Rachel Summers

Rachel Anne Summers (also known as Rachel Grey) is a fictional character appearing in American comic books published by Marvel Comics. The character was created by writer Chris Claremont and artist/co-writer John Byrne.
In her first appearance, the character's surname was not revealed; later publications and retcons further expanded her backstory to involve central characters of mainstream continuity. She is the daughter of the alternate future counterparts to Cyclops (Scott Summers) and Jean Grey-Summers from a harsh dystopia, the sister of Nate Grey and half sister of Cable, a niece of Havok and Vulcan, and a powerful mutant in her own right. She inherited her mother's vast telepathic and telekinetic talents, and code names Phoenix and Marvel Girl before later adopting her own code name Prestige. Although the character is considered unique to the Marvel Comics "multiverse", her name has been used to designate the mother of Marvel characters Hyperstorm and Dream Richards in respective timelines.
Publication history.
Rachel first appeared in "The Uncanny X-Men" #141 (January 1981) and has since been affiliated with several comic book superhero teams including the X-Men and Excalibur.
Fictional character biography.
Future adolescence.
Rachel Anne Summers comes from an alternate future Earth known as Earth-811, as seen in the "Days of Future Past" storyline from "The Uncanny X-Men" #141–142. In this reality, the assassination of Senator Robert Kelly provoked the ratification of the Mutant Registration Act, leading to a dystopian future where the mutant-hunting Sentinel robots rule North America. Rachel was abducted by operatives working for Ahab, who used drugs and hypnotherapy to turn Rachel into a "Hound", a mutant who tracks down other mutants. She fulfilled her duties, but her psychic powers linked her to her victims; fueling her grief and despair until she attacked Ahab and scarred him. In return, he sent her to the mutant concentration camps. There, she befriended the surviving mutant rebels, including Wolverine, Magneto, Colossus, Storm, Kate Pryde, and her lover, the adult Franklin Richards.
Rachel managed to send Kate's consciousness into the past to her younger self to prevent the assassination, but it did not change their time. Rachel sent her astral form into the past to find out why and discovered she had sent Kate into an alternate past. On the way back, she encountered the disembodied Phoenix Force and it followed her to her present. Rachel passed out from the strain of astral projection and the Phoenix Force revealed itself to Kate, who asked it to give Rachel a fresh start.
When Rachel and Kate broke into "Project: Nimrod" on a suicide mission to destroy a new model of Sentinel, they became trapped. When Kate spoke the words "Dark Phoenix," the Phoenix Force ripped Rachel from her timeline and sent her body back to the alternate past to which she had sent Kate's consciousness. This was a past where she learned Jean Grey was dead and that her father was married to someone else. Rachel experienced additional heartache and displacement trauma when she discovered that her father's new wife, Madelyne Pryor, was pregnant with a son (Nathan Summers), because in her timeline she was the first-born child of Scott Summers.
X-Men.
Rachel had a brief membership in the X-Men before finding a Shi'ar holoempathic crystal at her grandparents' home. The crystal was imprinted with a portion of Rachel's mother's essence inside it as a tribute to the family shortly after Jean Grey's death. After Rachel took a vow to remember her mother with the uniform and name of Phoenix, the Phoenix Force fully bonded with her. She was granted access to its power on a cosmic magnitude, albeit in a much more limited fashion than the Dark Phoenix. Soon after, the grudge which she had begun with Selene boiled over when Rachel secretly invaded the Hellfire Club. She did this with the intention of taking vengeance on Selene for the murders she had committed, particularly that of nightclub owner Nicholas Damiano, who had previously taken Rachel into his home after Selene had attacked her. Selene proved to be no match for Rachel's newly increased powers, but just as she was about to finish Selene, Wolverine arrived and was forced to stab Rachel in the chest. Critically injured, Rachel was lured into Spiral's "Body Shoppe."
Excalibur.
Months later, while recuperating from injuries on Muir Island, Shadowcat and Nightcrawler both had the same dream, where they were actors on a weird set and helped Rachel, who was trapped there, escape. Shortly thereafter, Rachel escaped from the alternate reality of Mojoworld. Rachel has once been cited having a flashback to her time there where she is held in chains and tortured. The three former X-Men were joined by Captain Britain and Meggan and founded the British superhero team Excalibur. While part of the team, she discovered that this universe's version of her mother, Jean Grey, was alive. She attempted to bond with Jean, but Jean, upon discovering Rachel was the present host for the Phoenix, rejected any contact with her. Jean still resented the Phoenix Force for stealing a portion of her life. She also rejected Rachel because she felt that Rachel's existence was a constant reminder of the dystopian future she feared could still come to pass. Eventually, however, Jean moved past those feelings and formally welcomed Rachel into her life.
Askani.
Rachel remained with Excalibur until an incident caused Captain Britain to be lost in the timestream. She exchanged places with the time-lost Captain Britain and emerged two thousand years in the future, in a world conquered by Apocalypse and crushed under his iron fist. She gathered together a group of rebels and founded the Askani. She trained one of her followers to travel back in time and bring her "brother" Nathan forward in time when he was infected with a techno-organic virus. The Askani cloned Nathan in case he was not able to survive the virus. Apocalypse's followers attacked the Askani and took the clone (who would later become the supervillain Stryfe), leaving Rachel critically injured. Hooked up to life support, she drew the minds of Scott and Jean into the future, as "Slym" and "Redd", to raise Nathan and tutor him in the use of his powers. Rachel finally died ten years later and sent Scott and Jean back to their original bodies seconds after they had left.
After Nathan, now known as Cable, had finally defeated Apocalypse, he went into the timeline to retrieve Rachel. There, he discovered a Rachel "sans" Phoenix Force. With the premature death of Apocalypse, the Askani timeline had been diverged from the mainstream Marvel Universe, Earth-616. As a result, she had been flung into the far future, yet subjectively a short time after she had been lost in the timestream, as the slave of a creature called "Gaunt", who had used her to lead Cable there for a "battle of the ages". Cable defeated Gaunt in the battle and Rachel, now free, was able to use her residual Phoenix Force to return them both to the present. She then decided to take a break from superheroics and enrolled in college after she made Cable promise he would not tell anyone she was back. Despite her efforts to live a normal life, however, she was kidnapped by the telepath Elias Bogan and subsequently rescued by the X-Men.
Rachel Grey.
She decided to rejoin the X-Men, taking the name "Marvel Girl" to honor her mother (who had recently died yet again) and wearing a costume her mother had designed but never worn: a variation on Jean's first green costume. She also changed her last name to "Grey", possibly to express disapproval at her father's betrayal of Jean, as well as his continuing relationship with Emma Frost; though she and Emma made a truce of sorts during one of the team's missions in Hong Kong. Rachel and Nightcrawler began to have an attraction towards each other, kissing at one point, but nothing came of it as Nightcrawler also had an attraction to Storm at the time, who was in somewhat of a romantic "friendship" with Wolverine. Her stint with this team also included a visit to the Savage Land. In the storyline "World's End", which was heavily criticized by readers, Rachel was subjected to the mind-control of a tribe of advanced dinosaur people, the Hauk'ka, causing her to believe she belonged to their species. Afterward, she subconsciously used her telekinesis to change her own genome in their image. This was eventually reverted. After "House of M" and "Decimation", where most of the world's mutants lost their powers, the government had Sentinels instituted at the X-Mansion to protect the mutants in case any enemies used this low point to attack. Though their intentions were good this time, it reminded Rachel too much of the previous timeline when Sentinels herded mutants into concentration camps.
Rachel had a short stint with newly re-formed Excalibur, reminiscent of the former team, shortly after the "House of M" events. She assisted the team in battling Shadow-X and the Shadow King (in the guise of Professor X).
End of Greys.
Rachel spent some time with her grandparents, bonding with her grandfather. At a family reunion with all her relatives, a commando unit under the order of the Shi'ar attacked the party, killing everyone in an effort to wipe out the Grey genome. The commando unit was unable to kill Rachel; instead, one member was able to graft a "deathmark" on her back that would allow them to find her wherever she went. It is assumed that the only remaining member of the Grey family now left on Earth besides Rachel is Cable. Afterward, at the graves of the Grey family, Rachel vowed a terrible vengeance on the Shi'ar and was quoted as saying, "I'm not my mom. I'm not the Phoenix. I'm my own woman. And by the time I'm done... they'll wish I "was" the Phoenix."
The Death Commandos later tracked Rachel to the psychiatry office of Dr. Maureen Lyszinski. Rachel, with the help of Psylocke, Nightcrawler, Bishop, and Cannonball, saved the doctor and took down the Death Commandos. She decided to imprison them, instead of killing them, by telling them, "I mean to find destiny in a way that brings us both [Jean Grey] honor." She is also sometimes referenced as "Starchilde" in this series.
Rise and Fall of the Shi'ar Empire.
