text:
Phoenix Wright: Ace Attorney/Episode 5: Rise from the Ashes/Day 3 - Investigation

Back at Wright & Co. Law Offices, Ema apologizes for what Lana said, and reveals that the night Marshall's brother was murdered, Joe Darke tried to kill her. Neil Marshall tried to save her, but lost the battle to the death, and so Ema became a witness in the case.

Wright & Co. Law Offices
Talk to Ema:

 * SL9 Incident: She was at her sister's office, waiting for Lana to finish work so they could go eat dinner. Suddenly, Darke barged in and tried to take her hostage. Neil came in and fought Darke. The lights went out due to a thunderstorm, and when the lightning flashed the picture of a man raising a knife up to stab the other was burned into her memory.
 * After the incident: She didn't remember the moment that Darke stabbed Prosecutor Marshall, so they didn't ask her about it — only about when she was attacked. That must be why her sister had to make up evidence to get a conviction. Ema says that even though he used it to get the Guilty verdict, Edgeworth didn't know the evidence was fabricated. Even so, the rumors about him still started that day.
 * Permanent picture: Ema says in that instant during the lightning she saw that Darke had knocked Neil down and was about to stab him. She doesn't remember what happened next as she passed out, later to awake in her sister's arms. She couldn't bring herself to testify about the incident, but managed to draw a picture.
 * Something puzzling: Why did Darke run all the way to Lana's office from the Police Department? It turns out that at the time, Lana was a detective! After the Darke case, she was transferred to the Prosecutor's Office and became the Chief Prosecutor.

Lana was a detective? Move to the Detention Center to talk to her.

Detention Center Visitor's Room
Talk to Lana:

 * Today's trial: Lana still won't talk about Goodman's murder; she says she isn't the only one involved.
 * Detective Lana Skye: Lana and Gant were brilliant detectives back then. Lana became a detective to gain experience investigating crime scenes so that she would make an effective attorney. After being promoted to Chief of Police for wrapping up the SL-9 case, Gant transferred her to the Prosecutor's Office so that she could follow her dream.
 * Darke investigation: She and Damon Gant shared the same occupation and office. They both led a team of detectives, two of which were Marshall and Starr, the other being Goodman. Jake worked with his brother for the first time. They brought Darke in for questioning, and that was when he escaped and tried to kill Ema.
 * First one at scene: Lana was the first person to the crime scene, where she found three bodies on the floor: Neil's body, and the unconscious forms of Ema and Joe Darke. Apparently, Neil had been able to strike a final blow, knocking out his killer. Lana panicked and took her sister out of the office to wake her up, after which she immediately placed Joe Darke back under arrest.

Phoenix doesn't believe it — that all the people involved in the current trial were also involved in SL-9 seems a little too convenient. Although the case was supposedly solved two years ago, there are still people like Jake Marshall that aren't satisfied either. Since Neil's murder took place there, you should probably go check the Chief's Office for more information.

Move to the Police Department Entrance.

Police Department Entrance
You'll find Marshall. He'll tell you that the "Evidence Room" (where you and Ema did the scientific investigation) is no longer accessible due to the trial yesterday. Also, he'll turn himself in after going to the Chief Prosecutor's Office. Talk to him:

 * Darke trial: All the detectives on the case thought something was fishy. Though the Switchblade Knife was Joe Darke's, it didn't match the wound on the victim. However, in the final report, the possibility that it wasn't the murder weapon had been erased.
 * Prosecutor Marshall: Neil had been awarded the highest honor that day — the same Prosecutor Trophy that Edgeworth just won. That means the day of the murder was also the day of the evidence transferrals two years ago.
 * Scars: All the detectives were taken care of in some way by Damon Gant and Lana Skye: Starr was fired and Marshall was demoted. Goodman was safe because if they got rid of him too, the commissioners might have become suspicious.
 * Gant & Skye: When Gant became Chief of Police, he promoted Lana to Chief Prosecutor. Since she moved to the Prosecutor's Office, Lana started acting completely different. Marshall doesn't know the reason, as Lana keeps her secret too well hidden.

Marshall is done with his story and tells you that although Edgeworth presented the evidence, it was Damon Gant who falsified it and planned the whole thing. When he leaves, move to the Criminal Affairs Dept.

Police Department Criminal Affairs
Not much is happening and Gumshoe isn't here, but the Chief Detective tells you everyone is in the conference room trying to deal with all the chaos of the last few days. You'll also learn where Gant's Office is. Move to the Chief's Office.

Police Department Chief's Office
Ema and you came in to see a huge church organ. After Ema plays a note, Gant shows up. He'll put a piece of paper in his desk after he sees you and her. Gant shows you a picture hanging on his wall. It's Lana, Neil, and Gant.

Something is weird in the picture, but you can't put your finger on it. The Gant Team Picture will be added to the Court Record. Gant then decides to lock up the room, and takes you with him, depositing you at the Police Dept. Entrance. Is he kicking you out? Against your request to investigate? He must be hiding something. Move to the Criminal Affairs Dept.

Criminal Affairs Department (2)
You meet Gumshoe, who was serving coffee in the meeting, saying that Edgeworth is now "under fire" since the rumors were true. Talk to Gumshoe.

 * Edgeworth's crisis: Things haven't been going well for Edgeworth since the rumors were proven true. Lana is the guilty party, but Edgeworth is the one responsible. Gumshoe is very worried about him.
 * SL9 Incident: Gumshoe was reading the SL-9 file and may have come up with something involving the murder weapon

Maybe you should jog his memory. Present the Switchblade Knife, then go back to talking to him to get some answers.

 * Murder weapon: The weapon was owned by Darke. The tip that was broken off was, in fact, found in Neil's body. You will receive Neil's Autopsy Report, and the Switchblade Knife will be updated in the Court Record.
 * Darke's crimes: Darke was actually just an average businessman. Until one day, when he accidentally hit someone with his car. Desperate, he killed a man who witnessed it, killed a lady who witnessed that crime, killed a child who witnessed that crime and killed a jogger who saw him bury the bodies. He finally gave himself up afterward.

Gumshoe admits that the whole scenario was conjecture, since they never found a shred of evidence. There was a good chance he would've gotten away with it, except for the incident at the Chief's Office, where he left a witness (Ema) to the murder of Neil Marshall.

You ask Gumshoe to let you into the Chief's Office, but he can't, as letting a civilian in there would result in him losing his job. Although any detective's ID card can get you into the office, the data on Goodman's ID was deleted the day he died. So you need to find something to get Gumshoe to help you. Move to the Underground Parking Lot.

Prosecutor's Office Underground Parking Lot
Everyone seems to be investigating the other crime scene. You've proven that there wasn't a murder there, but you've also made Lana seem even more guilty. Move to the High Prosecutor's Office.

High Prosecutor's Office Room 1202
You'll see Edgeworth, who throws whatever he was writing down on the ground upon seeing you come in. Talk to Edgeworth:

 * Forged evidence: Edgeworth blames himself for the forged evidence. Even though someone else in the department gave him the false evidence without his knowledge; as head prosecutor he is still responsible. The prosecutors and detectives share the same bond. If broken, everything would fall apart.
 * Tomorrow's trial: Now that the SL-9 Incident is coming back up, Edgeworth is worried because the evidence list for the case is only half as long as it should be.
 * The day of the crime: Edgeworth had finished his work at the office for the day, but Gant requested he take evidence from another case (the Screwdriver) back to the Prosecutor's Office.

Present Edgeworth with the Gant Team Picture. Ema notices that Neil's trophy is different than Edgeworth's.

 * Prosecutor trophy: Edgeworth then tells you the story behind the trophy. It comes from an ancient Chinese tale about the word "contradiction". In Chinese, the word is made up of two characters, the first meaning halberd and the second meaning shield.

An arms merchant presented the emperor a halberd that could penetrate any shield or armor, and a shield that could withstand any weapon. The emperor then spotted what would be called a contradiction, and the Chinese word for contradiction was born. Hence the broken knife and chipped shield in the Prosecutor Trophy symbolize the merchant's items.

But there is only a shield on Edgeworth's trophy. The halberd part was abolished two years ago by Damon Gant, who Edgeworth says you should talk to about it. The Prosecutor Trophy will be updated in the Court Record.

Examine the piece of paper under Edgeworth's desk. It's a letter of… resignation? Edgeworth feels as if something has died inside him. He can't forgive himself for the path on which he walked. The Letter of Resignation will be put in Phoenix's pocket.

Move to the Underground Parking Lot.

Prosecutor's Office Underground Parking Lot (2)
You'll meet Angel who is still harping on witnessing Lana stabbing Goodman, still believing that what she saw was true. Talk to Angel:

 * Darke investigation: They trailed Darke for a very long time, until the night Neil was killed. Everyone was very appalled by this, especially Jake, who was haunted by his brother's death, which in turn affected Lana.
 * After case closed: All detectives, except for Goodman, were relieved of their duties, without an explanation. Then Lana became Chief Prosecutor, although Angel says that Lana was merely a pawn.
 * Legendary duo: Gant was very famous back then, because he could produce evidence under any circumstances, so he had rumors around him too.

