or png() functions if you feel the need for a change!). # clearly not linear # also note use of the classic theme ggplot(data = squid) + geom_point(aes(x = DML, y = weight)) + theme_classic() # natural log and sqrt tranform weight squid$weight.sqrt <- sqrt(squid$weight) squid$weight.log <- log(squid$weight) ggplot(data = squid) + geom_point(aes(x = DML, y = weight.log)) + theme_classic() ggplot(data = squid) + geom_point(aes(x = DML, y = weight.sqrt)) + theme_classic() # the square root transformation looks # most appropriate # save plot as jpeg ggsave('output/gg_xy_weight.jpeg', device = 'jpeg') # save plot as png ggsave('output/gg_xy_weight.png', device = 'png') 1. When visualising differences in a continuous variable between levels of a factor (categorical variable) then a boxplot is your friend (avoid using bar plots - Google ‘bar plots are evil’ for more info). Create a boxplot to visualise the differences in DML at each maturity stage (don’t forget to use the recoded version of this variable you created in Q4) . Include x and y axes labels in your plot. Make sure you understand the anatomy of a boxplot before moving on - please ask if you’re not sure. An alternative to the boxplot is the violin plot. A violin plot is a combination of a boxplot and a kernel density plot and is great at visualising the distribution of a variable. # note: Fmaturity is the recoded maturity.stage variable cerated in Q4 ggplot(data = squid) + geom_boxplot(aes(x = Fmaturity, y = DML)) + theme_classic() + xlab("Maturity stage") + ylab("Dorsal mantel length") # violin plot ggplot(data = squid) + geom_violin(aes(x = Fmaturity, y = DML)) 1. To visualise the relationship between two continuous variables but for different levels of a factor variable you can create a conditional scatterplot. Use the ggplot() function to plot the relationship between DML and square root transformed weight (you created this variable in Q8) for each level of maturity stage. Does the relationship between DML and weight look different for each maturity stage (suggesting an interaction)? # use facet_wrap to produce a # panel plot ggplot(data = squid) + geom_point(aes(x = DML, y = weight.sqrt)) + facet_wrap(~Fmaturity) 1. To explore the relationships between multiple continuous variables it’s hard to beat a pairs plot. Create a pairs plot for the variables; DML, weight, eviscerate.weight, ovary.weight, nid.length, and nid.weight. If it looks a little cramped in RStudio then click on the ‘zoom’ button in the plot viewer to see a larger version. If you want to produce a pairs plot using ggplot then you will need to install the package GGally using the usual code: install.packages("GGally") and use the ggpairs() function. Don’t forget to make the package available with the library() function. library(GGally) ggpairs(squid[, c(5, 8, 9, 11, 12, 13)]) 1. Almost every aspect of the figures you create in R is customisable. Learning how to get your plot looking just right is not only rewarding but also means that you will never have to include a plot in your paper/thesis that you’re not completely happy with. When you start learning how to use R it can sometimes seem to take a lot of work to customise your plots. Don’t worry, it gets easier with experience (most of the time anyway) and you will always have your code if you want to create a similar plot in the future. Use the plot() function to produce a scatterplot of DML on the x axis and ovary weight on the y axis (you might need to apply a transformation on the variable ovary.weight). Use a different colour to highlight points for each level of maturity stage. Produce a legend explaining the different colours and place it in a suitable position on the plot. Format the graph further to make it suitable for inclusion into your paper/thesis (i.e. add axes labels, change the axes scales etc). # square root transform ovary weight squid$ovary.weight.sqrt <- sqrt(squid\$ovary.weight)
ggplot(data = squid) + geom_point(aes(x = DML, y = ovary.weight.sqrt, colour = Fmaturity), alpha = 0.8,
size = 2) + scale_colour_manual(values = c("deepskyblue3", "darkolivegreen3", "coral3", "lemonchiffon3",
"darkorchid3"), labels = c("stage 1", "stage 2", "stage 3", "stage 4", "stage 5")) + theme_classic(base_size = 12) +
labs(colour = "", x = "DML (mm)", y = "square root ovary weight (g)")

# OR
ggplot(data = squid) + geom_point(aes(x = DML, y = ovary.weight.sqrt, colour = Fmaturity), alpha = 0.8,
size = 2) + theme_classic() + labs(colour = "", x = "DML (mm)", y = "square root ovary weight (g)")

browseURL("https://giphy.com/gifs/RyXVu4ZW454IM/fullscreen", browser = getOption("browser"), encodeIfNeeded = FALSE)

1 Smith JM et al (2005) Seasonal patterns of investment in reproductive and somatic tissues in the squid Loligo forbesi, Aquatic Living Resources. 18, 341–351.