After Rachel was kidnapped, along with Cyclops, by her paternal uncle Vulcan, he freed Darwin from inside him. Later, Professor X recruited Rachel, along with Havok, Nightcrawler, Warpath, Darwin, and Polaris, for a space mission to stop Vulcan from laying waste to the Shi'ar empire. Xavier, who recently was stripped of his powers, recruited Rachel to serve as his telepathic "eyes and ears" during their mission. Aware of Rachel's vendetta against the Shi'ar, Xavier agreed to use their trip into space to find out who in the Shi'ar Empire gave the order to wipe out all members of the Grey family, and he warned Rachel that they will deal with the people responsible for her recent losses Xavier's way.
While in space, the team was attacked by Korvus, a Shi'ar warrior sent to kill Rachel. Korvus' ancestor, Rook'shir, was a previous host of the Phoenix Force, and a small portion of the Phoenix's power was left behind in his sword, the Blade of the Phoenix. With this power, Korvus made short work of the other X-Men, but when Rachel blocked the sword, their minds were involuntarily linked. Through this link, Rachel learned that Korvus' family was also murdered by the Shi'ar government because of their connection to the Phoenix. The remaining echo of the Phoenix power from the sword was then transferred to Rachel. Rachel claimed that rather than having taken the power, the power chose to go to her, saying, "The Phoenix knows me, remember? It likes me." When this happened, Rachel's normally gold energy aura turned blue, the same color as the Blade of the Phoenix. She then telekinetically disabled an explosive implant that the Shi'ar chancellor was using to force Korvus' obedience.
Due to Rachel's connection to Korvus through the sword, she discovers the Phoenix Force formerly in the blade is just an echo, a "blue shadow", of the Force. The shadow of the Phoenix begins influencing Rachel's behavior, causing her to design a new darker uniform and begin a romance with Korvus. She soon breaks off the relationship after she realizes their bond is only because of the residual Phoenix Force.
Leading up to the fight with Vulcan, Rachel is shown using her powers to kill the guards who stand in her way. Havok warns her not to, but Rachel tells him that they deserve to die after what they did to her family. When it comes to the big fight, Rachel shows just how powerful she is by protecting Korvus from one of Vulcan's blasts. Rachel is one of the X-Men stranded in Shi'ar space when their ship is sent back to Earth.
After the death of her other grandfather, Corsair, at the hands of Vulcan, she, along with Havok, Polaris, Korvus, Ch'od, and Raza, become the new Starjammers. They elect to remain in Shi'ar space and restore Lilandra to the throne or die trying. As her uncle states, "If they fail, he has no doubt that Vulcan will head for Earth."
Starjammers.
During the conflict, the Starjammers find another threat in the form of the Scy'ar Tal (translates as "Death to the Shi'ar"). Rachel makes contact with the eldest Scy'ar Tal and discovers their true origin. The Scy'ar Tal were originally called the M'Kraan. Early in their history, the Shi`ar attacked them, killing a great number of their people and making the rest flee for their lives. Eventually, the Shi'ar settled on their planet, took the M'Kraan Crystal as their own, and passed down the legend of the M'Kraan Crystal as a sacred gift from their deities, Sharra and K'ythri. The M'Kraan then changed their name to Scy'ar Tal and devoted their culture and society to the destruction of the Shi`ar Empire. With their first attack, they destroyed Feather's Edge by transporting a star to obliterate it. After which, Vulcan made contact with the Starjammers to call a temporary ceasefire.
During the ceasefire, Rachel comes into contact with the Death Commandos again and attempts to kill them to avenge the deaths of her relatives; however, she is deterred by Polaris. At the end, all the Starjammers are captured by the Shi'ar except Rachel, Korvus, and Lilandra.
X-Men: Kingbreaker and War of Kings.
Rachel and the Starjammers play a large role in the sequel to the "Emperor Vulcan" miniseries called "X-Men: Kingbreaker". She is also seen prominently in the "War of Kings" storyline, which features Vulcan, the Inhumans, Nova, and the Guardians of the Galaxy.
While with the Starjammers, in battle with Vulcan's new guard, the fragment of the "blue" Phoenix within her and Korvus' blade mysteriously leaves them. After the Phoenix echo leaves Rachel, she says "please... not now... Mom." From this frame onward, the "hound" markings reappear on Rachel's face.
In agreement with the Inhumans, the Starjammers and the Guardians of the Galaxy assault a Shi'ar vessel in order to free Lilandra, hoping to end the conflict while restoring her to the throne. Even without her Phoenix powers, Rachel is powerful enough to entrap Gladiator in an illusion in order to keep him distracted from battle. Their gambit pays off and the group is able to free Lilandra.
Rachel is next seen as Lilandra's bodyguard along with the rest of the Starjammers. On the home planet of the Shi'ar, Lilandra assumes her throne, but while making a ceremonial gesture is killed by the murderer known as Razor, who possesses the Darkhawk armor. The only person who perceives this is Rachel, since Razor is shielded from the perceptions of others.
After Lilandra is assassinated, Rachel fights alongside the Starjammers against the Shi'ar Guard and Araki, who has summoned the same Shi'ar commandos that killed Rachel's family and branded her with the Shi'ar death mark. Rachel uses her powers to implode Black Cloak's head, saying, "He was the one... He killed my family," though killing him does not make her feel happier. Gladiator finishes the job by killing Araki himself. Rachel, along with the rest of the Starjammers, regroup later on and mourn the Shi'ar, as they doubt that they will recover from this war.
Realm of Kings.
It is known through Ch'od, and apparently due to the incident where she and Korvus both lost the connection to the Phoenix Force, that Rachel and Korvus, along with Havok and Polaris, have departed for Earth.
Age of X.
While on the way back to Earth, Rachel attempted to contact Professor Xavier or Emma Frost with a message. However at that moment, Moira (a powerful alternate personality of the mutant Legion) warped reality taking Rachel's mind with it creating the amnesic Revenant. Once reality was restored, Rachel's mind is separated from her body which according to her is "half a universe away". Because of Moira's actions, Rachel no longer remembers the message and her mind retains the form of her Age of X counterpart. Scott promised her that they would return her home.
Schism and Regenesis.
Being actually a mental manifestation, Rachel soon began fading. She asks Rogue to connect with her to see what will happen. When the two of them touch, Rogue sees a vision of where Rachel, Havok and Polaris are as Rachel then returns to her body. When she awakens she is met by an unseen villain holding a gun and telling her he has killed her friends.
Borrowing one of Legion's manifold powers, Rogue teleports alongside Gambit, Magneto and Frenzy to rescue their teammates on the other side of the galaxy. Once there, Rachel is retrieved from a band of pirates as Rogue becomes their new leader. The remaining X-Men discover they've arrived at a space station called Gul Damar which is in a state of upheaval due to the rebellion of insectoid creatures called Grad Nan Holt against their Shi'ar enslavers. They are also being pulled into an exploding sun and the entire civil war is revealed to be orchestrated by a powerful Grad Nan Holt telepath known as "Friendless." A number of battles with the creature proves unsuccessful for Rachel, but with the combined efforts of Rogue and the X-Men they are able to defeat him and return home via the wormhole that was created from the collapsing star.
Rachel as part of a team of Wolverine's X-Men attempted to psychically battle Exodus in an attempt to prevent the assassination of Cyclops. The team is eventually beaten and the X-Men are saved by Generation Hope.
Rachel is then invited to hold a position as senior staff member of the "Jean Grey School for Higher Learning," which was rebuilt from the Xavier Institute and has Wolverine as acting Headmaster and Kitty Pryde as Headmistress.
Avengers vs. X-Men.
During the events of AVX, when the X-Men on Utopia escape from the invading Avengers, Rachel contacts Cyclops to provide Hope's location. Afterwards, Rachel states that she knows the Phoenix Force better than anyone else on Earth and that she is living proof that it can be controlled. She also says that if the Phoenix has chosen her and that is the destiny that Hope wants, she will do everything in her power to help her. She and Iceman tell Wolverine at the school that they are going to help Cyclops in the battle. However, when Cyclops and Emma Frost begin to be corrupted by the Phoenix's power, she, like the other X-Men, decide to oppose their rule. Rachel then battled against Cyclops and helped both the Avengers and X-Men in the final battle.
Marvel NOW! and Inhuman war.
In 2013, Marvel revealed an all new all female series simply named "X-Men". Written by Brian Wood with art by Olivier Coipel, "X-Men" will feature an all female cast including Storm, Jubilee, Rogue, Kitty Pryde, Rachel Grey and Psylocke.
The X-Men are in pursuit of Arkea who has attempted to revive the X-Men's long lost enemies. The X-Men are attacked by the newly formed Sisterhood and Rachel is telepathically locked down by Madelyne Pryor. Storm strikes a deal for Rachels freedom allowing Madelyne and Selene to walk free.
In the AXIS story arc, Storm sends the entire student body against the Red Onslaught in an attempt to save Earth. Rachel is shown to be reverted to her hound form by Dr. Doom and Wanda Maximoffs spell.
Rachel is part of Storm's team of Amazing X-Men who attempts to stop the Living Monolith from becoming the new Juggernaut.
Rachel Grey is distrusting of Spider-Man's newly appointed position as Jean Grey School guidance counseller. Using her telepathy she seeks to expose his secret identity only to lose that memory by being mind-wiped by Martha Johansson.
Rachel is shown as a teacher at the Jean Grey Institute in the battle against Kenji.
ResurrXion.
When Kitty Pryde returns from space to lead the X-Men, Rachel joins her team in "X-Men Gold" and adopts the unique, new code name Prestige.
Dawn of X.
In the new status quo for mutants post "House of X and Powers of X", Professor X and Magneto invite all mutants to live on Krakoa and welcome even former enemies into their fold. Rachel spends some time with her family on the Summer House, the new residence of the Summers Family, located on the Moon. Soon after X (Xavier) is killed by bioengineered human terrorists, Cyclops gathers Rachel and Kid Cable for a mission on the Atlantic Ocean regarding a piece of their new homeland of Krakoa.