Everyone liked Lana and wanted to be like her because she hated anything crooked and watched out for the other detectives. She felt especially bad for Jake and when his brother was killed she felt like she'd lost a brother herself. If it wasn't for her, Jake might not have gotten over the shock of the murder. But something happened in that incident that made her change and became so cold.

 * Being "used": Angel suspects that Lana was only given the Chief Prosecutor position so that Gant could gain control of her office. Angel doesn't know how Gant is controlling her, but it must have been something that happened two years ago that causes her demeanor to change.

Finally, you are getting closer to the bottom of this. Move to the Criminal Affairs Dept.

Police Department Criminal Affairs (3)
You'll meet Gumshoe, who still refuses to let you into the office. You need something to change his mind. You catch him making brewing coffee and copying 150 files for the chief, turning him into a desk jockey!

Present Edgeworth's Letter of Resignation to Gumshoe, who'll be shocked at seeing this. Edgeworth trusted detectives to give him sound evidence, but was betrayed. Gumshoe feels very sorry and would do anything for Edgeworth now. He then makes up his mind and gives you his ID card. Gumshoe's ID will be added to the Court Record. The Letter of Resignation will be taken off the Court Record at the same time.

Move to the Chief's Office.

Police Department Chief's Office (2)
You arrive at the Office… followed by Gumshoe, who was worried something might go wrong. Like yesterday in the "Evidence Room", Ema mistakes Gumshoe as a ghost and slaps him again! Why did he give you his ID card anyway? Gumshoe's ID will be removed from the Court Record. Actually, you crushed it and put it in your pocket. Gumshoe blames you for ruining his ID card! Talk to Gumshoe:

 * The Chief's office: The left side of the room was Lana's, and that's where Ema was waiting for her that night. Both Lana and Gant used to work together in this room. Now it's all Gant's, and he hasn't let anyone touch Lana's side in two years.
 * Chief Gant: Something had been bothering Gumshoe since you wanted to look around his office for the SL-9 Incident. Could Gant be a suspect for that case? You can't say anything about him… yet.

Examine the safe behind Gant's desk on the right side of the room. It looks like you need a code to open it. You have a choice, choose "Input number". At the bottom of the screen, you'll find a keypad like you would find on a phone. Tap the numbers "7777777" and the safe will open.

Note: Don't waste your time guessing what number will open the safe. Check the ID Card Record to find which number would open it.

Wii Remote Controls for WiiWare version: Select number with D-Pad. Confirm number with the A button. Press the B button to go back one digit.

7777777? That's the ID number of the executive that entered the Evidence Room on the day of the crime! It must be Gant's number! No time for chat though, examine the safe again and you'll find a broken piece of jar, and a piece of leather cloth with a handprint on it. Gumshoe wants to know if there is a connection between the two pieces and the case before he'll let you take them.

Start with the shard. Present the Unstable Jar to Gumshoe. Remember yesterday, you found pieces like this and put it together. Gumshoe then notices the possible connection. You have a choice, choose "Assemble fragments", and you'll put the final piece in there.

It fits! Gant was obviously hiding the fragment in there. He was hiding valuable evidence in there! But why? Gumshoe then points out something. There's blood on it! Could this be part of the reason? The Unstable Jar is updated in the Court Record.

Now for the cloth. Present the Fingerprinting Set. You choose a finger you want to test: the middle fingerprint in this instance. Once you start, sprinkle all around the touch screen and blow. After the fingerprint is clear, you get to choose its owner. It's Ema Skye's! This seems to be a very grim piece of evidence, but you can't tell Ema about this for now. The Strip of Cloth will be added to the Court Record.

In case you forget the controls on the WiiWare version, sprinkle powder all over the screen by pressing A. Then shake the Wii Remote to blow the powder out. You've done it before in the "Evidence Room". So it should be easy.

Examine Gant's desk. The first time you came in here, he hid something he was reading from you. It's an evidence list saying, "SL-9 Incident!". Remember, Edgeworth said his evidence list for SL-9 is twice as short as an average one. This must be the other half of the list; it was ripped in half! The Evidence List will be added to the Court Record. If you check the Evidence List and rotate it to show the back, you can examine it to find a crude drawing of what looks like a figure about to stab another person. Didn't Ema say she drew what happened that night? At any rate, it's best not to disturb her, so you keep it to yourself.

Gant is obviously hiding something! Speaking of Gant, uh-oh! He somehow sneaked in to see you three in his room. He has you and Gumshoe leave, but wants Ema to stay for questioning. Also, Gumshoe has to leave his id card behind.

You and Gumshoe would be at the Police Department Entrance. Gumshoe blames you for taking this risk. So Gumshoe goes back in to try and smooth things over with the chief.

You automatically go to the Detention Center.

Detention Center Visitor's Room (2)
What Lana has told you over the past few days is nothing useful. Talk to Lana:

 * Keeping quiet: She said that she stabbed Goodman, but there is no supporting evidence. She could be doing it for someone else — somebody she is afraid of. Lana asks you who that person is, so present Damon Gant's profile when prompted.
 * Damon Gant: Lana wasn't the only one who could have forged evidence. Before Lana will let you accuse Gant, however, she wants to see some proof that he falsified evidence.

Present to her the Evidence List or the Unstable Jar. Lana finally gives in, and tells you that she does follow Gant's orders and three days ago she got orders concerning Detective Goodman.

 * "Orders": She had orders from Gant to dispose of Goodman's body, which he said was in the trunk of Edgeworth's car.

If that is true, then Lana wasn't the one who killed Goodman! When Lana opened Edgeworth's trunk, she found Darke's knife plunged inside him, so she took the knife out, and plunged Edgeworth's Knife inside instead. That's when Angel saw her. Why did she have to hide the Switchblade Knife so badly?

 * Darke's Knife: Lana didn't want the SL-9 Incident opened up again, so after she stabbed Goodman's body, she called her sister to tell her to hide the knife for her.

When she got the orders from Gant, the first thing Lana did was call Marshall, thinking she could trust him to help her hide the body. He had plans of his own to keep the case from dying, and after Lana's phone call, he resolved to use the stolen ID card to get the evidence from the Evidence Room.

Lana is finished talking with you, and doesn't want you to pursue this further in court. It looks like you're done for today. Tomorrow you'll have to prove the mystery behind Goodman's murder, the SL-9 Incident, and Lana's secret.


text:
Phoenix Wright: Ace Attorney/Episode 5: Rise from the Ashes/Day 4 - Trial Former

You start out at the Defendant Lobby, but you don't find Lana or Ema. Edgeworth comes to tell you that the reason the trial didn't reach a verdict yesterday is that there was still doubt in the ID Card Record. Once it comes out that the last number is Gant's, there will be no more doubt in the court's mind and Lana will be found Guilty.

You have to get Lana to tell the truth about Gant. You suggest that Edgeworth should help him out. Otherwise, he'll never know what happened during the SL-9 Incident. It looks like he'll think about it. It's time to start the trial. Remember the "Turnabout Samurai" with Dee Vasquez? This will be the second time you'll be working together with Miles with one goal: To find the truth.

The trial begins
As the trial starts, Gant takes the stand to make a proposal to have Lana speak to the court.

Lana wants the trial to end immediately and once again confesses to the murder of Bruce Goodman. She even tries to fire you and forfeits her right to an attorney. The judge agrees and reaches his verdict.

Before he can do so, however, Edgeworth objects, stating he didn't prove Lana's guilt beyond the shadow of doubt. Gant, hearing this, tells Edgeworth to keep quiet. Edgeworth refuses and suspects that the sudden confession must be due to some deal being made behind the scenes.

Edgeworth calls a new witness to the stand: Ema. Lana objects, not wanting to hear Ema's testimony, but Edgeworth tells Lana they are doing this to find the truth, no matter how tragic it may be. However, it would be even more tragic if one averted his eyes away from it. The court proceeds to bring her up to the stand and she testifies about two years ago. Gant warns Edgeworth he'll regret this.

You realize that Edgeworth will go through this with you after all, in bringing out the truth and end the SL-9 case once and for all!

Two Years Ago: Cross-examination
Press Ema's fourth statement. As Darke jumped on Neil, a terrible storm outside caused a blackout. Ema told Goodman about the image left in her mind. You have a choice to move on or not. Choose "Hear more". Ema couldn't say anything, so she instead drew a picture. When given the choice, pick "Ask about the picture". The judge asks Ema to add it to her testimony.
 * "I drew a picture of that scene once… but it seems to have been lost."

Present the Evidence List on that statement. You hand the judge the list and ask him to turn it over. It's the drawing! Edgeworth scolds Phoenix for having what only the prosecution should have access to. The judge notices the list Edgeworth was given during the SL-9 case is the other half. If Edgeworth was handed the sheet he has, then he was only given half of what they found. If that is true, then there could be more of the drawing on the back of Edgeworth's list. There is, and it looks like the head of the Blue Badger! The Evidence List will be updated in the Court Record.

The judge has Ema testify about the drawing. As Ema looks at the picture, she seems to be thinking about something, but what?

