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All About Eve

All About Eve is a 1950 American drama film written and directed by Joseph L. Mankiewicz, and produced by Darryl F. Zanuck. It is based on the 1946 short story (and subsequent 1949 radio drama) "The Wisdom of Eve" by Mary Orr, although Orr does not receive a screen credit.

The film stars Bette Davis as Margo Channing, a highly regarded but aging Broadway star, and Anne Baxter as Eve Harrington, an ambitious young fan who maneuvers herself into Channing's life, ultimately threatening Channing's career and her personal relationships. The film co-stars George Sanders, Celeste Holm, Gary Merrill, and Hugh Marlowe, and features Thelma Ritter, Marilyn Monroe in one of her earliest roles, Gregory Ratoff, Barbara Bates and Walter Hampden.

All About Eve held its world premiere in New York City on October 13, 1950. Highly praised by critics at the time of its release, it received a record 14 nominations at the 23rd Academy Awards, becoming the only film in Oscar history to receive four female acting nominations (Davis and Baxter as Best Actress, Holm and Ritter as Best Supporting Actress). It went on to win six awards, including Best Picture, as well as Best Director and Best Adapted Screenplay, Mankiewicz's second consecutive wins in both categories.

Widely considered as among the greatest films of all time, in 1990, All About Eve became one of 25 films selected for preservation in the United States Library of Congress's National Film Registry, deemed "culturally, historically, or aesthetically significant". The film was ranked No. 16 on AFI's 1998 list of the 100 best American films.

Plot
Broadway star Margo Channing, recently turned 40, worries about what advancing age will mean for her career. After a performance of Margo's latest play, Margo's close friend, Karen Richards, wife of the play's author Lloyd Richards, brings besotted fan Eve Harrington backstage to meet Margo. In Margo's dressing room, Eve tells Karen, Lloyd, and Margo's maid Birdie, that she followed Margo's last theatrical tour to New York City after seeing her perform in San Francisco. She tells an engrossing story of growing up poor in Wisconsin and losing her young husband, Eddie, in the South Pacific during World War II. Moved by Eve's story, Margo takes her into her home as her assistant, upsetting Birdie. Eve quickly manipulates her way into Margo's life, acting as both secretary and adoring fan.

Adding to Margo’s concerns about aging is her romantic relationship with the play’s director, Bill Sampson, 8 years her junior. Without telling Margo, Eve places a long-distance phone call “on Margo’s behalf” to Bill when Margo forgets his birthday. This results in Margo’s embarrassment, exposing her to Bill's realizing that Margo forgot his birthday, while Eve remembered and even sent her own greetings. Margo grows increasingly distrustful toward Eve, particularly after catching Eve taking a bow in front of a mirror while pretending to wear Margo's costume. Margo asks producer Max Fabian to hire Eve at his office, but instead, Eve becomes Margo's understudy without Margo's knowledge. As Margo's irritation grows, Karen sympathizes with Eve, believing that Margo is overdramatizing her resentment of Eve out of temperamental peevishness. Hoping to humble Margo, Karen conspires for her to miss a performance so that Eve can perform. Eve invites the city's theater critics to attend the performance – including the acerbic and influential Addison DeWitt. Eve's performance is a triumph. Later that night, Eve attempts to seduce Bill, who as a successful Broadway producer might be useful to her career. Bill rejects Eve, letting her know that he sees her manipulations. Instead, Addison takes an interest in Eve. He interviews her for a column, harshly criticizing Margo for resisting younger talent.

Margo and Bill announce their engagement at dinner with Lloyd and Karen. Eve summons Karen to the ladies' room and, after first appearing regretful, delivers an ultimatum: Karen must recommend her to Lloyd to play Cora, the lead role in Lloyd's new play. Otherwise, she will reveal Karen's part in Margo's missed performance. When Karen returns to the table—to her relief—Margo surprisingly announces that she does not wish to play Cora, saying she is too old for the ingénue role.