Later, she joins Polaris, Daken, Northstar and former X-students Eye-Boy and Prodigy in a new initiative in Krakoa: they are to investigate any mutant death and prepare a report for The Five as part of the Resurrection Protocols of Krakoa. Their first case involves the supposed death of Northstar's twin sister, Aurora.
Powers and abilities.
Rachel possesses various psionic abilities, generally exhibited through telepathy, psychometry, telekinesis, and limited time manipulation.
Telepathy.
Marvel Girl's "virtually unlimited" telepathy allows her to receive, broadcast, and manipulate cognitive processes (such as thoughts) in an intricate manner. Examples of Rachel's aptitudes for this include creating durable mind-links across distances, projecting blasts of psionic energy that disrupt aspects of brain functioning, shielding her mind from other telepaths, creating illusions, and rendering someone imperceptible to the five senses. In addition, Rachel has demonstrated the ability to telepathically suppress superpowers; control, repair, and exchange minds; as well as safely editing memories. Rachel has also harnessed her telepathy to sense, locate, and track other sentient beings based on their thought patterns, but has a moral apprehension about using this skill due to her experiences as a Hound.
It has been suggested that Rachel's telepathy, although immeasurable in raw power, is mitigated by her limited training and finesse. Emma Frost was able to outflank an incredulous Rachel in a contest on the astral plane. In the same issue, Emma offered her educative services; and later still, Rachel received training from Professor Charles Xavier (while he was depowered), giving her access to his vast knowledge and expertise in telepathy.
Telekinesis.
By using telekinesis, Marvel Girl can remotely manipulate matter even on a sub-atomic level. She can channel this ability to create protective force fields and blasts of concussive force. By using her telekinesis to levitate herself, Marvel Girl can fly at incredible speeds. Rachel has been able to create a micro black hole, levitate an entire city for a time, sustain shields that withstood Jovian atmospheric pressures, and direct blows from Thor's hammer, Mjolnir. Moreover, Rachel's telekinetic fine-motor control has allowed her to alter molecular valences, mentally alter clothing with ease, create a telekinetic/psionic sword (much like Psylocke's telekinetic katana), a telekinetic hammer powerful enough to knock Thor off his feet, and even rewrite human genomes.
While all depictions portray Rachel as an extremely powerful telekinetic, the limitations of her hard-psi capabilities have been somewhat inconsistent. Some instances have depicted Rachel's telekinetic potential to be nigh-unlimited, whereas others have shown her struggling against, and even outmatched by, lesser developed telekinetics such as Psylocke.
Chronoskimming.
Marvel Girl utilizes her psionic talents in conjunction with her ability to manipulate time in a variety of ways. "Chronoskimming" describes her ability to temporarily transplant a person's mind and send it through time into a younger/older version, a close ancestor/descendant, or as a disembodied astral form.
Rachel unconsciously emanates a fourth dimensional pulse, effectively creating a chrono-shield that protects her from changes in the timeline. She can also sense and manipulate residual psychic energy in the form of psychometry.
Phoenix Force.
When Rachel was bonded to the cosmic entity known as the Phoenix Force, she demonstrated heightened psi-powers, the ability to manipulate energy and life-forces, and limited cosmic awareness. Rachel's connection to the Phoenix power was lost in the distant future and did not return with her when she traveled back to the early 21st century (present) of Earth-616 (Marvel's mainstream universe).
Most recently, Marvel Girl absorbed a residual echo of the Phoenix Force left in the sword of a previous host, a Shi'ar named Rook'shir. It was revealed that this energy source was a less powerful (but easier to wield) form of the Phoenix Force. The echo was powerful enough to allow Rachel to survive in and fly through the vacuum of space without the need for additional protection, as well as being able to hold her own in combat against the tremendous physical power of Gladiator. These demonstrations were short lived, however, due to its disappearance, which Rachel attributes to Jean Grey. She now exhibits her standard power levels.
Power signature.
As a host for the Phoenix Force, Rachel manifested the cosmic firebird in displays of great power. During her 2000s "Uncanny X-Men" appearances, Marvel Girl also exhibited a Phoenix emblem over her left eye whenever she demonstrated psionic feats. It was at first accompanied by a "shadow form" (similar to the one Jean Grey manifested when she absorbed the telepathic powers of Psylocke). However, the illustration of this shadow form ceased without explanation. After regaining a small portion of the Phoenix Force (echo), the emblem over her eye changed from a gold Phoenix shape to a static version made of electric blue flame. Her display of power was once more altered in "X-Men: Emperor Vulcan" #3, where she produced the familiar fiery raptor with which the Phoenix Force is commonly associated (see profile image).
Skills and abilities.
At times, Rachel has been shown to learn new skills extremely quickly. For example, she mastered a set of "demon ninja" sword skills simply by watching her teammate Shadowcat perform them. Along with sword fighting, Rachel has experience in lock-picking, vehicular repair (such as engines), and use of advanced technology and weaponry. However, these abilities have not been evident in her more recent appearances.
Potential and limitations.
Rachel's power level and scope of abilities have been inconsistently described by writers over the years. However, she is usually depicted with "virtually unlimited" potential in her dual psionic talents. In most cases, she displayed greater feats as the Phoenix and even matched Gladiator's strength with the aid of a "Phoenix echo". Rachel is considered by many to be an Omega-level mutant (like her mother), but the only literary reference to this attribute is when the future Sentinel, Nimrod, classified Rachel as an "Omega class subject" several years before the term was established in Marvel canon.
Even with the omnipotent strength of the Phoenix, magic and magical objects are able to resist Rachel's powers. When the Soulsword appeared near the Excalibur lighthouse headquarters seeking Kitty Pryde to become its new wielder, Rachel attempted to remove it from bedrock to alleviate her friend's apprehension. Despite using the full extent of power permitted by the Phoenix Force, Rachel was unable to remove the sword, surmising that only Kitty could remove it.
Other versions.
In the very first issue of the "Uncanny X-Men" story arc "Season of the Witch", Rachel and Psylocke were transported to the White Hot Room as an indirect result of the reality-shift performed by a mentally unstable Scarlet Witch. While there, it was established that the Rachel appearing in Earth-616 (originally from Earth-811) has no true alternate counterparts within the Marvel multiverse. Rather, all other incarnations of "Rachel Summers" that exist in parallel timelines (see below) are linked only by having the same name, or attributes.
House of M.
The subsequent issues had Rachel and Psylocke return to their dimension and thus be engrossed in the Scarlet Witch's reconstruction of the world. In this reality, Rachel was the bodyguard and traveling companion to Psylocke, who was crowned British royalty after her brother, Brian, became ruler of all England. Rachel then became involved with Captain Britain's mission to seal the breach in reality (rift) that was created by the Scarlet Witch's manipulations.
Variations of Days of Future Past.
In at least three alternate future timelines derived from "Days of Future Past", a Rachel Summers married Franklin Richards and procreated mutant children. One such child was the time-traveling supervillain Hyperstorm (Jonathan Richards). Hyperstorm was responsible for causing the Fantastic Four to think that Mister Fantastic (Jonathan's grandfather) and Doctor Doom were dead; he was only defeated when he was trapped in another dimension by Galactus. The second child was Dream Summers, who possessed empathic superpowers. She was a superhero who appeared in the "Spider-Man/X-Men: Time's Arrow" trilogy of novels (although Marvel Comics novels tend to be considered non-canon). In a third reality, they produced a child named David Richards, who was rescued from a concentration camp by the interdimensional traveling Exiles and raised by the "Age of Apocalypse" version of Sabretooth. David's traumatic experiences at the camp motivated him to become a fanatical murderer.
In another variation of the "Days of Future Past", shown in "Weapon X: Days of Future Now", a Rachel Summers was captured by Weapon X and detained in the "Neverland" concentration camp.
Exiles.
In the so-called "Legacy Earth" reality, in which the Legacy Virus mutated into a techno-organic plague, a Rachel Summers was a member of the Avengers, the last superhero group. At Morph's behest, she contacted Thor and the Asgardians to help them against the Vi-Locks, a race of beings infected with the techno-organic virus.
"X-Men: The End".
The miniseries "" (written by Chris Claremont) details the last adventures of the X-Men in a possible future. In this reality, Rachel played a central role, and was the political campaign manager of Kitty Pryde (Chicago mayoral candidate). However, she abandoned this position after Cassandra Nova led a series of attacks on the X-Men and their allies. She traveled alongside Cable into Shi'ar space. After a Professor X was taken hostage by Nova, and she entered his psyche in order to free him. However, Nova traps her and Lilandra within. She seemingly kills them both, but Rachel used the power of the Phoenix to save herself and incapacitates Nova. They leave Xavier's mind and continue the battle in Rachel's mind, where Rachel is overpowered. Cassandra, using Rachel's body, kills Jean Grey and Cyclops, leaving Rachel forced to watch the ordeal. Nova then leaves Rachel's body, stealing the Phoenix force. The Phoenix being the only thing keeping her alive after her Cassandra destroyed her mental form, she simply died.