Ema's Picture: Cross-examination
On Ema's fourth statement, present Neil's Autopsy Report. The judge wants you to find the contradiction. When you examine the picture, present the tip of the knife held by its wielder. The Autopsy Report says Neil was only stabbed once, and yet the picture shows that the knife was already broken. It couldn't have been broken beforehand either, because its tip was found in the victim. If the knife's tip was found in Neil's body despite the facts, then the tip may be the falsified evidence given to Edgeworth!

Phoenix then recites the story. After winning the King of Prosecutor's Trophy at the ceremony, Neil took Darke in for questioning, along with Gant. During the questioning, Darke fled to the Chief's Office. Neil chased after Darke, but was killed when he got there.

If the story was told this way, then one of the facts could be a lie. Ema denies it. Edgeworth concludes if that is true, then there could be another broken knife that killed Neil. When choosing to agree or not, choose "There is another one". When asked what the weapon is, present the Gant Team Picture from the Court Record. Edgeworth immediately sees what you are showing. The Prosecutor Trophy Neil was holding had the broken knife on it, with its tip broken off! Neil must have taken the blade out of the trophy and followed Darke to the Chief's Office; since he wasn't a detective, he didn't have a gun. However, Edgeworth concludes that if that is true, the man Ema saw raising the knife was actually Neil! But Neil was the one who died, not Darke!

Before you can continue, Ema stops you, as she now remembers everything that happened that moment. She recalls drawing the other half of the picture — the side with the Blue Badger's head on it. The Evidence List wasn't ripped in half when she drew the picture! The judge has Ema testify about her recollection.

Ema's Testimony: Ema's Recollection
Edgeworth objects, saying that the Blue Badger hadn't been designed until just recently. If so, what was it? Looks like a cross-examination would bring an answer.

Before you begin, however, Lana tries to stop you, not wanting to go any further. But Edgeworth tells Lana it's already too late. The judge has the bailiff detain Lana. With such an outburst like that, you may be getting close to her secret. You begin the cross-examination.

Ema's Recollection: Cross-examination
If that wasn't the Blue Badger Ema saw, then what is it? Press Ema's fifth statement. She's sure about it. When asked about what it really was, select "I just might know...". Present the Unstable Jar. It looks like it doesn't resemble the Blue Badger, but if you look at it from a different angle, you may see the resemblance. You automatically check the Unstable Jar. Alternatively, you can Present the Unstable Jar on the fifth statement to jump straight into the puzzle.

The bottom of the jar, the small side, has to become the top ball on The Badger's head. The two handles go on either side of it. The top hole on the bigger end should be pointing towards you, and downward. If you see the resemblance, present it to the court.

If you've forgotten the button layout on the WiiWare version, lie the Wii Remote flat while turning the object around. Face the Wii Remote towards you to flip around the object. Once you match it with the picture, press the Minus (-) or A button to confirm it.

Edgeworth wants to see your point. It doesn't look like it changes the case. You'll be given a choice of answers: "The location", "The murder weapon" and "The murderer". It doesn't matter what you choose, because all of those actually changes the scenario completely! Remember the Gant Team Picture. It looks like there is a knight statue there with the jar. Remember when you visited the room. You saw that statue on Gant's side of the room. This means that the fight didn't take place on Lana's side of the office, but Gant's!

If that is true, then could the murderer have moved the bodies to Lana's side? If so, then there has to be a reason, and you have to show it. There's no going back now.

The tragic truth?
Recall Ema's testimony. She said she pushed the man holding a knife after seeing him raise it up. When she did, the man knocked over the jar. And if that man was pushed in the direction he was going, he might have ran into the suit of armor holding the sharp sword. It opens up another possibility for what happened.

If what you said was true, then Neil's death may have been caused by Ema's actions! Ema screams and faints upon hearing this.

Lana yells at you for pinning the crime on Ema and wants proof that she did so. It seems like you don't have any but Edgeworth tells Lana that Neil didn't die instantly, so it's a possibility that he left a message before dying. Lana says it's impossible and you are given a choice of whether you agree or not. Say the victim's message is in the evidence. Then present the Unstable Jar. There's blood on it, and it looks like Neil tried to write something, but it's been wiped off! Is it the killer's name? Looks like you'll have to connect the dots, literally.

Just press on a dot and drag a line to the correct dot. Make sure it spells "EMA". You can press "Back" to erase the last line you created if you make a mistake.

With the WiiWare version, there's two ways of doing this: use the D-Pad to move the cursor and press the A button to connect the dots. Press the B button if you have made a mistake. The other is using the "motion controls" as moving the Wii Remote on the screen the same way. When you finish, press the Minus (-) button to Present it.

On the iOS version of the game, you have to do the lines in a certain order (many report this as a bug). First the line from the top left dot to the next right dot (the top of the 'E'), then from the top left dot down to the bottom left dot (the side of the 'E'), then from the bottom left dot to the next right dot (the bottom of the 'E'). Next do from the middle of the 'M' to the top right corner of it, then down (the right dot of the 'M'). After that start at the bottom left corner of the 'A' and go to the top middle of the 'A'. Next from the top middle of the 'A' down to the bottom right dot (the right side of the 'A'). Next, do the line from left to right in the middle of the 'A'.

Once you have the name, present it to the court. As a defense attorney, you are defending someone. However, you have proven her loved one to be a murderer. Gant takes the stand. He reminds Edgeworth that two years ago, he sentenced an innocent man to death, and to do that, he forged evidence to ensure his sentence. This leads the crowd into such an uproar, the judge calls a recess.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Phoenix Wright: Ace Attorney/Episode 5: Rise from the Ashes/Day 4 - Trial Latter

Recess
As you apologize to Edgeworth for what happened, Gumshoe comes in. Lana wanted Gumshoe to give you a book. The cover reads, "Evidence Law", and the book will be added to the Court Record. Lana also wanted him to tell you, "If you're planning to take HIM on, you're going to need this book."

Lana claimed to have stabbed Goodman, but he was already dead. So the real killer is still out there somewhere. This case has hurt many people. Edgeworth unknowingly used forged evidence to have an innocent man sentenced to death, and you have implicated your client's little sister as a murderer. This is a complete nightmare for everyone, and it's now up to you to end it.

Back in court, the judge tells Edgeworth that due to his actions, he will receive harsh penalties. Edgeworth heartily accepts… and turns over his right to call witnesses to you. Present Damon Gant's profile to call him to the stand. He led the SL-9 case first-hand, and you probably want to hear what he has to say.

Gant takes the stand and warns you he has "weapons" at his disposal. Gant begins to testify. He'll warn the court if his testimony is a waste of time, he'll be allowed to refuse further testimony. Edgeworth and the judge, however know the consequences behind it. So, you have to hit him hard and fast with evidence before he decides to refuse to go any further.

Gant's Testimony: SL-9 Incident
Gant says before Darke was arrested, he was knocked unconscious when Ema pushed Neil.

SL-9 Incident: Cross-examination
Gant is clearly lying. Present the Unstable Jar or Evidence List on Gant's 5th statement. Remember when you were at Gant's office. You found this in Gant's safe, and the second half of the Evidence List in Gant's desk. Gant denies it, saying you are probably forging the evidence. Even though Gumshoe was there, Gant doesn't think this would have stopped you.

The judge has Gant testify to prove his point.

Note: You can Present a Strip of Cloth on the final statement, but in case you didn't get the memo via Evidence Law, rule 2 states that "unregistered evidence presented must be relevant to the case in trial". So, for now, hold on to it until a Strip of Cloth is proven to be connected to this case.

Gant's Testimony: Evidence & Forgery
Gant warns you that consequences may be at hand when you slip up.

Evidence & Forgery: Cross-examination
Press on Gant's 5th statement. He probably rearranged the crime scene to get his Chief of Police position. However, Gant tells you that even if he didn't, he was still going to get the position anyway. So, he probably did it to help someone else. Gant denies it, saying he only helps himself. The judge has Gant add that statement to his testimony.
 * "I wouldn't be anyone's 'accomplice' if there was nothing in it for me."

Press on that statement. There was "nothing in it" for him? Gant didn't care about Ema at all, and Edgeworth backs up this claim that he only looks out for himself. Something else is going on here. When asked if Gant would help someone out, choose "Point out accomplice". The judge wants to know who. Present Lana Skye's Profile. If he appointed Lana Chief Prosecutor, he must have done it for his own profit. He had Lana transferred so he could control the Prosecutor's Office through her.

Gant wants proof of your claim. Edgeworth states that in the current case, Lana has been acting strangely, as if she was being "controlled". If that is true, then Gant probably forced Lana to take the fall for the murder. If that is true, then he must be blackmailing Lana!

The judge wants you to show evidence tying Chief Gant to the murder. Present the ID Card Record. The last remaining card number on the list is "7777777". That is the same number that unlocks Gant's safe in his office. This proves that Gant was at the Evidence Room on the day of the crime. Gant admits to going in the Evidence Room, but as he's allowed to go anywhere, this is immaterial. You claim that he met with Goodman that day, but the judge wants you to show some proof.