End of Exercise 4


text:
Accelerating the pace of engineering and science

# modred

Model order reduction

## Syntax

rsys = modred(sys,elim)
rsys = modred(sys,elim,'method')

## Description

rsys = modred(sys,elim) reduces the order of a continuous or discrete state-space model sys by eliminating the states found in the vector elim. The full state vector X is partitioned as X = [X1;X2] where X1 is the reduced state vector and X2 is discarded.

elim can be a vector of indices or a logical vector commensurate with X where true values mark states to be discarded. This function is usually used in conjunction with balreal. Use balreal to first isolate states with negligible contribution to the I/O response. If sys has been balanced with balreal and the vector g of Hankel singular values has M small entries, you can use modred to eliminate the corresponding M states. For example:

[sys,g] = balreal(sys)  % Compute balanced realization
elim = (g<1e-8)         % Small entries of g are negligible states
rsys = modred(sys,elim) % Remove negligible states

rsys = modred(sys,elim,'method') also specifies the state elimination method. Choices for 'method' include

• 'MatchDC' (default): Enforce matching DC gains. The state-space matrices are recomputed as described in Algorithms.

• 'Truncate': Simply delete X2.

The 'Truncate' option tends to produces a better approximation in the frequency domain, but the DC gains are not guaranteed to match.

If the state-space model sys has been balanced with balreal and the grammians have m small diagonal entries, you can reduce the model order by eliminating the last m states with modred.

## Examples

expand all

### Order Reduction by Matched-DC-Gain and Direct-Deletion Methods

Consider the following continuous fourth-order model.

To reduce its order, first compute a balanced state-space realization with balreal.

h = tf([1 11 36 26],[1 14.6 74.96 153.7 99.65]);
[hb,g] = balreal(h);

Examine the gramians.

g'

ans =

0.1394    0.0095    0.0006    0.0000

The last three diagonal entries of the balanced gramians are relatively small. Eliminate these three least-contributing states with modred, using both matched-DC-gain and direct-deletion methods.

hmdc = modred(hb,2:4,'MatchDC');
hdel = modred(hb,2:4,'Truncate');

Both hmdc and hdel are first-order models. Compare their Bode responses against that of the original model.

bodeplot(h,'-',hmdc,'x',hdel,'*')

The reduced-order model hdel is clearly a better frequency-domain approximation of h. Now compare the step responses.

stepplot(h,'-',hmdc,'-.',hdel,'--')

While hdel accurately reflects the transient behavior, only hmdc gives the true steady-state response.

## Limitations

With the matched DC gain method, A22 must be invertible in continuous time, and IA22 must be invertible in discrete time.

expand all

### Algorithms

The algorithm for the matched DC gain method is as follows. For continuous-time models

$\begin{array}{l}\stackrel{˙}{x}=Ax+By\\ y=Cx+Du\end{array}$

the state vector is partitioned into x1, to be kept, and x2, to be eliminated.

$\begin{array}{l}\left[\begin{array}{c}{\stackrel{˙}{x}}_{1}\\ {\stackrel{˙}{x}}_{2}\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& {A}_{12}\\ {A}_{21}& {A}_{22}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]+\left[\begin{array}{c}{B}_{1}\\ {B}_{2}\end{array}\right]u\\ y=\left[\begin{array}{cc}{C}_{1}& {C}_{2}\end{array}\right]x+Du\end{array}$

Next, the derivative of x2 is set to zero and the resulting equation is solved for x1. The reduced-order model is given by

$\begin{array}{l}{\stackrel{˙}{x}}_{1}=\left[{A}_{11}-{A}_{12}{A}_{22}{A}_{21}\right]{x}_{1}+\left[{B}_{1}-{A}_{12}{A}_{22}{B}_{2}\right]u\\ y=\left[{C}_{1}-{C}_{2}{A}_{22}{A}_{21}\right]x+\left[D-{C}_{2}{A}_{22}{B}_{2}\right]u\end{array}$

The discrete-time case is treated similarly by setting

${x}_{2}\left[n+1\right]={x}_{2}\left[n\right]$


text:
# How to evaluate solid angle subtended by a segmented circle?