6582 Flagsymphony


Quantitative analysis of behavior

Quantitative analysis of behavior is the application of mathematical models, conceptualized from a robust corpus of environment-behavior-consequence interactions in the experimental analysis of behavior, with the aim to describe and/or predict relations between a dependent variable and all possible levels of an independent variable. The parameters in the models hopefully have theoretical meaning beyond being used to fit models to data. The field was founded by Richard Herrnstein (1961) when he introduced the matching law to quantify the behavior of organisms working on concurrent schedules of reinforcement. The field has integrated models from economics, zoology, philosophy, and other branches of psychology, especially mathematical psychology of which it is a branch. The field is represented by the Society for Quantitative Analysis of Behavior. Quantitative analysis of behavior addresses the following topics among others: behavioral economics, behavioral momentum, connectionist systems or neural networks, integration, hyperbolic discounting including the delay reduction hypothesis, foraging, hunting, errorless learning, creativity, learning, and the Rescorla-Wagner model, matching law, melioration, scalar expectancy, signal detection, neural hysteresis, and reinforcement control.


Annual equivalent rate


Cinetheodolite

A cinetheodolite (a.k.a. "kinetheodolite") is a photographic instrument for collection of trajectory data. It can be used to acquire data in the testing of missiles, rockets, projectiles, aircraft, and fire control systems; in the ripple firing of rockets, graze action tests, air burst fuze tests, and similar operations. Cinetheodolites provide angular measurements of the line of sight to the vehicle. This permits acquiring accurate position data. Together with timing systems, velocity and acceleration data can be developed from the position measurements. Cinetheodolites can serve as primary sources of position and velocity data to about 30 km slant range.
These instruments were developed from a family of optical devices known as theodolites by the addition of a movie camera, thus adding the ability to track the vehicle in flight and to obtain continuous trajectory data.
Introduction.
One of the objectives of testing missile and rocket systems is to determine the actual "in-flight" performance of the vehicles themselves. One of the prime requirements for establishing the performance of vehicles in flight is to obtain accurate data which will reveal the position in space and the attitude of the vehicle during its trajectory. The employment of optics at a missile range may become highly significant in obtaining these data, if the atmosphere permits reasonably unobstructed observation, and if, moreover, an all-land test area makes possible optimum siting of instruments for most desirable look angles. Under these conditions, optics in general, and photogrammetry in particular, correlated with other instrumentation systems, can provide effective and accurate data of target trajectory.
Description.
The cinetheodolite is a combination photo-recording and surveying instrument which tracks and photographs targets (in flight vehicles, etc.) at selectable frame rates. Some cinetheodolites have rate-aided tracking control, whereby an open loop servomechanism in conjunction with operator actuated hand wheels match the angular rates of the tracking axis with the angular rates of the target line of position.
Cinetheodolites consist of a stable base and bearing, a vertical gimbal or trunnion carrier which rotates about a vertical axis normal to the plane of the base; a central drum or housing which contains the system telescopic lenses, plus a camera and film assembly; a horizontal trunnion shaft on which the central drum is mounted so that it can rise or dip about the horizontal
axis; and the sighting telescopes, which are also mounted on the horizontal trunnion shaft.
Manufacturers.
Notable cinetheodolite manufacturers include J. W. Fecker Division, American Optical Co., Pittsburgh, Pa. (USA), iMAR Navigation GmbH, St. Ingbert (Germany), "Askania Werke Rathenow" (Germany), Rheinmetall Air Defence (formerly Contraves AG, Switzerland), and BELOMO (Belarus).
[[PhotoSonics, Inc. (California, USA)
References.
Cinetheodolites, Materiel Test Procedure 5-1-031, [[White Sands Missile Range]], 31 March 1969
[[Category:Measuring instruments]]
[[Category:Articles containing video clips]]


Microphone connector

Many different electrical connectors have been used to connect microphones to audio equipment—including PA systems, radios, tape recorders, and numerous other devices.
Phone connectors.
Photo: 2.5 mm mono (TS), 3.5 mm mono and stereo (TRS), and 1/4″ (6.35 mm) stereo (TRS) phone connectors
The most common microphone connector in consumer use is the venerable phone connector, in 1/4" (6.35 mm), 3.5 mm, and 2.5 mm sizes, and in both mono and stereo configurations.
Not all microphone plugs with 3-segment TRS (Tip-Ring-Sleeve) connectors are actually stereo. Most computer microphones use the third segment to carry bias power for the microphone.
Type "O" 1930's Connectors.
These round connectors are about 1 inch in diameter. They were in common use in the later 40's and thru the 50's at WELI, New Haven Connecticut. see: http://terryking.us/OldRadio/OP-4-RCA-RemoteAmplifier-1934-Rear.jpg The 3 connectors on the left are 1934 type "P" round microphone connectors. They look like this: http://terryking.us/OldRadio/P-CG-12S.jpg
Type "UA" 1950's - 1960's Professional connectors.
These were "D" shaped upgrades of the old P connectors, with gold plated pins and sockets. They looked like this: http://terryking.us/OldRadio/UA-3-12.jpg
SwitchCraft 2501F.
This is the part number for a Swichcraft connector used with high impedance microphones up through about 1980. It was intended for coax style cable up to outside diameter.
Attachment is made with a 5/8-27 threaded ring that can be unthreaded over the body of the connector body allowing the female connector to become a male connector.
Wire connection was via solder. The center conductor is soldered to a hole resulting in a flat center contact. The cable shield is soldered to the strain-relief spring which is held in the body of the connector with a set screw.
With no provision for spring loading of the connection ring, vibration tends to loosen the threaded ring and create intermittent contact.
Amphenol 80-series and 91-Series.
Popular microphone connector from 1930s to 1980s, originally manufactured by Amphenol.
Amphenol sold this production to WPI Interconnect in early 1980s.
The Amphenol 80-MC2M microphone connector was popular with American radio manufacturers (1950s to 1970s).
XLR connectors (Mid-50's to today).
Most commonly used on professional microphones, the common 3-pin XLR connector is a standard for transferring balanced audio among professional audio equipment.
The 4-pin XLR is the standard connector for intercom headsets, such as systems made by ClearCom and Telex. Two pins are used for the mono headphone signal and two pins for the unbalanced microphone signal.
The 4-pin XLR connector is also commonly used on amateur radio microphones, but transferring unbalanced audio instead, and using the 4th pin (with the common ground) for a push-to-talk (PTT) circuit activated by a button on the microphone. Some broadcast radio professionals also use these microphones, instead using the PTT circuit as a cough button (a momentary mute button) so that they may cough, clear their throat, or make comments to others in the studio without the sounds being broadcast. It is also used for dmx512 digital stage controls.
Multi-pin Circular Connector (Marushin, MCC, CBC-series, Foster, GX-16).
First used in the 1970s by Japanese radio manufacturers, as an alternative to the German DIN connector standard for microphone connectors on Citizens Band and Amateur Radio equipment,
these screw-on circular connectors were available with 2, 3, 4, 5, 6, 7, or 8 pins. They often were used for other purposes as well, such as 1970s amateur radio transceivers (microphone), 1970s Kenwood mobile radios (2-pin, DC power), Eastern Asian manufacturers built soldering stations in 1990s using the conne tor fir the iron/handpiece connector.
As reported by Philmore-Datel (LKG Industries of Rockford, IL), the USA importer and distributor for these connectors, the original Japanese mfg. (Foster) ceased production. After the 2011 Tōhoku earthquake and tsunami (loss of factories), connector manufacturing moved from Japan to Taiwan and mainland China manufacturers.
Marushing Electric Mfg. Co., Ltd. in Kanagawa, Japan the original 1950's producer of the connector type still continues manufacturing.
Japanese communications equipment manufacturers as Icom, Yaesu Musen etc. have been using Marushin connectors in their products.
These connectors are sold under a variety of generic names: "CB connector", "Mobile connector", "Microphone connector", "Power connector" — even sometimes misidentified as XLR connectors.
The Chinese mainland (and Taiwan) manufacturers adopted this connector as the “GX16” (16mm), for “Power” or “Avionics” usage. You can find the Chinese “GX series” used as a multi-voltage connector (soldering stations, avionics, marine, etc.). Google or eBay search “GX16” , and you will find hundreds of vendors, such as Renhotecpro, based in Guangdong, China.
http://www.renhotecpro.com/gx16-connector-data-sheet/
Renhotecpro has standardized the manufacturing of GX-series, in a variety of sizes,
by the diameter of the connector (in mm):
GX12, GX16, GX20, GX25, GX30, GX35, GX40.
Even though this circular connector is largely standardized, there was NO industry standard for wiring or pin usage/assignments. The individual pin assignments to specific functions varied from manufacturer to manufacturer.
For example, in amateur radio, Icom, Kenwood, and Yaesu radios have all used the 8-pin screw-on connector, yet microphones from one brand are incompatible with the other two brands, at least without the use of an adapter cable in between the microphone and the radio.
Modular connectors.
Although originally designed for telephone usage and later adopted for computer networking, the Modular connector has been used for microphone connections extensively on modern amateur radio equipment, including models from the three major manufacturers (Icom, Kenwood, and Yaesu), but primarily used on mobile radio equipment (designed to be mounted in vehicles).
Like the screw-on round microphone connectors, there is little standardization among manufacturers, and even among different models from the same manufacturer, leading to significant interoperability problems when connecting non-OEM or homebrew microphones and other equipment.
Most radios with modular microphone connectors use the 8P8C (vernacular: RJ45) connector, but some use 6P6C or 4P4C connectors.