Present Goodman's Lost Item Report. In yesterday's court session, you found out that Marshall stole Goodman's ID Card on the day of the murder. Goodman would have had to file the report with the Chief, and he definitely filed it, because he needed to transfer the evidence from the SL-9 Incident. Since all the numbers on the ID Card Record are accounted for, Goodman and the murderer must have used the last one left — Gant's ID! Furthermore, if Gant had just lent the card to Goodman, it would have been found on his body.

You have Gant on the ropes now. He killed Goodman, then he called Lana to dispose of the body. But how could he have done so? Goodman was killed in the Evidence Room, but was found at the Prosecutor's Office, and Gant and Lana never left their respective offices that day.

The judge wants you to show how Gant managed to move the body. Present the Screwdriver or the Parking Stub. On the day of the crime, Gant told Edgeworth to take the Screwdriver to the Prosecutor's Office. Edgeworth did so, and parked at the Underground Parking Lot. Goodman's body was found in the trunk of Edgeworth's car, so Gant must have put Goodman's body in the trunk and Edgeworth moved the body to Gant's accomplice, Lana Skye.

Gant then decides to take a break. Remember, as the Chief of Police, Gant has the authority to refuse to testify. He can leave the courthouse, unless you have concrete proof that he killed Goodman. You have a choice, choose "I have no proof yet". Gant then leaves, and the judge penalizes you for accusing the Chief of murder.

Before the judge can do so, however, Edgeworth objects, saying there is someone else that could testify. You have the authority to call up witnesses, so when given the choice, present Lana Skye's Profile. The judge calls for a recess. Before he can do so however, Gant takes the stand, warning Lana that if she agrees to help with Phoenix's claim, her sister will be found Guilty of Neil's murder.

Recess (2)
Gumshoe comes in saying he is now taking your assistant's place. Does he mean Maya? Edgeworth then explains about the "risks" he was talking about concerning Gant refusing to testify. If he refuses to testify, that also means he gives up his right to testify later if he wants to. Ema then walks in looking somewhat happy. She heard everything about the case. Lana was blackmailed by Gant to protect her little sister. She is also sad, because if it wasn't for her accidentally killing Neil, this probably wouldn't be happening right now. You step into the courtroom now, where hopefully Lana will tell the truth.


text:
Phoenix Wright: Ace Attorney/Episode 5: Rise from the Ashes/Day 4 - Trial Latter 2

The judge calls Lana to the stand and has her testify about her relationship with Gant.

Gant & The Fabrication: Cross-examination
Press Lana's fourth statement. Lana says she arranged the crime scene, because she was the first to get there. She saw the knife from the Prosecutor's Trophy inside Neil's body. But didn't he die from accidentally stumbling into the suit of armor? Lana stands by her statement that Darke killed Neil. But if that's true, then what happened to Darke's knife? Lana says it was lying on the ground. The judge has Lana add what she did after she found the crime scene.
 * I broke off the tip of Darke's knife, planted it inside the wound, then moved the body.

She did what? Press on that statement. You are then given a choice of questions to ask. Ask "Why did you move the body?". She adds the answer in her testimony.
 * The pieces of the jar that shattered during the events threatened my plan.

Present the Unstable Jar on that statement. The name could only have been written on the jar before it was shattered. Yet it was broken in the struggle! The judge has Lana testify on Phoenix's claim.

Jar & Message in Blood: Cross-examination
On Lana's fourth statement, present the Unstable Jar. There was still one piece left that Lana didn't wipe off, and that piece was found in Gant's locker. When Lana hears this, she is stunned, just as you hoped. Since she testified to gathering every last piece, that means Gant's piece must have been collected before Lana got there. If that is true, then Gant was the first person at the scene, not Lana!

What was Gant doing there? Why was he hiding this for two years? Edgeworth says that he helped Lana protect Ema. Lana still denies it, and scolds Phoenix for telling foul lies. Hearing this, Phoenix gets suspicious. What if this is what Gant is planning?

Phoenix then tells Lana to testify once more, saying that if Gant fabricated the evidence before Lana got there, then Ema's involvement may also be a lie. Lana is scared, but with Edgeworth's encouragement, she finally agrees to tell the truth.

Lana's Testimony: Actual Crime Scene
Lana has proof that she saw Neil on the armor, she had taken a picture, which she sent to Phoenix this morning. But you don't recall ever receiving the photo, although she did give something to you.

Check the Evidence Law Book. Turn it to the back of the cover and examine it. You'll find the picture inside. Lana was right, she did see Neil on the armor's sword. Lana's Picture will be added to the Court Record. Ema sees something stunning. A piece of cloth from Neil's vest had been removed. This could be where the cloth piece you found in Gant's safe earlier came from! You begin the cross-examination.

Actual Crime Scene: Cross-examination
"Hold it!"

Gant, out of nowhere, comes to the stand as you start. Edgeworth tells Gant he has already forfeited his chance to make further statements. Gant says he doesn't have to, as he claims you have the evidence that says who the real murderer is. You have a choice to present it or not. The Piece of Cloth has Ema's handprint on it, and you don't want to finger her as the murderer, so choose "Cannot show evidence".

(If you were to show evidence at this point, Ema would be implicated and you would receive an automatic Game Over.)

Solving the case
When the court realizes that Gant hid evidence in his safe, Gant decides to come clean. He was at the scene before Lana, and saw the opportunity to control Lana. He knew she would come to him for aid, so he helped her conceal her sister's involvement after hiding two of the pieces of evidence from the scene. He put the evidence in his safe as insurance, in case something in his plan went awry down the road.

To ensure that he would never get fingered for the crime, he kept the most legible piece of the broken jar, although he hadn't counted on Lana wiping off the other pieces. Edgeworth wants to know how they can believe he didn't just forge Ema's name on the jar along with everything else. That's why he kept the other piece of evidence — the Piece of Cloth. Gant knows you have it, and the judge asks you again to present it, so choose "Show evidence", then present the Piece of Cloth. When asked who the fingerprints belong to, present Ema Skye's profile.

Before you regret this, however, Phoenix finds a contradiction on the cloth. You should probably catch it by now as well. When asked to show the contradiction, present Lana's Picture. Blood is splattered everywhere on Neil's chest, even on the spot the piece of cloth should have been. If the piece of cloth was there, blood would have gotten on it. However, there's no blood on the cloth! The fingerprints prove that Ema shoved Neil, but the cloth must have been cut off before he was impaled, or it would be covered in blood.

Gant was there before Lana, and he also admitted to cutting out the Strip of Cloth, and arranging the scene a little. When Gant arrived on the scene he lifted Neil's body and impaled it, then wrote Ema's name on the jar in the victim's blood and shattered it. Leaving behind the clues led Lana to believe her sister was responsible, which set her up to be blackmailed.

It looks like it's finally over, but Gant persists. He accuses you of presenting illegal evidence, which can't be used to convict a suspect. Gant says that when you concealed the Strip of Cloth the first time he asked you to present it, it became illegal evidence.

When asked if you admit to purposefully concealing evidence, say "I did not". It is true that you refused to present the cloth at one point, but it's because you "couldn't", not that you "didn't". When asked to prove that the evidence isn't illegal, present the Evidence Law book.

There are two certain rules in this book:
 * 1) No evidence shall be shown without the approval of the Police Department. Two years ago, Gant cut off the cloth from Neil's vest because it was decisive evidence, and hid it in his safe. It is obvious that you didn't have Gant's approval to show it, until Gant admitted that he had removed the Strip of Cloth.
 * 2) Unregistered evidence must be relevant to the case in trial. When Gant first wanted you to present the cloth, it was impossible for you to prove its relevance to the case. Until Lana's Picture was presented, there was no way for Phoenix to present the evidence.

Hearing this, Gant finally gives in to your claim. Not only that, he also explains what he was doing in the current case. Jake Marshall never let SL-9 go, and continued to investigate behind his back, first recruiting Starr, then Goodman. Gant also reveals the reason he killed Goodman — Goodman wanted Gant to reopen the SL-9 case, and he wanted to bring some evidence out for Marshall.

Lana takes the stand again. Phoenix explains to Lana that Ema did nothing wrong, and Gant betrayed her trust. She no longer has a reason to stay silent and will reveal everything from the time she and Gant forged evidence, up until today. Poor Phoenix didn't get her out of all the trouble… but Lana believes that one day Phoenix will make it to the top. She thanks both Phoenix and Edgeworth for their diligence. Unfortunately, she's still an accomplice to Damon Gant's crimes, and as such, a trial for those crimes are to be set for a later date.

Strange, though, Lana is still smiling. She's just happy to finally be freed from the chains of her past.

Lana Skye is found...

What really happened
It all started two years ago. Joe Darke, a businessman, was driving home when he met with an unfortunate accident. He ran into a pedestrian, killing him. Darke panicked, and killed a man who witnessed the hit and run. He then killed a woman who witnessed that killing. He then killed a child, who witnessed that third killing, and as he buried the bodies, he killed a jogger that caught him.

Darke seemingly left all evidence hidden, and it looked like he would get away with it. However, he was overridden with guilt and eventually turned himself in. Thus the trial of the "Joe Darke Killings" began.