The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane along x & y axes. Thus four identical segments of the circular plane are left over.

Question: How to calculate the solid angle subtended by the remaining plane (consisting of four identical segments) at a point $P (0, 0, 4cm)$ lying on the z-axis (normally outwards to the plane of paper)?

Solid angle $(\omega)$ subtended by the circular plane, with a radius $r$ at any point lying on the geometrical axis at a distance $h$ from the center, is given as $$\omega=2\pi \left(1-\frac{h}{\sqrt{h^2+r^2}}\right)$$ And the solid angle $(\omega)$ subtended by any rectangle of size $l*b$ at any point lying on the perpendicular axis at a distance $d$ from the center is given by standard formula as $$\omega=4sin^{-1}\left(\frac{lb}{\sqrt{(l^2+4d^2)(b^2+4d^2)}}\right)$$

But how to apply these formula on the non radial segments of a circle? Any help is greatly appreciated.


text:
# Invariant functions on the dual Lie algebra

Let $G$ be a real Lie group and $\mathfrak{g}$ the corresponding Lie algebra. Let $\mathfrak{g}^*$ be the dual of the Lie algebra. Then we have the coadjoint action of $G$ on $\mathfrak{g}^*$.

Consider a $G$-invariant (w.r.t. the coadjoint action) embedded submanifold $U \subset \mathfrak{g}^*$, such that the maximal-dimensional orbits in $U$ are embedded submanifolds. Let $$k := \dim U, \qquad \ell := \max_{x \in U} \dim (G.x),$$ and fix a point $z \in U$ such that $\dim (G.z) = \ell$.

Can we find $G$-invariant polynomials $f_1, \dots, f_{k-\ell} \colon U \to \mathbb{R}$ such that the function $F :=(f_1, \dots, f_{k-\ell}) \colon U \to \mathbb{R}^{k-\ell}$ is a submersion at $z$?

Edit: By polynomials on $U$, I mean polynomials on $\mathfrak{g}^*$ restricted to $U$. And by $G$-invariant, that the restricted polynomials are $G$-invariant on $U$ and not necessarily on the whole of $\mathfrak{g}^*$.

• Having a $G$-invariant polynomial on $\mathfrak{g}^*$ and then restricting it to $U$, shouldn't we still get a $G$-invariant function? So my polynomials don't need to be $G$-invariant on all of $\mathfrak{g}^*$, only when restricted to $U$. Although I think it doesn't hurt, if they are $G$-invariant on all of $\mathfrak{g}^*$. Jun 8, 2016 at 12:14


text:
Followers 0

# limit of ((x(sqrt(x+2)))/sqrt(x+1))-x is 1/2? Why?

## 5 posts in this topic

I need this limit ((x(sqrt(x+2)))/sqrt(x+1))-x to calculate the asymptote of this function: ((x(sqrt(x+2)))/sqrt(x+1)).

which, according to the class notes: y=x+1/2

with a = 1, b = 1/2

However, the online math calculators say that currently no steps are supported to show for this kind of problem.

I calculated this limit as x*sqrt(1)-x = 1, but apparently the correct answer is 1/2.

What is my mistake?

r

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$\frac{x(\sqrt{x+2})}{\sqrt{x+1}}-x=\frac{x}{\sqrt{x+1}}(\sqrt{x+2}-\sqrt{x+1})= \frac{x}{\sqrt{x+1}(\sqrt{x+2}+\sqrt{x+1})}(x+2-(x+1))$

Limit = 1/2, since the denominator ->2x.

1

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You should use l'Hôpital's rule for a rigorous proof:
$\lim_{x\to\infty} {\frac{x\sqrt{x+2}}{\sqrt{x+1}}-x}=\lim_{x\to\infty} {\frac{x}{\sqrt{(x+1)(x+2)}+x+1}}$ (see the previous post).