Degree (music)

In music theory, the term scale degree refers to the position of a particular note on a scale relative to the tonic, the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals and chords and whether they are major or minor.
In the most general sense, the scale degree is the number given to each step of the scale, usually starting with 1 for tonic. Defining it like this implies that a tonic is specified. For instance, the 7-tone diatonic scale may become the major scale once the proper degree has been chosen as tonic (e.g. the C-major scale C–D–E–F–G–A–B, in which C is the tonic). If the scale has no tonic, the starting degree must be chosen arbitrarily. In set theory, for instance, the 12 degrees of the chromatic scale usually are numbered starting from C=0, the twelve pitch classes being numbered from 0 to 11.
In a more specific sense, scale degrees are given names that indicate their particular function within the scale (see table below). This definition implies a functional scale, as is the case in tonal music.
This example gives the names of the functions of the scale degrees in the seven note diatonic scale. The names are the same for the major and minor scales, only the seventh degree changes name when flattened:
\override Score.TimeSignature #'stencil = ##f
 #(set-global-staff-size 18)
 \set Score.proportionalNotationDuration = #(ly:make-moment 1/8)
\relative c' {
 \clef treble \key c \major \time 8/1
 c1
 d e f g a \override ParenthesesItem.padding = #1.5 \parenthesize bes b
 \time 1/1 c \bar "||"
</score>
The term scale step is sometimes used synonymously with scale degree, but it may alternatively refer to the distance "between" two successive and adjacent scale degrees (see steps and skips). The terms "whole step" and "half step" are commonly used as interval names (though "whole scale step" or "half scale step" are not used). The number of scale degrees and the distance between them together define the scale they are in.
In Schenkerian analysis, "scale degree" (or "scale step") translates Schenker's German "Stufe", denoting "a chord having gained structural significance" (see Schenkerian analysis#Harmony).
Major and minor scales.
The degrees of the traditional major and minor scales may be identified several ways:


Henry (unit)

The henry (symbol: H) is the SI derived unit of electrical inductance. If a current of 1 ampere flowing through a coil produces flux linkage of 1 weber turn, that coil has a self inductance of 1 henry. The unit is named after Joseph Henry (1797–1878), the American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday (1791–1867) in England.
Definition.
The inductance of an electric circuit is one henry when an electric current that is changing at one ampere per second results in an electromotive force of one volt across the inductor:
where "V"("t") denotes the resulting voltage across the circuit, "I"("t") is the current through the circuit, and "L" is the inductance of the circuit.
The henry is a derived unit based on four of the seven base units of the International System of Units: kilogram (kg), metre (m), second (s), and ampere (A).
Expressed in combinations of SI units, the henry is:
where: , , , , , , , , , , , , , Hz = hertz
Use.
The International System of Units (SI) specifies to write the symbol of a unit named for a person with an initial capital letter, while the name is not capitalized in sentence text, except when any word in that position would be capitalized, such as at the beginning of a sentence or in material using title case.
The United States National Institute of Standards and Technology recommends users writing in English to use the plural as "henries".
Applications.
The inductance of a coil depends on its size, the number of turns, and the permeability of the material within and surrounding the coil. Formulas can be used to calculate the inductance of many common arrangements of conductors, such as parallel wires, or a solenoid. A small air-core coil used for broadcast AM radio tuning might have an inductance of a few tens of microhenries. A large motor winding with many turns around an iron core may have an inductance of scores or hundreds of henries. The physical size of an inductance is also related to its current carrying and voltage withstand ratings.


Ljubov Rebane

Ljubov A. Rebane (née Chagalova) (September 6, 1929 Leningrad – June 13, 1991 Tallinn) was an Estonian physicist. She graduated from Leningrad University in 1952 and received a PhD in Physics and Mathematics in 1961 from the same university.
She received the USSR State Prize (Russian: Госуда́рственная пре́мия СССР) in 1986, together with Rein Avarmaa, Anšel Gorohhovski, Roman Personov, Boris Harlamov, Jevgeni Alšits, Ljudmila Bõkovskaja, Vladimir Maslov, Jaak Kikas and Konstantin Solovjov, for the cycle of articles "High-spectral-resolution spectroscopy and for the persistent spectral hole burning of molecules and solids" (Russian: Авармаа, Рейн Арнольдович, зав. сектором, Гороховский, Аншель Александрович, Кикас, Яак Вернерович, ст. н. с., Альшиц, Евгений Иосифович, Быковская, Людмила Анатольевна, мл. н. с. Института физики АН ЭССР; Маслов, Владимир Григорьевич, ст. н. с. ГОИ имени С. И. Вавилова; Ребане, Любовь Александровна, ст. н. с. ИХБФАН ЭССР; Соловьёв, Константин Николаевич, зав. лабораторией Института физики АН БССР, — за цикл работ «Фотовыжигание стабильных спектральных провалов и гелективная спектроскопия сложных молекул» (1972—1984)).
Family.
She married the physicist Karl Rebane. Their children Aleksander and Inna Rebane both became physicists themselves.


Kron Stone


WM Recordings

WM Recordings is a Heerlen, the Netherlands based record label founded in 2004, by Marco Kalnenek, who was also involved with Comfort Stand Recordings.
Kalnenek also is the webmaster of Weirdomusic.com, a website nominated for a Webby Award in 2004.
WM Recordings focuses on music that is, in their own words, "a little different".
They do not specialize in one style, but release anything from free jazz to experimental soundscapes, world music and other styles that are often neglected by other netlabels.
In 2007, WM Recordings started adding commercial releases to their catalog. These are available from most major download stores.
August 2008 saw the release of their first physical CD: "Handle With Care - Might Panic" by Danish band Panicphobia.