His trial involved a team of investigators (Jake Marshall, Angel Starr, and Bruce Goodman, with Neil Marshall as the case's prosecutor) led by Damon Gant and Lana Skye, who were very famous back then. The team was very desperate for answers and couldn't obtain good enough evidence to find Darke Guilty.

One night, the team took him for questioning at the Police Department. During the questioning, however, a terrible storm caused a blackout in the building. Darke took this advantage and ran off. Neil followed, carrying the broken knife from the Prosecutor's Trophy he'd received at a ceremony earlier that day.

The chase led to Lana and Gant's office. There, Ema Skye was waiting for her sister Lana, so they could have dinner. Just then, Darke barged in, noticed Ema, and was about to make her his next victim. Before he could do so, however, Neil entered the office and they both fought a short battle on Gant's side.

Neil eventually gained the upper hand, and as he lifted his knife, a bolt of lightning left an image that would haunt Ema for years to come. Seeing someone about to be stabbed, Ema panicked, and, assuming the one holding the knife (Neil) was Darke, pushed Neil aside. Darke and Neil were knocked unconscious when Ema pushed them, and Ema passed out from shock. Right before she lost consciousness, another flash of lightning illuminated the jar sitting on one of Gant's shelves.

Shortly afterward, Gant arrived at the scene. He saw two men and a girl lying unconscious. Gant took advantage of the opportunity to cut off the piece of Neil's vest with Ema's handprint on it. He then did the unthinkable, lifting Neil's body and impaling it on the sword carried by a nearby suit of armor, killing him, with lots of blood spilling onto his chest. Gant then wrote Ema's name on the jar with Neil's blood. He broke the jar, and took the piece with the most legible part of Ema's name, and hid it in his locker with the cloth.

Shortly afterwards, Lana came in, and seeing the crime scene (unaware that it had changed), was horrified to find that Ema may have killed Neil. Being Ema's sister, Lana asked Gant to cover this up, and promised him she would do anything he asked. Gant started by telling Lana to arrange the crime scene. Lana wiped Ema's name off the jar fragments, broke off the tip of Darke's blade and placed it in Neil's wound (after taking his body off the armor), and moved the unconscious bodies to her side of the office. By doing this, Darke would be sure to get the Guilty verdict.

Neil was the prosecutor of the case, so Lana took the forged evidence to Miles Edgeworth, who replaced Neil. Edgeworth was unaware that the decisive evidence he was given was forged, the real evidence having been taken away by Gant.

Before the trial, Ema was too affected to talk about the crime, so she drew a picture of the events on the back of the evidence list and gave it to Goodman. Gant, however, got to it first and took the half of the picture that could identify on which side of the office the murder took place. Without any evidence to the contrary, Darke, framed for the murder of Neil, was found Guilty and was sentenced to death.

Gant and Lana, for different reasons, didn't want the detectives to investigate further. Since they were suspicious of what happened, Gant fired Angel and demoted Jake to patrolman. Goodman however, was able to keep his position, because firing or demoting him as well would have left the commissioners very suspicious.

The detectives were persistent however. They knew something was up. The decisive evidence they presented did not fit what they had at all, which hurt the three. Jake was the one suffering the most, since his brother had died. The trio decided to investigate behind Gant's back.

Gant, who had been promoted Chief of Police thanks to the trial, promoted Lana to become the Chief Prosecutor so he could control the Prosecutor's Office through her. Lana did just about anything Gant told her to do to protect Ema from going to jail, no matter how cruel and harsh it may have been. Lana developed a very cold personality since.

The "Joe Darke Killings" case was closed and was then known as the infamous "SL-9 Incident".

Two years later...
It was the morning of evidence transferrals, when the Police Department and the Prosecutor's Office got together for an awards ceremony, before transferring evidence from two-year-old cases down in the vaults, never to be seen again.

Bruce Goodman's ID was stolen (by Jake Marshall) that day, so he went to the Police Chief to fill out a missing item report. Gant then went with Goodman to the Evidence Room and they used his ID card to get in.

In the Evidence Room, Goodman asked Gant to reopen the SL-9 case so the evidence wouldn't be lost forever. He also said he wanted to take some of the evidence to Jake Marshall. Hearing this, Gant panicked, took Joe Darke's knife, and stabbed Goodman with it, killing him instantly.

Gant realized he was in trouble, so he grabbed what he could out of Goodman's evidence locker in a rush, and put Goodman's body inside the trunk of Miles Edgeworth's car. He left behind pieces of the broken jar and a rubber glove was caught between the locking sensor and the door, so it didn't close all the way. Then, he took a screwdriver from a random old case, and told Edgeworth himself to take it to his office. Wondering why, Edgeworth drove his car back to the Underground Parking Lot of the Prosecutor's Office, unknowingly carrying Goodman's body with him! Gant then contacted Lana Skye and told her to go to the Parking Lot and dispose of the body. Lana obeyed and walked to the Parking Lot.

Meanwhile, police officer Mike Meekins was assigned to protect the Blue Badger that day. He moved the cardboard figure to the Evidence Room, unknowingly covering a handprint Gant had left.

Jake Marshall stole Goodman's ID Card on the morning of the crime, because that day was his last chance to get at the evidence for the SL-9 case. He took Goodman's clothing and went to the Evidence Room just after the ceremony (where Edgeworth won the Prosecutor's Trophy). He opened Goodman's locker, unaware that there was a fingerprint sensor in place and the only reason the locker opened for him was that a glove had jammed the sensor. Unfortunately, nearly all the evidence was gone. He was then startled by Meekins, who was retrieving the Blue Badger from the room. Meekins, not recognizing Marshall, asked for his ID card. Fearing his cover would be blown, Marshall pulled out his knife instead. They fought a short battle, with Marshall the victor. Meekins' blood was splattered all over his disguise, so he threw the coat inside his locker to hide what happened.

Since Marshall was a security patrolman, he was aware of the camera in the Evidence Room, and he managed to leave the room undetected.

Around the same time, Lana got to the Underground Parking Lot and opened up the trunk of Edgeworth's car to find Goodman's lifeless body. She had to get rid of Darke's knife so that the SL-9 case wouldn't be brought back up, so she took Edgeworth's knife, which she found in his trunk, and stabbed the corpse in the same place Gant had stabbed him. She then took Darke's knife, wrapped it in her scarf, and put it inside the car's exhaust pipe. Lana then called Ema to tell her to dispose of the evidence in her muffler.

Angel Starr noticed the "crime" and captured the moment with her camera. Angel then proceeded to arrest Lana, who decided to pin herself for the crime, as speaking out would spell certain doom for Ema.

Epilogue
Now sit back as you enjoy the ending. At some point, you are given an opportunity to show something (evidence that neither Edgeworth nor Phoenix would have found on their own). Present the Evidence List, and when given the chance, yell "Objection!".

During the Credits, you'll notice that the green grid is on the touchscreen. However, instead of finding fingerprints, you'll find pictures!

After the credits, you are given another book to check. Examine the back cover for a cute photo.

Congratulations, you have just completed Phoenix Wright: Ace Attorney!


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text:
South Park: The Stick of Truth/The Bard

Exit Kupa Keep and you will be taken to the Inn of the Giggling Donkey. After a cutscene, Cartman will send you and Butters down to the basement to flush out the Bard. Make your way to the far left side nearest to the screen, where you will encounter the Bard (Jimmy). Defeat his elves, then use Cup-a-Spell to blow up the lamp. Shoot the window to let Craig in. Use Cup-a-Spell again on the candle to reach Craig, then lure the other elves into the electrified water. Use Butters to heal Craig, and he will unlock the basement door.

Exit the basement, then enter the kitchen on the left. Shoot the light on the ceiling to take out the elves attacking Cartman. He informs you the elves have taken Princess Kenny and instructs you to follow them upstairs and rescue her. Head back into the living room and shoot the light hanging from the ceiling, then use Cup-a-Spell on the candle to take out the spikes on the floor. Shoot the lamp by the front door over, then use Cup-a-Spell on the flame to let Token and Tweek in. Destroy the dresser, then defeat the elves blocking the stairs. The Elven Anti-Magician possesses a large amount of health and armor points, and is immune to magic attacks and the disgusted status affect, so take him out first.

After defeating the elves, head upstairs and enter the second room. Shoot the shelf with the bowling ball, then the light hanging from the ceiling, then the art on the wall. Break the desk on the right, walk up the ramp to the dresser, climb the piece of art like a ladder, head to the top of the cupboard, then slide down the wire onto the bed. Use Kenny's buddy ability to get the elf in the attic to drop down the ladder.

Enter the attic and use Cup-a-Spell on the flames to clear the room of hostiles. Shoot the chest on the right, then drop through the hole to enter the Bard's room.

The Bard has 1620 health points. He has four main attacks, detailed below:

 * Pied Piper's Song - Jimmy summons 3 rats to assist him in battle. Kenny's Furry Friends ability will dispose of them quickly.
 * Ranged Attack - Jimmy shoots you with his crossbow.
 * Lullaby - Puts Jimmy's enemies to sleep. You can mash the button onscreen to stay awake, but your buddy is unable to resist falling asleep.
 * The Brown Note - causes your Mana bar to fill up halfway each time if unblocked. If your Mana bar is full, you'll defecate uncontrollably and lose all of your Mana. Stun Jimmy with Dragonshout during the Song of Protection to avoid being hit with the Brown Note.