Using l'Hôpital's rule, this equals $\lim_{x\to\infty} {\frac{1}{(\sqrt{(x+1)(x+2)})'+1}}=\lim_{x\to\infty} {\frac{1}{\frac{2x+3}{2\sqrt{(x+1)(x+2)}}+1}}$. Assuming the limit exists, this is equal to $\frac{1}{\lim_{x\to\infty} {\frac{2x+3}{2\sqrt{(x+1)(x+2)}}}+1}$.

You then evaluate $\lim_{x\to\infty} {\frac{2x+3}{2\sqrt{(x+1)(x+2)}}}$: Using l'Hôpital again, this is equal to $\lim_{x\to\infty} {\frac{2}{\frac{2(2x+3)}{2\sqrt{(x+1)(x+2)}}}}=\lim_{x\to\infty} {\frac{2\sqrt{(x+1)(x+2)}}{2x+3}}$. It follows that this limit is 1, and therefore $\lim_{x\to\infty} {\frac{x\sqrt{x+2}}{\sqrt{x+1}}-x}=\frac{1}{1+1}=\frac{1}{2}$.

Edited by renerpho
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Why use L'Hopital's rule? Mathematic's answer is much simpler.

0


text:
# The difference of symmetric difference of sets

In Kirillov,Gvishiani, Theorems and Problems in Functional Analysis(Springer,1982) Exercise 71 (b) states

$(A_1\cap A_2)\Delta(B_1\cap B_2)\subset(A_1\Delta B_1)\cap(A_2\Delta B_2)$.

I tried with different methods, but I could prove only $(A_1\cap A_2)\Delta(B_1\cap B_2)\subset(A_1\Delta B_1)\cup(A_2\Delta B_2)$ (union instead of intersection in the right-hand-side). It can be a typo, but they remark that this inclusion means that the operation of intersection is continuous with respect to the "distance" $d(A,B):=A\Delta B$. I would appreciate for any proof or disproof. I don't understand this type of continuity.

• You are right. If $A_1=A_2=B_1$ then the right-hand side is empty, but the left-hand side not necessarily. – njguliyev Sep 4 '13 at 13:33
• By the way, the second Russian edition has the correct form of this formula (with $\cup$). – njguliyev Sep 4 '13 at 13:38

$$(A_1\cap A_2)\Delta(B_1\cap B_2)\subset(A_1\Delta B_1)\cup(A_2\Delta B_2)$$
DeMorgan's comes into play when expanding the definition of the symmetric difference of sets, and introduces "disjunctions" in set membership (and hence, introduces the need for set union), for example $x \notin B_1 \cap B_2 \iff (x\notin B_1 \lor x\notin B_2)$, and likewise $x \notin A_1 \cap A_2 \iff (x \notin A_1 \lor x\notin A_2)$. In the end, this, plus "unpacking" the other clauses implies $$(x \in A_1 \Delta B_1) \lor (x \in B_1\Delta B_2) \iff x \in (A_1\Delta B_1)\cup (A_2\Delta B_2)$$


text:
# sylvester: Solve Sylvester Equation In maotai: Tools for Matrix Algebra, Optimization and Inference

## Description

The Sylvester equation is of form

AX + XB = C

where X is the unknown and others are given. Though it's possible to have non-square A and B matrices, we currently support square matrices only. This is a wrapper of armadillo's sylvester function.

## Usage

 1 sylvester(A, B, C)

## Arguments

 A a (p\times p) matrix as above. B a (p\times p) matrix as above. C a (p\times p) matrix as above.

## Value

a solution matrix X of size (p\times p).

## References

\insertRef

  1 2 3 4 5 6 7 8 9 10 11 12 13 ## simulated example # generate square matrices A = matrix(rnorm(25),nrow=5) X = matrix(rnorm(25),nrow=5) B = matrix(rnorm(25),nrow=5) C = A%*%X + X%*%B # solve using 'sylvester' function solX = sylvester(A,B,C) pm1 = "* Experiment with Sylvester Solver" pm2 = paste("* Absolute Error : ",norm(solX-X,"f"),sep="") pm3 = paste("* Relative Error : ",norm(solX-X,"f")/norm(X,"f"),sep="") cat(paste(pm1,"\n",pm2,"\n",pm3,sep=""))


text:
location:  Publications → journals → CMB
Abstract view

# The initial and terminal cluster sets of an analytic curve

Published:2017-04-05

• Paul M Gauthier,
Département de mathématiques et de statistique, Université de Montréal, Pavillon André-Aisenstadt, 2920, chemin de la Tour, Montréal H3T 1J4, Québec
 Format: LaTeX MathJax PDF