976 Benjamina

976 Benjamina ("prov. designation:" "or" ) is a dark background asteroid from the outer regions of the asteroid belt, approximately in diameter. It was discovered on 27 March 1922, by Russian-French astronomer Benjamin Jekhowsky at the Algiers Observatory in North Africa. The large X/D-type asteroid has a rotation period of 9.7 hours and is likely regular in shape. It was named after the discoverer's son.
Orbit and classification.
"Benjamina" is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. It orbits the Sun in the outer main-belt at a distance of 2.9–3.5 AU once every 5 years and 9 months (2,092 days; semi-major axis of 3.2 AU). Its orbit has an eccentricity of 0.10 and an inclination of 8° with respect to the ecliptic.
The asteroid was first observed as () at Heidelberg Observatory on 8 January 1910. The body's observation arc begins at the Crimean Simeiz Observatory in September 1930, more than 7 years after its official discovery observation at Algiers Observatory in March 1922.
Naming.
This minor planet was named after Benjamin Jekhowsky Jr., son of discoverer Benjamin Jekhowsky. The was mentioned in "The Names of the Minor Planets" by Paul Herget in 1955 ().
Physical characteristics.
In the Tholen classification, and based on a noisy spectra (:), "Benjamina" is an X-type asteroid, somewhat similar to that of a dark D-type (XD), which are common in the outer asteroid belt and among the Jupiter trojan population.
Rotation period and pole.
In September 2004, a rotational lightcurve of "Benjamina" was obtained from photometric observations by French amateur astronomer Laurent Bernasconi. Lightcurve analysis gave a well-defined rotation period of hours with a brightness variation of magnitude (). Follow-up observation by his college René Roy in March 2018, determined a concurring period of hours but with a much higher amplitude of magnitude (). The objects first lightcurve was obtained by Colin Bembrick at Mount Tarana Observatory in Australia in March 2003. It showed a period of hours with an amplitude of magnitude (). The overall amplitude suggest a rather regular shape with a ratio of 0.86 for the length of the a and b axes.
In 2018, Czech astronomers Josef Ďurech and Josef Hanuš published a modeled lightcurve using photometric data from the "Gaia" probe's second data release. It showed a sidereal period of hours, and gave a spin axis at (354.0°, 80.0°) in ecliptic coordinates (λ, β).
Diameter and albedo.
According to the surveys carried out by the Japanese Akari satellite, the Infrared Astronomical Satellite IRAS, and the NEOWISE mission of NASA's Wide-field Infrared Survey Explorer (WISE), "Benjamina" measures , and kilometers in diameter with an albedo of its surface of , and , respectively. Additional measurements by the WISE telescope were published giving a mean-diameter as low as , and as high as .
The "Collaborative Asteroid Lightcurve Link" adopts the result from IRAS, that is, an albedo of 0.0559 and a diameter of 80.53 kilometers based on an absolute magnitude of 9.22.
The asteroid had been observed in 7 stellar occultation events since 2003. On 19 July 2003 the mag. 5.7 star HIP 88816 was occulted by the asteroid, and was observed at 11 stations; 1 in Argentina, 3 in New Zealand, and 7 in Australia. From these observations, the best-fit ellipse measures 85.2 x 56.2 +/-12.4 kilometres.


2430 Bruce Helin

2430 Bruce Helin ("prov. designation": ) is a stony Phocaea asteroid and slow rotator from the inner regions of the asteroid belt. It was discovered by American astronomers Eleanor Helin and Eugene Shoemaker at the U.S. Palomar Observatory in California, on 8 November 1977. It was later named after Bruce Helin, son of the first discoverer. The S-type asteroid (Sl) has an exceptionally long rotation period of 128 hours and measures approximately in diameter.
Orbit and classification.
"Bruce Helin" is a member of the Phocaea family (). It orbits the Sun in the inner main-belt at a distance of 1.9–2.9 AU once every 3 years and 8 months (1,326 days). Its orbit has an eccentricity of 0.21 and an inclination of 23° with respect to the ecliptic. It was first observed at at Heidelberg Observatory in 1908. The body's observation arc begins with a precovery taken at Crimea–Nauchnij in 1976, or one year prior to its official discovery observation at Palomar.
Naming.
This minor planet was named after Bruce Helin, son of the first discover Eleanor Helin, in an expression of gratitude for "the many years he tolerated his mother's preoccupation with extraterrestrial objects". The discoverer has also honoured her daughter-in-law and wife of Bruce, Nancy Coker Helin, by the minor planet 4222 Nancita. The official was published by the Minor Planet Center on 13 October 1981 ().
Physical characteristics.
In the Tholen classification, "Bruce Helin" is a common, stony S-type asteroid, while in the Bus–Binzel SMASS taxonomy, it is an Sl-subtype, which transitions from the S-type to the uncommon L-type asteroid.
Rotation period.
The first rotational lightcurve of "Bruce Helin" was obtained from photometric observations by Czech astronomer Petr Pravec in September 2006. The lightcurve showed a rotation period of hours with a brightness variation of 0.60 in magnitude (). Later observations rendered a similar rotation period of 129.75 and hours, respectively ().
Diameter and albedo.
According to the surveys carried out by the Japanese Akari satellite and NASA's Wide-field Infrared Survey Explorer with its NEOWISE mission, "Bruce Helin" measures 11.8 and 12.5 kilometers in diameter, and its surface has an albedo of 0.18 and 0.24, respectively. The "Collaborative Asteroid Lightcurve Link" (CALL) assumes an albedo of 0.23 and calculates a diameter of 12.1 kilometers with an absolute magnitude of 11.8.


Bernard Ziegler

Bernard Ziegler (12 March 1933 – 4 May 2021) was a French pilot and engineer, who served in Airbus as senior vice president for engineering, well known for his evangelical zeal for the application of the fly-by-wire system in the Airbuses. He was the son of Airbus founder Henri Ziegler.
Biography.
Bernard Ziegler was born in Boulogne sur Seine, France. He was educated in the École Polytechnique (1954), as an engineer and École de l'Air (1955), for his pilot training and military commission. Ziegler began his career as a fighter pilot in the French air force. He saw action in the Algerian War and was decorated twice. Principal medals and decorations received include the Officier de l’Ordre National du Mérite, Médaille de l'Aéronautique, Croix de la Valeur Militaire and Officier de la Légion d'honneur.
In August 1961 the Vallee Blanche Aerial Tramway was badly damaged when an aeroplane of the French Air Force piloted by Bernard Ziegler tore its hauling cable. Three cars crashed and six people were killed.
He continued his studies in École nationale supérieure de l'aéronautique et de l'espace (Supaéro) (1961) and was later posted to École du personnel navigant d'essais et de réception (EPNER), the French Air Force test pilot establishment in 1964. He was the chief test pilot for the Dassault Mirage G in 1968. At Airbus, his career spans from chief test pilot (1972), to senior vice president for flight and support and then, to senior vice president for engineering. As a test pilot, he flew the first flights of the Airbus A300, A310, A320 and the A340.
Bernard Ziegler was the most influential figure in developing the cockpit design and fly-by-wire control system for the Airbus airliners. He proposed that numerous technological innovations be applied to Airbus aircraft; for example, using composites, twin-engine configuration for the A300, fly-by-wire and many others. He was the guiding force in the creation of the flight envelope protection, incorporated in the Airbus flight-control software. This innovation allows the pilot to apply the maximum control forces considered necessary, while preventing inadvertent inputs that could place the aircraft outside the safety margin. This feature is considered by many to be highly beneficial in avoiding unusual altitudes in flight and in safely maximizing the effectiveness of evasive maneuvers in response to GPWS warnings.
For his efforts in advancing the fly-by-wire cause, he was honoured by the Flight Safety Foundation in 1998. He retired from Airbus after 25 years of service in 1997.


Random variate

In the mathematical fields of probability and statistics, a random variate is a particular outcome of a random variable: the random variates which are other outcomes of the same random variable might have different values.
A random deviate or simply deviate is the difference of random variate with respect to the distribution central location (e.g., mean), often divided by the standard deviation of the distribution.
Random variates are used when simulating processes driven by random influences (stochastic processes). In modern applications, such simulations would derive random variates corresponding to any given probability distribution from computer procedures designed to create random variates corresponding to a uniform distribution, where these procedures would actually provide values chosen from a uniform distribution of pseudorandom numbers.
Procedures to generate random variates corresponding to a given distribution are known as procedures for "random variate generation" or pseudo-random number sampling.
In probability theory, a random variable is a measurable function from a probability space to a measurable space of values that the variable can take on. In that context, those values are also known as random variates or random deviates, and this represents a wider meaning than just that associated with pseudorandom numbers.
Definition.
Devroye defines a random variate generation algorithm (for real numbers) as follows:
The distinction between "random variable" and "random variate" is subtle and is not always made in the literature. It is useful when one wants to distinguish between a random variable itself with an associated probability distribution on the one hand, and random draws from that probability distribution on the other, in particular when those draws are ultimately derived by floating-point arithmetic from a pseudo-random sequence.
Practical aspects.
For the generation of uniform random variates, see Random number generation.
For the generation of non-uniform random variates, see Pseudo-random number sampling.


List of presidents of the New York Stock Exchange

This is a list of presidents of the New York Stock Exchange.


Mode Records

Mode Records is an American record label in New York City that concentrates on contemporary classical music and other forms of avant-garde music. The label was founded by Brian Brandt in 1984, with a goal of releasing music composed by John Cage.
Composers featured include John Cage, Morton Feldman, Iannis Xenakis, Giacinto Scelsi, and Harry Partch. Performers include Steve Lacy, Aki Takahashi, Martine Joste, the Arditti Quartet, and Gerry Hemingway. The label also has a commitment to younger composers with releases featuring Jason Eckardt, Joshua Fineberg, and Lei Liang.
An earlier unrelated Mode Records existed for a short time in the 1950s and was involved West Coast jazz. It is now controlled by VSOP.