Upon defeating the Bard, you will regain possession of the Stick of Truth.

title: South Park: The Stick of Truth/The Bard


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Template:Phoenix Wright Testimony

 Testimony - -

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title: Template:Phoenix Wright Testimony


text: Category:Book:Phoenix Wright: Ace Attorney/Templates


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text:
The Legend of Zelda: Link's Awakening (2019 video game)/Dream Shrine

Dream Shrine
Head to Dream Shrine in north Mabe Village. Go inside and go to bed to enter the true shrine. To beat the enemies without breaking the cracks, use the Pegasus boots to dash through. In the center right of the chamber, the right chest contains a silver rupee worth 100 rupees. Break the crystal with the dash and head left to pick up the ocarina. Currently Link doesn't know how to play, but you can still use the ocarina to hear a comical tune.

Marin's Song
Head outside to meat Marin by the rooster statue. Talk to her and she will ask Link to accompany her in a song. Marin vocalizes the Ballad of the Wind Fish and Link manages to follow amazingly well. After confirming Link knows the song, Marin asks Link to remember it and to play it every so often.

On the phone Old Man Ulrira mentions the Angler Key can be found in the Yarna Desert.

Optional Tasks
After Key Dungeon visit Dampe's shack to update the chamber components.

Just east of Mable Village is a bombable cracked wall behind the trees next to the cliff ladder. The cracks are located the east side of the tree line, bordering the west field of holes with insect enemies that pop out to attack Link. Bomb the cracked rock to reveal a cave entrance. With the Pegasus boots Link can pop the crystals to make a way to the chest. To actually get to the chest, push the rock to the northwest of the chest to the west. Pop open the chest for a cool 50 rupees. Note the cracked wall on the east side of the room. Bomb it open and enter to find a chamber with a Piece of Heart.

In Mabe village Mamasha mentions that her husband is starving up in the mountains, where he is lost. This is just as he said earlier.

Tarin can be found recuperating from the honey incident at home, having lost his desire for honey.

The Trendy Game Shop has been updated.

With the Pegasus Boots in tow, visit the village library. Dash into the bookshelf to knock the red book down. This book is entitled The Hidden Power of Color and contains a hint for unlocking an optional dungeon.


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User:CompletePerson/minerva.css

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text:
Fractals/Iterations in the complex plane/analysis

How to analyze discrete dynamical system based on the rational map $$f$$ of one complex variable defined on the Riemann sphere ?

$$z_{n+1} = f(z_n)$$ $$f : \hat{\mathbb{C}} \to \hat{\mathbb{C}}$$

Find non-repelling periodic points

=Goals=
 * find critical points ( finite and infinite) = critical set
 * find periodic points (cycles), it's periods and stability
 * find relation between critical point, attracting points and their basin (domain of attraction)

=dictionary=
 * rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius transformations in both domain and range.

=Algorithm=

Algorithm by pauldelbrot "For rational maps, I'd suggest the following procedure would work:
 * Follow all critical orbits for some large number of iterations.
 * For each one: Run it a few thousand more iterations and see if it gets close to the landing points of any previous ones. If it does, discard it.
 * Divide the Riemann sphere into small regions in a grid; e.g. by keeping two square bitmaps, representing in one the inside of the unit circle and in the other the outside of it by representing the inside of the unit circle for w = 1/z.
 * Color the pixels in these bitmaps containing the remaining critical orbit landing points red, green, blue, etc.; maybe use a conservative distance estimate at each one to widen this to a small disk of pixels.
 * Apply the original algorithm, only instead of looking for pixels to hit white or not, you look for them to hit all those various colors, and propagate those colors.

In the end you should have colorful filled-in basins of attraction along with grey along the Julia set itself. Conversion of the two bitmaps into a suitable visual representation of a sphere is left as an exercise for the reader."

If local dynamics near fixed point is to hard
 * move fixed point to zero
 * (to do)

=steps=
 * compute degree of the map
 * compute first derivative wrt the variable z : $$f'(z) = \frac{d}{dz}f(z) $$
 * compute critical points
 * compute attractors ( attracting periodic cycles) and their periods as a limits of critical orbits
 * compute multiplier for each attractor
 * make image

degree
The degree d of rational map f is the maximum of degrees of its denominator and of its numerator, provided they are relatively prime:

$$ f(z)=\frac{p(z)}{q(z)}$$ $$d = deg(f) = max\{deg(p), deg(q)\} \ge 2$$

Definition
Critical point on the Riemann sphere is a
 * point z where f is not locally one-to one = fails to be injective in any neighbourhood of z, and f is not constant
 * points at which valency ( order) is grater then 1

A critical point of $$f(z)$$ is a:
 * finite point z satisfying $$f'(z) = 0$$
 * poles of f of order 2 or higher
 * point at infinity z = ∞
 * if the degree d of $$f(z)$$ is at least two
 * if $$f(z) = 1/g(z) + c$$ for some c and a rational function $$g(z)$$ satisfying this condition.

The Riemann-Hurwitz formula
Number of critical points n counted with appropriate multiplicity on the Riemann sphere  is

n = 2*d-2 Where d = degree of the function

This is an upper bound (maximal number because of counting multiplicities), so the function can have less critical points

How to check if infinity is critical point ?
Evaluate function f

$$ f(z) = \frac{p(z)}{q(z)} $$

at point in infinity

There are 2 cases
 * $$f(\infty) = \infty$$ so infinity is a fixed point of f
 * $$f(\infty) \ne \infty$$ so infinity is not a fixed point of f

The degree of
 * of numerator $$p(z)$$ is $$n$$
 * of denominator $$q(z)$$ is $$m$$.

fixed point
The infinity is a fixed point of f

$$f(\infty) = \infty$$

then the $$\infty$$ is a critical point of $$f$$ if

$$n > m + 1$$ Example

$$f(z) = \frac{z^3 +2z +3}{z - 1}$$

kill(all); remvalue(all); display2d:false;

define(f(z), (z^3+ 2*z + 3)/(z - 1));

/* first derivativa wrt z */ define( d(z), diff(f(z),z,1));

/* hipow does not expand expr, so hipow (expr, x) and hipow (expand (expr, x)) may yield different results */ n : hipow(num(f(z)),z); m : hipow(denom(f(z)),z);

/* check if infinity is a fixed point */ limit(f(z),z,infinity);

/* finite critical points */

s:solve(d(z)=0)$ s : map(rhs,s)$ s : map('float,s)$ s : map('rectform,s)$

not a fixed point
The infinity is not a fixed point of f

$$f(\infty) \ne \infty$$ look at the function

$$w \mapsto f(1 / w)$$. The derivative is given by

$$-\frac{f' (1/w)}{w^2} = -\frac{p'(1/w) q(1/w) - p(1/w) q'(1/w)}{w^2 q(1/w)^2}$$

then if:
 * $$n = m = 1$$ then $$\infty$$ is not a critical point of $$f$$.
 * $$n = m > 1$$ then if
 * the numerator is of degree $$2 m - 2$$ and $$\infty$$ is not a critical point of $$f$$.
 * the numerator is of degree $$< 2 m - 2$$ then $$\infty$$ is a critical point of $$f$$.
 * $$n = m - 1$$ then $$\infty$$ is not a critical point of $$f$$.
 * $$n < m - 1$$ then $$\infty$$ is a critical point of $$f$$.

Attractors

 * Find attractor ( attracting periodic points, cycle) as a limit set of critical orbit
 * find period of a cycle

How to check if infinity is fixed point ?
Here infinity has period 2 = is not a fixed point a: -3-3*%i; /* d */ c: 0.0; define(f(z), 1/(z^3+ a*z + c));

(%i4) limit(f(z),z,infinity); (%o4)                                 0 (%i5) limit(f(z),z,0); (%o5)                             infinity

Here infinity is not a fixed point remvalue(all); display2d:false; define(f(z), 1/(z^3+ 2.099609375*z + 0.349609375)); (%i5)limit(f(z),z,infinity); (%o5) 0 (%i6) limit(f(z),z,0); (%o6) 2.860335195530726

Here infinity is a fixed point define(f(z), (z^3+ 2*z + 3)/(z - 1)); n : hipow(num(f(z)),z); m : hipow(denom(f(z)),z); (%o3) f(z):=(z^3+2*z+3)/(z-1) (%i4) (%o4) 3 (%i5) (%o5) 1 (%i6) limit(f(z),z,infinity); (%o6) infinity

Basins = Fatou domains

 * Each of the Fatou domains ( basin of attraction) contains at least one critical point of $$f(z)$$
 * each attracting, superattracting and parabolic cycle attracts a critical point
 * there is only a finite number of Fatou domains :
 * rational function of degree d cannot have more than 2(d—1) cycles of stable regions
 * "the basins of attractions of distinct periodic cycles are disjoint. As a rational map of degree d has at most 2d−2 critical points, the number of attracting and parabolic cycles of f is bounded by 2d − 2"
 * A Fatou domain can contain several critical points
 * Each domain of the Fatou set of a rational map can be classified into one of four different classes.
 * number of componnets