## Abstract

For an analytic curve $\gamma:(a,b)\rightarrow \mathbb C,$ the set of values approached by $\gamma(t),$ as $t\searrow a$ and as $t\nearrow b$ can be any two continuua of $\mathbb C\cup\{\infty\}.$
 Keywords: analytic curve, cluster set
 MSC Classifications: 30B40 - Analytic continuation

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text:
## Sunday, May 22, 2011

### Ejs Open Source Newton's Mountain Projectile Orbits Model java applet « on: May 20, 2011, 11:44:32 pm »

Ejs Open Source Newton's Mountain Projectile Orbits Model java applet
update: 14 june 2913
 http://weelookang.blogspot.sg/2011/05/ejs-open-source-newtons-mountain.htmlwith firing and retarding thrusters to get in circular orbit or otherwise https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejs_NewtonsMountainwee06.jar author: timberlake and lookang

 http://weelookang.blogspot.sg/2011/05/ejs-open-source-newtons-mountain.htmlwhite out background for inserting  pictures into tutorials with y = 4.23E07 m and vi = 3070.98 m/shttps://dl.dropbox.com/u/44365627/lookangEJSworkspace/export/ejs_NewtonsMountainwee.jar author: timberlake and lookang
white out background for inserting  pictures into tutorials with y = 4.23E07 m and vi = 3070.98 m/s

new interface with controls all at the bottom

27 May 2011 version with better Newton's mountain picture thanks to leongster, x control selectable

23 May 2011 version with multiple representation of versus time and radius from center of Earth
« on: May 20, 2011, 11:44:32 pm »

Ejs Open Source Newton Mountain Model java applet by Todd Timberlake remixed by lookang
Ejs Open Source Newton's Mountain Projectile Orbits Model java applet
reference:
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=398.msg7167#msg7167

Simulation above is kindly hosted by NTNUJAVA Virtual Physics Laboratory by Professor Fu-Kwun Hwang
http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=2204.0
alternatively, go direct to http://www.phy.ntnu.edu.tw/ntnujava/index.php?board=28.0
Author: Todd K. Timberlake, this remixed version is by lookang

General Description by Todd K. Timberlake (ttimberlake@berry.edu)
This simulation illustrates the motion of a projectile launched from the top of a VERY tall mountain on Earth. The diagram shown in the simulation is taken from Newton's A Treatise on the System of the World, which he wrote after the Principia, but the basic idea is found in the Principia itself. Newton concluded that a projectile launched horizontally with sufficient speed would orbit Earth rather than crashing to Earth's surface. Thus the motion of a projectile fired on Earth was not qualitatively different from that of the moon orbiting Earth.
The simulation allows the user to adjust the initial speed and launch angle of the projectile using sliders. The projectile is launched from the top of the mountain (note that the mountain shown in the diagram is ridiculously tall - such a mountain would stick out above Earth's entire atmosphere). The motion of the projectile is calculated using Newton's Second Law of motion and Newton's Universal Law of Gravitation.
The user has the option to allow the projectile to pass through Earth. The Earth is treated either as a homogeneous sphere (the default) or as a point mass located at its center (if selected in the Model Options menu). The homogeneous sphere is more realistic, but using the point mass can help illustrate the "true" shape of the projectile's path before it hits Earth.