Marcia Gumpertz

Marcia Lynn Gumpertz is an American statistician known for her research on agricultural statistics, spatial analysis, the design of experiments, and plant disease epidemiology. She has also studied employment issues for women and members of underrepresented minorities in science and technology. She is a professor of statistics at North Carolina State University.
Education and career.
Gumpertz earned a bachelor's degree in philosophy from the University of California, Berkeley in 1973, and a master's degree in statistics from Oregon State University in 1979. She completed her Ph.D. at North Carolina State University in 1989. Her dissertation, "Simple Estimators for the Linear Random Coefficient Regression Model and the Nonlinear Model with Variance Components", was jointly supervised by Sastry Pantula and John Rawlings.
She worked as a scientist for Northrop Services Inc., in Oregon, from 1980 to 1984, doing research for the Environmental Protection Agency, and joined the North Carolina State University on completing her doctorate in 1989. At North Carolina State University, she has also served as assistant vice provost for faculty diversity.
Book.
With Francis G. Giesbrecht, Gumpertz is the co-author of the book "Planning, Construction, and Statistical Analysis of Comparative Experiments" (Wiley, 2004).
Recognition.
Gumpertz was president of Mu Sigma Rho, the US national statistics honor society, for 2004–2006. She was elected as a Fellow of the American Statistical Association in 2006, and given the Distinguished Achievement Award of the American Statistical Association Section on Statistics and the Environment in 2008.


203 Pompeja

Pompeja (minor planet designation: 203 Pompeja) is a quite large main-belt asteroid.
It was discovered by C. H. F. Peters on September 25, 1879, in Clinton, New York, and named after Pompeii, the Roman town destroyed in volcanic eruption in AD 79.
Based upon Photometric observations taken during 2011, it has a synodic rotation period of 24.052 ± 0.001 h, with a peak-to-peak amplitude of 0.10 ± 0.01 in magnitude. Because the rotation period nearly matches that of the Earth, it required coordinated observations from multiple observatories at widely spaced latitudes to produce a complete light curve.


Hypodorian mode

The Hypodorian mode, a musical term literally meaning 'below Dorian', derives its name from a "tonos" or octave species of ancient Greece which, in its diatonic genus, is built from a tetrachord consisting (in rising direction) of a semitone followed by two whole tones. The rising scale for the octave is a single tone followed by two conjoint tetrachords of this type. This is roughly the same as playing all the white notes of a piano from A to A: A | B C D E | (E) F G A. Although this scale in medieval theory was employed in Dorian and Hypodorian, from the mid-sixteenth century and in modern music theory they came to be known as the Aeolian and Hypoaeolian modes.
The term Hypodorian came to be used to describe the second mode of Western church music. This mode is the plagal counterpart of the authentic first mode, which was also called "Dorian". The ecclesiastical Hypodorian mode was defined in two ways: (1) as the diatonic octave species from A to A, divided at the mode final D and composed of a lower tetrachord of tone–semitone–tone, ending on D, plus a pentachord tone–semitone–tone–tone continuing from D, and (2) as a mode whose final was D and whose ambitus was G–B (that is, with B below the final and B above it). In addition, the note F, corresponding to the reciting note or "tenor" of the second psalm tone, was regarded as an important secondary center.


Test of Russian as a Foreign Language

The Test of Russian as a Foreign Language (TORFL) ( or ) is a contemporary test of Russian language for foreign citizens compliant with the best European quality standards.
Levels.
Levels of the Test of Russian as a Foreign Language are matched to the Common European Framework of Reference for Languages: Learning, Teaching, Assessment (CEFR).
The TORFL test consists of 6 different levels:
The Testing Center of the Herzen State Pedagogical University conducts testing in accordance with the Federal Law of the Russian Federation № 115 from the 25th of June 2002. On the legal status of the foreign citizens in the Russian Federation and the order of the Ministry of Education and Science of the Russian Federation № 1156 from the 28th of August 2014. On the approval of the form and procedure of the examination in Russian language, history of Russia and fundamentals of legislation of the Russian Federation and requirements on the minimum level of knowledge requisite for the stated exam.
Test Content.
The Test of Russian as a Foreign Language comprises 5 parts examining language competences: writing, vocabulary/grammar, reading, listening and speaking.
Usually the exam is held over a period of 2 days. On the first day candidates take the "Writing", "Vocabulary/Grammar" and “Reading" parts, on the second day – "Listening" and "Speaking" parts. The TORFL test assesses foreign citizens' skills to express various communicative intentions in the writing form using correct words and phrases, to understand different types of written information in Russian, to comprehend information in Russian given aurally, to participate in dialogues and to produce the monologue according to the proposed subject.
The question papers for the TORFL test are compiled by the experts of each educational organization authorized to run the Test, e.g. Herzen University question papers are compiled by Herzen University experts, Moscow State University question papers are compiled by MSU experts and so on.
Procedure of issuing certificates.
After successfully passing a test, a foreign citizen receives the TORFL certificate confirming the person's level of language competence in Russian as a foreign language. The form and procedure of issuing the TORFL certificates are determined by the Order of the Ministry of education and science of the Russian Federation of 25 April 2014 No. 412.
This Order states that the 10 days is the maximum period for issuing a certificate, but the testing organizations seek to do this faster. For example, Herzen University issues a TORFL certificate on the next day after exam.
Testing Centres.
By the Order of the Ministry of education and science of the Russian Federation of 20 February 2018 No. 130 the list of educational organizations conducting testing of Russian as a foreign language includes 13 educational organizations, for example: Herzen University, Moscow State University, Pskov State University, St. Petersburg State University, RUDN University, Pushkin State Russian Language Institute.
These Universities are the only organizations authorized to conduct the TORFL test. However, they can outsource the provision of technical support to other organizations in various regions of Russia and in overseas countries. By that SPbU cooperates with more than 50 organizations around the world.
The full lists of test centres where the TORFL test is conducted can be found on the Universities’ websites.


Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.
A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a random variable is called a "sample space," defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
Although the idea was originally introduced by Christiaan Huygens, the first person to think systematically in terms of random variables was Pafnuty Chebyshev.
Definition.
A random variable formula_1 is a measurable function formula_2 from a set of possible outcomes formula_3 to a measurable space formula_4. The technical axiomatic definition requires formula_5 to be a sample space of a probability triple formula_6 (see the measure-theoretic definition). A random variable is often denoted by capital roman letters such as formula_1, formula_8, formula_9, formula_10.
The probability that formula_1 takes on a value in a measurable set formula_12 is written as
Standard case.
In many cases, formula_1 is real-valued, i.e. formula_15. In some contexts, the term random element (see extensions) is used to denote a random variable not of this form.
When the image (or range) of formula_1 is countable, the random variable is called a discrete random variable and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of formula_1. If the image is uncountably infinite (usually an interval) then formula_1 is called a continuous random variable. In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous, a mixture distribution is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
Extensions.
The term "random variable" in statistics is traditionally limited to the real-valued case (formula_19). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
However, the definition above is valid for any measurable space formula_20 of values. Thus one can consider random elements of other sets formula_20, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a "random variable of type formula_20", or an "formula_20-valued random variable".
This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of formula_20, using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space formula_5, which allows the different random variables to covary). For example:
Distribution functions.
If a random variable formula_38 defined on the probability space formula_6 is given, we can ask questions like "How likely is it that the value of formula_1 is equal to 2?". This is the same as the probability of the event formula_41 which is often written as formula_42 or formula_43 for short.
Recording all these probabilities of output ranges of a real-valued random variable formula_1 yields the probability distribution of formula_1. The probability distribution "forgets" about the particular probability space used to define formula_1 and only records the probabilities of various values of formula_1. Such a probability distribution can always be captured by its cumulative distribution function
and sometimes also using a probability density function, formula_49. In measure-theoretic terms, we use the random variable formula_1 to "push-forward" the measure formula_51 on formula_5 to a measure formula_49 on formula_54.
The underlying probability space formula_5 is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space formula_5 altogether and just puts a measure on formula_54 that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on quantile functions for fuller development.
Examples.
Discrete random variable.
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum formula_58.
In examples such as these, the sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If formula_59 are countable sets of real numbers, formula_60 and formula_61, then formula_62 is a discrete distribution function. Here formula_63 for formula_64, formula_65 for formula_66. Taking for instance an enumeration of all rational numbers as formula_67, one gets a discrete distribution function that is not a step function or piecewise constant.
Coin toss.
The possible outcomes for one coin toss can be described by the sample space formula_68. We can introduce a real-valued random variable formula_8 that models a $1 payoff for a successful bet on heads as follows:
If the coin is a fair coin, "Y" has a probability mass function formula_71 given by:
Dice roll.
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers "n"1 and "n"2 from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable "X" given by the function that maps the pair to the sum:
and (if the dice are fair) has a probability mass function "ƒ""X" given by:
Continuous random variable.
Formally, a continuous random variable is a random variable whose cumulative distribution function is continuous everywhere. There are no "gaps", which would correspond to numbers which have a finite probability of occurring. Instead, continuous random variables almost never take an exact prescribed value "c" (formally, formula_75) but there is a positive probability that its value will lie in particular intervals which can be arbitrarily small. Continuous random variables usually admit probability density functions (PDF), which characterize their CDF and probability measures;
such distributions are also called absolutely continuous; but some continuous distributions are singular, or mixes of an absolutely continuous part and a singular part.
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, X = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any "range" of values. For example, the probability of choosing a number in [0, 180] is . Instead of speaking of a probability mass function, we say that the probability "density" of X is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
More formally, given any interval formula_76, a random variable formula_77 is called a "continuous uniform random variable" (CURV) if the probability that it takes a value in a subinterval depends only on the length of the subinterval. This implies that the probability of formula_78 falling in any subinterval formula_79 is proportional to the length of the subinterval, that is, if , one has
formula_80
where the last equality results from the unitarity axiom of probability. The probability density function of a CURV formula_81 is given by the indicator function of its interval of support normalized by the interval's length: formula_82Of particular interest is the uniform distribution on the unit interval formula_83. Samples of any desired probability distribution formula_84 can be generated by calculating the quantile function of formula_84 on a randomly-generated number distributed uniformly on the unit interval. This exploits properties of cumulative distribution functions, which are a unifying framework for all random variables.
Mixed type.
A mixed random variable is a random variable whose cumulative distribution function is neither piecewise-constant (a discrete random variable) nor everywhere-continuous. It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the will be the weighted average of the CDFs of the component variables.
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
Measure-theoretic definition.
The most formal, axiomatic definition of a random variable involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.
The measure-theoretic definition is as follows.
Let formula_86 be a probability space and formula_87 a measurable space. Then an formula_87-valued random variable is a measurable function formula_89, which means that, for every subset formula_90, its preimage formula_91 where formula_92. This definition enables us to measure any subset formula_93 in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of formula_5 is a possible outcome, a member of formula_95 is a measurable subset of possible outcomes, the function formula_51 gives the probability of each such measurable subset, formula_20 represents the set of values that the random variable can take (such as the set of real numbers), and a member of formula_98 is a "well-behaved" (measurable) subset of formula_20 (those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
When formula_20 is a topological space, then the most common choice for the σ-algebra formula_98 is the Borel σ-algebra formula_102, which is the σ-algebra generated by the collection of all open sets in formula_20. In such case the formula_87-valued random variable is called an formula_20-valued random variable. Moreover, when the space formula_20 is the real line formula_54, then such a real-valued random variable is called simply a random variable.
Real-valued random variables.
In this case the observation space is the set of real numbers. Recall, formula_86 is the probability space. For a real observation space, the function formula_109 is a real-valued random variable if
This definition is a special case of the above because the set formula_111 generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that formula_112.
Moments.
The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted formula_113, and also called the first moment. In general, formula_114 is not equal to formula_115. Once the "average value" is known, one could then ask how far from this average value the values of formula_1 typically are, a question that is answered by the variance and standard deviation of a random variable. formula_113 can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of formula_1.
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables formula_1, find a collection formula_120 of functions such that the expectation values formula_121 fully characterise the distribution of the random variable formula_1.
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function formula_123 of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable "X" that can take on the nominal values "red", "blue" or "green", the real-valued function formula_124 can be constructed; this uses the Iverson bracket, and has the value 1 if formula_1 has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.
Functions of random variables.
A new random variable "Y" can be defined by applying a real Borel measurable function formula_126 to the outcomes of a real-valued random variable formula_1. That is, formula_128. The cumulative distribution function of formula_8 is then
If function formula_131 is invertible (i.e., formula_132 exists, where formula_133 is formula_131's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain
With the same hypotheses of invertibility of formula_131, assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to formula_137, in order to obtain
If there is no invertibility of formula_131 but each formula_137 admits at most a countable number of roots (i.e., a finite, or countably infinite, number of formula_141 such that formula_142) then the previous relation between the probability density functions can be generalized with
where formula_144, according to the inverse function theorem. The formulas for densities do not demand formula_131 to be increasing.
In the measure-theoretic, axiomatic approach to probability, if a random variable formula_1 on formula_5 and a Borel measurable function formula_126, then formula_149 is also a random variable on formula_5, since the composition of measurable functions is also measurable. (However, this is not necessarily true if formula_131 is Lebesgue measurable.) The same procedure that allowed one to go from a probability space formula_152 to formula_153 can be used to obtain the distribution of formula_8.
Example 1.
Let formula_1 be a real-valued, continuous random variable and let formula_156.
If formula_158, then formula_159, so
If formula_161, then
so
Example 2.
Suppose formula_1 is a random variable with a cumulative distribution
where formula_166 is a fixed parameter. Consider the random variable formula_167 Then,
The last expression can be calculated in terms of the cumulative distribution of formula_169 so
which is the cumulative distribution function (CDF) of an exponential distribution.
Example 3.
Suppose formula_1 is a random variable with a standard normal distribution, whose density is
Consider the random variable formula_173 We can find the density using the above formula for a change of variables:
In this case the change is not monotonic, because every value of formula_8 has two corresponding values of formula_1 (one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
The inverse transformation is
and its derivative is
Then,
This is a chi-squared distribution with one degree of freedom.
Example 4.
Suppose formula_1 is a random variable with a normal distribution, whose density is
Consider the random variable formula_173 We can find the density using the above formula for a change of variables:
In this case the change is not monotonic, because every value of formula_8 has two corresponding values of formula_1 (one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
The inverse transformation is
and its derivative is
Then,
This is a noncentral chi-squared distribution with one degree of freedom.
Equivalence of random variables.
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
Equality in distribution.
If the sample space is a subset of the real line, random variables "X" and "Y" are "equal in distribution" (denoted formula_191) if they have the same distribution functions:
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.
Almost sure equality.
Two random variables "X" and "Y" are "equal almost surely" (denoted formula_193) if, and only if, the probability that they are different is zero:
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
where "ess sup" represents the essential supremum in the sense of measure theory.
Equality.
Finally, the two random variables "X" and "Y" are "equal" if they are equal as functions on their measurable space:
This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment is rarely explicitly characterized or even characterizable.
Convergence.
A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.
There are various senses in which a sequence formula_197 of random variables can converge to a random variable formula_1. These are explained in the article on convergence of random variables.


Wegberg-Wildenrath Test and Validation Centre

The Wegberg-Wildenrath Test and Validation Centre () is a railway test centre owned by Siemens Mobility near Wildenrath in North Rhine-Westphalia, Germany.
History.
The centre is located on the site of the former RAF Wildenrath British military airfield. After the airfield closed in 1992, it was taken over by Siemens who opened the test and validation centre in June 1997. By 2007, the railway test tracks had taken over considerable areas of the airfield, with all but the western threshold and overrun of the runway obliterated. The north-east dispersal is completely taken over by sidings, workshops and loops. Of the southern dispersals, the central and eastern are bisected by the main railway oval test track.
There are ovals of track for testing trains, and various electrification systems. The test track is connected to the rest of the German railway network by the German portion of the Iron Rhine freight railway, branching off near Wegberg.
Testing.
As well as Siemens the test tracks are also used by competitors such as Stadler, Alstom and Bombardier Transportation.
There are standard gauge and metre gauge tracks. The facility is used for technical acceptance tests and approvals of locomotives, rolling stock and trams under operating conditions, and personnel training. It has been equipped for trials with Galileo.


Test equipment

Test equipment is a general term describing equipment used in many fields. Types of test equipment include:
Electrical and electronic test equipment.
Electronic test equipment.
Electronic test equipment is used to create signals and capture responses from electronic devices, to prove proper operation or trace faults. Types of electronic test equipment include:


3090 Tjossem


Transient grating spectroscopy

Transient grating spectroscopy is an optical technique used to measure quasiparticle propagation. It can track changes in metallic materials as they are irradiated.


Hierarchical INTegration

Hierarchical INTegration, or HINT for short, is a computer benchmark that ranks a computer system as a whole (i.e. the entire computer instead of individual components). It measures the full range of performance, mostly based on the amount of work a computer can perform over time. A system with a very fast processor would likely be rated poorly if the buses were very poor compared to those of another system that had both an average processor and average buses. For example, in the past, Macintosh computers with relatively slow processor speeds (800 MHz) used to perform better than x86 based systems with processors running at nearly 2 GHz.
HINT is known for being almost immune to artificial optimization and can be used by many computers ranging from a calculator to a supercomputer. It was developed at the U.S. Department of Energy's Ames Laboratory and is licensed under the terms of the GNU General Public License.
HINT is intended to be "scalable" to run on any size computer, from small serial systems to highly parallel supercomputers.
The person using the HINT benchmark can use any floating-point or integer type.
HINT benchmark results have been published comparing a variety of parallel and uniprocessor systems.
A related tool ANALYTIC HINT can be used as a design tool to estimate the benefits of using more memory, a faster processor, or improved communications (bus speed) within the system.