=Basin data =
 * function ( input)
 * period of attracting cycle
 * one critical point which falls into the attracting cycle and is in the same component as the attracting point ( see below)
 * attractor = one point from the attracting cycle, which is in the same component as above critical point
 * multiplier of the attracting cycle

If one have above data for each basin then it is possible to make image of dynamic plane

=Images=
 * critical orbits ( forward orbit of critical points)
 * periodic points ( mostly attracting cycles )
 * basins of attraction

=Examples=
 * Rational maps
 * Invariants and Parameter Space Models for Rational Maps by Anastasia Shepelevtseva 2022

=Programs=
 * m-describe from mandelbrot-numerics library by Claude Heiland-Allen: gives description of point c from the parameter plane of complpex quadratic polynomial
 * Describe iterated map (C++) : determine numerically the cycles of a polynomial Julia set by marcm200
 * C program for analyzing and drawing dynamical systems based on the rational maps by Michael Becker, 8/2003

GRPF: Global complex Roots and Poles Finding algorithm

 * GRPF: Global complex Roots and Poles Finding algorithm
 * Matalb code by Piotr Kowalczyk
 * Julia code

visualisation

 * Mark McClure - JavaScript
 * Jux - Julia fractal explorer by xenodream - binary program without src
 * Mandel by Wolf Jung - c++ code
 * Xaos
 * | Wolfram Alfa : julia set f(z) = (z^2-0.2+0.7*I ) /(z^2 + 0.917)
 * RatioField by Gert Buschmann - pascal code
 * Luis Javier Hernández Paricio
 * Plotting basins of end points of rational maps with Sage by Luis Javier Hernandez Paricio, Miguel Maranon Grandes, Marıa Teresa Rivas Rodrıguez
 * Plotting Basins of a Univariate Rational map in Julia by Luis Javier HERNÁNDEZ PARICIO
 * Global Visualization of Basins of Univariate Rational Functions with Julia by Luis Javier Hernández Paricio
 * An implementation in Julia language for the iteration of rational functions on the Riemann sphere. Applications to numerical methods - Extremiana José Ignacio, José Manuel Gutiérrez, Luis Javier Hernández Paricio, María Teresa Rivas Rodriguez

Maxima CAS code
Steps
 * computes first derivative d(z) using symbolic methods
 * computes ( only finite) critical points, as a roots of equation d(z) = 0 using symbolic methods
 * computes forward orbit of critical points and make images
 * visual inspection of images gives approximated period of the attractor

kill(all); remvalue(all); display2d:false;

/* map */

define(f(z), (z^2)/(z^9 -z + 0.025));

/* first derivativa wrt z */ define( d(z), diff(f(z),z,1));

GiveOrbit(z0,iMax):= /*   computes (without escape test) (forward orbit of critical point ) and saves it to the list for draw package */ block( [z,orbit,temp], z:z0, /* first point = critical point z:0+0*%i */ orbit:realpart(z),imagpart(z), for i:1 thru iMax step 1 do        ( z:f(z), z:float(z), z:rectform(z), z:float(z), if (cabs(z)>3) then break, /*if (cabs(z)< 0.00001) then break, */ orbit:endcons([realpart(z),imagpart(z)],orbit)), return(orbit) )$

GiveAttractor(z0,iMax):= /*   computes (without escape test) (forward orbit of critical point ) and saves it to the list for draw package */ block( [z,orbit,temp], z:z0, /* first point = critical point z:0+0*%i */ orbit:[], for i:1 thru iMax step 1 do        ( z:f(z), z:float(z), z:rectform(z), z:float(z) ), for i:1 thru 10 step 1 do       ( z:f(z), z:float(z), z:rectform(z), z:float(z), if (cabs(z)>3) then break, /*if (cabs(z)< 0.00001) then break, */ orbit:endcons([realpart(z),imagpart(z)],orbit)), return(orbit) )$

GiveScene(attractor):= gr2d(title= "Possible cycle",	user_preamble = "set nokey;set size square;set noxtics ;set noytics;",       point_type    = filled_circle,	points_joined = false,        point_size    = 0.7,        /* */	color		  =red,	points(attractor)        )$

/* critical points

[-0.8366600265340756*%i,0.8366600265340756*%i]

s:solve(d(z)=0)$ s : map(rhs,s)$ s : map('float,s)$ s : map('rectform,s)$

s2 : allroots(s[2])$ s2 : map(rhs,s2)$ s2 : map('float,s2)$ s2 : map('rectform,s2)$ s : cons(s[1], s2);

MyOrbits:[]; for z in s do ( 	print(i,z),    	orbit : GiveOrbit(z,30),  	MyOrbits:endcons([discrete,orbit], MyOrbits)

)$

MyAttractors:[]; for z in s do (   	attractor : GiveAttractor(z,300),    	attractor : GiveScene(attractor),  	MyAttractors:endcons(attractor, MyAttractors)

)$

load(draw); path:""$ /* pwd, if empty then file is in a home dir, path should end with "/" */ fileName: sconcat(path, "cycles")$ draw(	terminal = png,	file_name = fileName,	columns= 4,	MyAttractors );

=Difference between polynomial and rational maps=

Infinity
 * for polynomials infinity is superattracting fixed point. So in the exterior of Julia set (basin of attraction of infinity) the dynamics is the same for all polynomials. Escaping test ( = bailout test) can be used as a first universal tool.
 * for rational maps infinity is not a superattrating fixed point. It may be periodic point or not.

Critical points
 * The theorem of Lucas states that the critical points of a polynomial in the complex plane lie within or on the convex hull of the zeros.
 * The critical points of B(Z) in the interior of the disc lie within or on the (non-Euclidean) convex hull of the zeroes of B(Z), with respect to the Poincar´e metric.

Visualisation
 * "In order to graphically study these sets which, as we shall see are four dimensional, we employ a range of techniques for visualizing high dimensional objects more commonly reserved for studying 3D volumes arising from MRI or CT scans, techniques known as volume visualization."

=Problems=
 * overflows caused by denominators close to zero. Solution : "The use of normalized homogeneous coordinates avoids overflow and underflow errors " Luis Javier Hernández Paricio.
 * "the representation of a rational function by a pair of homogeneous polynomials of two variables and with the same degree allows us to compute the numerical value of the function at any pole point and at the point at infinity" Luis Javier Hernández Paricio.

=How can you read location from the image ? =
 * m-location-analysis ( parameter plane images) by Claude Heiland-Allen

=See also=
 * How to construct a map with desired properities ? ( inverse process)
 * Counting Hyperbolic Components by Jan Kiwi, Mary Rees. We give formulas for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.
 * Hyperbolic components of rational maps: Quantitative equidistribution and counting by Thomas Gauthier, Yûsuke Okuyama, Gabriel Vigny
 * On the classification of critically fixed rational maps by Kristin Cordwell, Selina Gilbertson, +2 authors Samantha Pinella Published 2013 Mathematics arXiv: Dynamical Systems, DOI:10.1090/S1088-4173-2015-00275-1Corpus ID: 51788955
 * On Rational Maps with Two Critical Points J. Milnor Published 2000 Mathematics, Computer Science Experimental Mathematics, DOI:10.1080/10586458.2000.10504657Corpus ID: 18618158
 * mathoverflow.net question: what-are-the-shapes-of-rational-functions?
 * Remarks on quadratic rational maps by John W. Milnor 1992
 * GLOBAL MAPPING PROPERTIES OF RATIONAL FUNCTIONS by Cristina Ballantine1 and Dorin Ghisa
 * analyzing polynomial functions

=References=
 * ON THE ZEROS AND CRITICAL POINTS OF A RATIONAL MAP by XAVIER BUFF
 * Iteration of rational functions (Talk Notes) by Erin Pearse
 * Symmetries for Julia sets of rational maps by Gustavo Rodrigues Ferreira
 * Remarks on Quadratic Rational Maps. by J. Milnor Stony Brook, August 1992
 * On Rational Maps with Two Critical Points by J. Milnor
 * Bounded Hyperbolic Components of Quadratic Rational Maps by Adam Lawrence Epstein September 1, 1997


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South Park: The Stick of Truth/It's Late

Exit Kupa Keep and head left. Enter your house and climb into bed.


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Donkey Kong Country 2: Diddy's Kong Quest

Donkey Kong Country 2: Diddy's Kong Quest is the sequel to the best-selling Super Nintendo game, Donkey Kong Country. The game, like its predecessor, was developed by Rare Ltd. (a second party developer for Nintendo at the time) and published by Nintendo. The game was first released in November 1995. In 2004, a remake of Donkey Kong Country 2 was released for the Game Boy Advance, dropping the Diddy's Kong Quest subtitle.

In the backstory, Donkey Kong was kidnapped by K. Rool, who this time dons a pirate outfit and has a new alias, Kaptain K. Rool. He demands the Kong Family to give him the Banana Hoard in exchange for Donkey Kong's freedom. They refuse, so Diddy his girlfriend Dixie Kong go on a journey to rescue him.