Newton's Mountain
Show Center of Earth: show a red dot at Earth's center.
Show Velocity Vector: show a blue arrow representing the projectile's velocity.
Show Force Vector: show a red arrow representing the force on the projectile (or the projectile's acceleration, since its mass is an arbitrary constant).
Trace Projectile's Path: trace the path followed by the projectile.
Show Newton's Diagram: show the mountain diagram in the background.
Let Projectile Pass Through Earth: allows the projectile to pass into and through Earth, treating Earth as either a homogeneous sphere or a point mass.
Treat Earth As Point Mass: model Earth as a point mass. If not selected, Earth is modeled as a homogeneous sphere.
Visual Elements
The background is a diagram from Newton's A Treatise on the System of the World showing Earth with a VERY tall mountain at the top. The lines on the diagram show the paths of projectiles launched from the mountain top, or satellites (moons) orbiting Earth.
Green disk: the projectile.
Red spot: the center of Earth, labeled C in the diagram.
Blue arrow: represents the projectile's current velocity.
Red arrow: represents the force on the projectile, (or the projectile's current acceleration).
Magenta trace: trace of the object's path.
Controls
Play/Pause: start and stop the simulation.
Step: advance the simulation by one time step.
Reset: reset the simulation to its initial state.
Clear Traces: clears all traces of the object's path.
Initialize: read in values and place the projectile back on the mountain top. This will NOT clear any existing traces, but it will start a new trace unconnected to the old traces. Note that any changes to the parameter values will not take effect until the Initialize button is clicked.
Initial Speed: sets the initial speed of the object (in km/s).
Launch Angle: sets the launch angle for the projectile (in degrees). The angle is measured counterclockwise with zero degrees corresponding to a horizontal launch to the right.

changes by lookang:
made sliders and fields conform to my design preference
redo for SI units for closer to sg syllabus instead of arbitrary units originally made
22May2011
add panel of x,y,vx,vy,ax,ay for calculating values to coincident with Earth data
add trace instead of the older trail that can display older data
rotate newton's diagram by 5 degree to coincident with x=0, instead of the older 0.1*R
change to acceleration instead of force

question.
what is the velocity of the projectile for circular motion to occur if it is launch horizontally at the surface of the Earth?

hint: use circular motion equation to find the v!

enjoy!

7920.05 m/s

try it with the this customized with Earth data simulation :)

A newton's mountain with editing done by leongster, thanks bro!

Orbits round the Earth
An object projected horizontally near the Earth’s surface follows a parabolic trajectory. As the velocity of projection increases, there will be an occasion in which the trajectory follows the curvature of the Earth’s surface. If air resistance is negligible, the object will orbit round the Earth and will never meet the Earth’s surface.
 speed of cannonball at 0 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 1 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 2 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 3 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 4 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 5 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 6 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that hits Earth surface.

 speed of cannonball at 7 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that does NOT hit the Earth surface.

 speed of cannonball at 7 300 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory orbit that does NOT hit the Earth surface that is a approximately a circular orbit.

 speed of cannonball at 8 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory orbit that does NOT hit the Earth surface.

 speed of cannonball at 9 000 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory orbit that does NOT hit the Earth surface.

 speed of cannonball at 11 200 km/hr launched horizontally from newton's very tall mountain, resulting in trajectory that does NOT hit the Earth surface and escape from the Earth's gravitational pull

 plot of kinetic energy $E_{K}$, gravitational potential energy, $E_{P}$ and total Energy $E$versus distance away from centre of earth, r at R= Re

 plot of kinetic energy $E_{K}$, gravitational potential energy, $E_{P}$ and total Energy $E$versus distance away from centre of earth, r at R= Re, R= 2*Re

 plot of kinetic energy $E_{K}$, gravitational potential energy, $E_{P}$ and total Energy $E$versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re

 plot of kinetic energy $E_{K}$, gravitational potential energy, $E_{P}$ and total Energy $E$versus distance away from centre of earth, r at R= Re, R= 2*Re, R=3*Re and lastly R = geostationary radius

### Question01

An earth satellite of mass 200 kg lost energy slowly through atmospheric resistance and fell from an orbit of radius 8.0 x 106 m to 7.8 x 106 m. Calculate the changes in the potential, kinetic and total energies of the satellite during this transition period.
Answers: ( -2.57E8 J, 1.28E8 J, -1.28E8 J ) where E denotes x10

Solution:
a)

b)
Assuming the orbits are in circular motion as it decays from to
equation can be assumed to be valid to approximate this motion
Thus,

c)
since

check:
This answer can also be verified from the formula relationship on page Newton's Mountain Projectile Orbits which is not intended to be memorized.

Using the simulation:

 using R = 8.0X10^6 , v = 7072.8 m/s for circular motion orbit, you can get for satellite m = 1kg, KE = 2.501X10^7 J, PE = -5.002X10^7 J and TE = -2.501X10^7 J

 using R = 7.8X10^6 , v = 7162.9 m/s for circular motion orbit, you can get for satellite m = 1kg, KE = 2.565X10^7 J, PE = -5.131X10^7 J and TE = -2.565X10^7 J
Thus,
ΔKE = 200(2.565X10^7 - 2.501X10^7) = 1.28X10^8 J

ΔPE = 200(-5.131X10^7 - (-5.002X10^7)) = -2.58X10^8 J

ΔTE = 1.28X10^8 + (-2.58X10^8) = 1.30X10^8 J

where the slight difference is due to carry over error from the significant figures of the computer model.

reference:
http://commons.wikimedia.org/wiki/File:R_%3D_geo_Re_2012-10-08_1809.png#.7B.7Bint:filedesc.7D.7D

also contributed to wikimedia media commons
http://en.wikipedia.org/wiki/Newton%27s_cannonball#The_experiment

Date Name Thumbnail Size User Description
05:46, 8 October 2012Newtonsmountainv=2000.gif (file)758 KBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=1000.gif (file)724 KBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=5000.gif (file)999 KBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=4000.gif (file)873 KBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=3000.gif (file)799 KBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=6000.gif (file)1.26 MBLookang(User created page with UploadWizard)
05:46, 8 October 2012Newtonsmountainv=8000.gif (file)8.59 MBLookang(User created page with UploadWizard)
05:45, 8 October 2012Newtonsmountainv=9000.gif (file)10.68 MBLookang(User created page with UploadWizard)
05:45, 8 October 2012Newtonsmountainv=7000.gif (file)7.21 MBLookang(User created page with UploadWizard)
05:45, 8 October 2012Newtonsmountainv=7300.gif (file)7.58 MBLookang(User created page with UploadWizard)
05:45, 8 October 2012Newtonsmountainv=10000.gif (file)7.3 MBLookang(User created page with UploadWizard)
05:45, 8 October 2012Newtonsmountainv=0.gif (file)717 KBLookang(User created page with UploadWizard)

Date Name Thumbnail Size User Description
10:15, 8 October 2012R=Re 2012-10-08 1808.png (file)151 KBLookang(User created page with UploadWizard)
10:15, 8 October 2012R = geo Re 2012-10-08 1809.png (file)47 KBLookang(User created page with UploadWizard)
10:15, 8 October 2012R = 3Re 2012-10-08 1809.png (file)79 KBLookang(User created page with UploadWizard)
10:15, 8 October 2012R = 2Re 2012-10-08 1809.png (file)127 KBLookang(User created page with UploadWizard)
There was an error in this gadget


text:
Lemma 89.27.5. Let $(U, R, s, t, c)$ and $(U', R', s', t', c')$ be minimal smooth prorepresentable groupoids in functors on $\mathcal{C}_\Lambda$. If $\varphi : [U/R] \to [U'/R']$ is an equivalence of categories cofibered in groupoids, then $\varphi$ is an isomorphism.

Proof. Let $\psi : [U'/R'] \to [U/R]$ be a quasi-inverse to $\varphi$. Then $\psi \circ \varphi$ and $\varphi \circ \psi$ are isomorphisms by Lemma 89.27.4, hence $\varphi$ and $\psi$ are isomorphisms. $\square$

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text:
How do you solve the equation 3(x-3)^2+2=26?

Sep 26, 2017

$x = 3 \pm 2 \sqrt{2}$

Explanation:

$3 {\left(x - 3\right)}^{2} = 26$

$\rightarrow 3 {\left(x - 3\right)}^{2} = 24$

$\rightarrow {\left(x - 3\right)}^{2} = 8$

$\rightarrow {x}^{2} - 6 x + 9 = 8$

$\rightarrow {x}^{2} - 6 x + 1 = 0$

$\textcolor{w h i t e}{\text{XXX}} x = \frac{- \left(- 6\right) \pm \sqrt{{\left(- 6\right)}^{2} - 4 \left(1\right) \left(1\right)}}{2 \left(1\right)}$
$\textcolor{w h i t e}{\text{XxXX}} = \frac{6 \pm \sqrt{32}}{2}$
$\textcolor{w h i t e}{\text{XxXX}} = \frac{6 \pm 4 \sqrt{2}}{2}$
$\textcolor{w h i t e}{\text{XxXX}} = 3 \pm 2 \sqrt{2}$