Table of contents

 * General information
 * /Barrels/

 * Supporting locations
 * Monkey Museum
 * Kong Kollege
 * Funky's Flights II
 * Swanky's Bonus Bonanza
 * Klubba's Kiosk
 * Gangplank Galleon
 * Pirate Panic
 * Mainbrace Mayhem
 * Gangplank Galley
 * Lockjaw's Locker
 * Topsail Trouble
 * Boss level: Krow's Nest


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Donkey Kong Country 2: Diddy's Kong Quest/Supporting locations/Kong Kollege

Wrinkly Kong, Cranky Kong's wife, runs a small school named Kong Kollege, where you can save your game and get helpful information.

Saving Your Game
Saving your game is one of the most critical actions in the game. Saving usually costs 2 Banana Coins, but sometimes if it's your first time saving, you can save for free!

Information
You can get a lot of information from Wrinkly Kong at each save point. The information she usually has is about how to perform a certain move, what a certain item does, how to perform moves with an animal, or how to defeat the upcoming boss of the area you are in.


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Donkey Kong Country 2: Diddy's Kong Quest/Barrels

There are many different types of barrels in Donkey Kong Country 2: Diddy's Kong Quest. The ones in the game are listed below.

Basic Barrel
This normal type of barrel is just made for throwing at enemies. It has no fancy artwork on the front of it.

Invincibility Barrel
Like the name says, gives Diddy or Dixie invincibility for a limited amount of time. Has an exclamation point on the front.

Donkey Kong Barrel
If one of your partners is missing, throw this barrel to get them back. Has a DK symbol on the front

Animal Barrel
If you jump into one of these barrels, it will morph you into the animal. Like if there was a picture of Enguarde on the front, you would morph into Enguarde.

+ Barrel
The + Barrel is used in the level Haunted Hall. It is used to add time before Kackle will attack you. Has a + sign on the front of it.

- Barrel
The - Barrel is almost identical to the + Barrel, except that it removes time before Kackle attacks you. Has a - sign on the Front.

Bonus Barrel
The Bonus Barrel sends you to a bonus game when you jump into it. These barrels are usually hidden somewhere in the level, so you will have to look. It has a B button on the front.

Check Barrel
The Check Barrels are used in the level Target Terror. You must try to jump and hit these barrels to open the path ahead. Has a checkmark on the front.

X Barrel
Like the Check Barrel, except blocks your path. Has an X on the front.

Halfway Barrel
This barrel marks the halfway point of the level, and if you die, you can restart here. Has stars moving around the barrel.

TNT Barrel
This barrel will blow up if you throw it at things. If you set it down, it will blow up by itself. Also, it can open up secret passages in walls. Has TNT written on the front of it.


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Donkey Kong Country 2: Diddy's Kong Quest/Supporting locations/Funky's Flights II

Funky's Flights II is the same thing as Funky Flights from Donkey Kong Country. The only difference is that you have to pay money most of the time. Funky's Flights II usually costs 2 Banana Bunch Coins, except for the first world, which is always provided free of charge. With Funky's Flights II, you can travel to any world that you have already been to without beating any bosses. This is very useful if you need some extra lives or something for a harder world.


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Donkey Kong Country 2: Diddy's Kong Quest/Supporting locations/Swanky's Bonus Bonanza

Gangplank Galleon
There are three quizzes, which check your general knowledge of the game. The reward is extra lives.

Swankey's treasure (free, reward 1 life):
 * 1) Swankey
 * 2) Rat
 * 3) Gangplank Galleon (French: Galion de la Galère)

Pirate puzzle (1 token, reward 2 lives)
 * 1) Y button to run
 * 2) Pirate Panic (French: Panique à bord)
 * 3) Kaptain K Rool

Challenge of chimps (2 tokens, reward 3 lives)
 * 1) 4 windows in Kaptain's Krool's cabin
 * 2) Mainbrace Mayhem (French: Meli-nelo dan les mats)
 * 3) A pistol


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Pirate Panic

The first level of the game isn't too difficult, nor complex. It is a simple run from left to right.

Near the start of the level is a door to your left. This room contains a 1-up balloon and a letter saying that Donkey Kong was Kidnapped. The room also serves a secondary purpose for patient players that wish to farm bananas and banana tokens, which are regenerated when you exit the room.

As you continue right, you will encounter various features of the map, which only pose limited difficulty:

 * The letter K is in plain sight, where you see two groups of two barrels.
 * A banana token can be found above and to the right of the large A spelled in bananas. You need to throw your companion upward to reach it.
 * The letter O is in plain sight, when the deck lowers.
 * When you see a break in the deck, jump down to reach the find the token minigame. There is a banana token here as well, but the main token is found at the top right and is reached by just jumping along the barrels.
 * The letter N is also in plain sight on the main deck, but you'll have grabbed it through the minigame entrance.
 * You can find a rhino companion at the top of a stack of barrels. If you bring him to the end of the level, you can get a 1-up balloon.
 * When you see a left arrow, use the rhino to break down the door by charging it (hold down the A button. In this minigame, eliminate all enemies from left to right - the fastest way to do it is to charge and jump to clean them out on one sweep.
 * There is also a column of bananas to the right. Throw your companion upward to collect the DK Token.
 * G is also visible.


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Mainbrace Mayhem

When you head right, you may see a right arrow pointing towards a net. The net contains bananas, but you require using a long jump to reach it (using Dixie Kong). Head across the net, and jump further to the right for a Bonus Barrel.

Jump up and to the right, and climb the first rope. You should see the letter K there. Head left, until you reach a rope ladder, where you see a banana token guarded by a climbing enemy. To hit this enemy, you need to jump off the side of the cargo net, and move to the left after the jump to land on top of him.

The letter O should be visible before you reach the halfway barrel.

Left of the halfway barrel is a series of eight rats, which are easily killed. If you did get injured, there is also a DK barrel to revive your companion.

Climb up the rope, and reach the top, where you see a horizontal rope. When you are at the mast, jump to be launched to a platform containing a 1-up balloon. Return and continue to the right, and look for the letter N on the far right of that level before continuing up.

When you see a cannonball, pick it up, and carry it to the left. Place it in the cannon to enter a minigame to defeat five rope climbers within a time limit for a Krool token.


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Gangplank Galley

A Bonus Barrel to the find the token minigame is located above the starting position. In this minigame, you need to pickup each chest, and throw it at the bee to break it open. The token position is randomly determined.

Head to the right. The first chest you see contains bananas.

When you see a hook, you can use it to reach the letter K.

The second chest has a 1-up balloon.

The second set of hooks leads to a DK barrel.

There is a barrel that will launch you to a high platform. Jump across the hooks here to reach to reach the letter O.

The third chest has a banana token. You will find it just before the halfway barrel.

The letter N is above the stack of barrels.

When you reach the gap, you will have to use the hook to jump across. Alternatively, you can use Dixie Kong spin while jumping to reach a launcher located to the left of the gap, where you will collect some bananas and a banana token.

When you get the invulnerability barrel, run to the right. Skip the chest. When you see a tall stack of barrels, take advantage of the invulnerability and reach the top of the stack for a bonus barrel. This token minigame requires you to jump from hook to hook. Once you resolve the minigame, or if you need breakable items to take out the blue guards, you can use the chest to collect the letter G.


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Lockjaw's Locker

When entering this stage, you can collect bananas if you follow the left/right wave.

After landing, proceed to the right. In the second pool of water, you will find the letter K. Directly below the K is a secret tunnel that contains two banana tokens. It is slightly longer than taking the direct approach, but still helps. Once the chamber is flooded, you may backtrack for some additional bananas and tokens.

You can find a swordfish, and the letter O if you follow the ground. Once you turn up and head to the left, you will have to leave the swordfish behind, but can get a 1-up balloon if you bring him to the sign. Break the halfway barrel.

The letter N is directly above the half-way barrel. Proceed left, and if you head down, you can find a replacement swordfish.

After the second upward trip, you can find the letter G at the top-left corner.

At the top of the level, exercise caution when heading to the right. When the corridor narrows just before entering a large room, have the swordfish perform a thrust as far as possible. This will bring you to the DK coin.


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Topsail Trouble

With the snake, you can reach bananas above the start, as well as the letter K to the left. Jump up to the next platforms. then head right to jump up again. At the top of this mast, you can reach a bonus game to kill dragonflies, but you need to jump off the snake and throw your companion to reach this barrel.

The letter O also requires a snake jump.

Past the half-way barrel, you will start to climb ropes. On the second set of ropes, head to the right and climb down for the letter N.

The letter G is available at the end of the level, by timing your jump correctly.


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Donkey Kong Country 2: Diddy's Kong Quest/Levels/Krow's Nest

This is a boss battle against the Krow.

The Krow will pick up an egg and drop it. Run under it as it bounces towards you, then pick it up and throw it back at the Krow. Repeat this once.

Next, the Krow will ram its next, causing the eggs to fall by themselves. One egg will stop, allowing you to grab it and throw it at the Krow as it flies around in circles.

Once the Krow is hit four times, it will fall down.

If you are able to jump on the Krow's nest itself, you will find two banana tokens.


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