Eve is cast as Cora. Just before the new play's premiere in New Haven, Eve reveals her next plan to Addison: to marry Lloyd—who she has seduced despite his wife Karen’s championship of her—so that Lloyd can write plays for her to star in. Angered with Eve's audacity, Addison says he knows her backstory is entirely lies; her real name is Gertrude Slescynski, she was never married, and she was paid to leave town over an affair with her married boss. He also says Lloyd would never leave Karen for Eve. Addison then blackmails Eve, saying she now "belongs" to him.

Months later, Eve is a Broadway star headed for Hollywood. While accepting an award at a banquet, she thanks Margo, Bill, Lloyd, and Karen as all four coldly stare back. Eve skips the after-party and returns home, where she encounters Phoebe, a teenage fan who slipped into her apartment and fell asleep. Phoebe professes her adoration and ingratiates herself with Eve, then begins packing her trunk. Eve invites her to stay over rather than take the long subway ride back to Brooklyn. While Eve is resting, Addison brings Eve's forgotten award to the door and is greeted by Phoebe, who admits she chose her own theatrical name. Addison realizes Phoebe will do to Eve what Eve did to Margo. When she is alone, Phoebe puts on Eve's elegant cloak and poses in front of a floor-length mirror, holding the award and bowing.

Cast

 * Bette Davis as Margo Channing
 * Anne Baxter as Eve Harrington
 * George Sanders as Addison DeWitt
 * Celeste Holm as Karen Richards
 * Gary Merrill as Bill Sampson
 * Hugh Marlowe as Lloyd Richards
 * Thelma Ritter as Birdie Coonan
 * Gregory Ratoff as Max Fabian
 * Marilyn Monroe as Claudia Casswell
 * Barbara Bates as Phoebe
 * Walter Hampden as Aged Actor (Sarah Siddons Award presenter)
 * Randy Stuart as Girl
 * Craig Hill as Leading Man
 * Leland Harris as Doorman
 * Barbara White as Autograph Seeker
 * Eddie Fisher as Stage Manager
 * William Pullen as Clerk
 * Claude Stroud as Pianist (at party)
 * Eugene Borden as Frenchman
 * Helen Mowery as Reporter
 * Steven Geray as Captain of Waiters

Development
The story of All About Eve originated in an anecdote related to Mary Orr by actress Elisabeth Bergner. While performing in The Two Mrs. Carrolls during 1943 and 1944, Bergner allowed a young fan to become part of her household and employed her as an assistant, but later regretted her generosity when the woman attempted to undermine her. Referring to her only as "the terrible girl", Bergner related the events to Orr, who used it as the basis for her short story "The Wisdom of Eve" (1946). In the story, Orr gives the girl an even more ruthless character and allows her to succeed in stealing the older actress's career and the husband of the unnamed female narrator. Bergner later confirmed the basis of the story in her autobiography Bewundert viel, und viel gescholten (Greatly Admired and Greatly Scolded).

In 1949, Joseph Mankiewicz was considering a story about an aging actress and, upon reading "The Wisdom of Eve," felt that the conniving girl would be a useful element. He sent a memo to Darryl F. Zanuck saying it "fits in with an original idea [of mine] and can be combined. Superb starring role for Susan Hayward." Mankiewicz presented a film treatment of the combined stories under the title Best Performance. He changed the main character's name from Margola Cranston to Margo Channing and retained several of Orr's characters – Eve Harrington, Lloyd and Karen Richards and Miss Casswell – while removing Margo's husband completely and replacing him with a new character, Bill Sampson. The intention was to depict Margo in a new relationship and allow Eve to threaten Margo's professional and personal lives. Mankiewicz also added the characters Addison DeWitt, Birdie Coonan, Max Fabian and Phoebe.

Zanuck was enthusiastic and provided numerous suggestions for improving the screenplay. In some sections, he felt that Mankiewicz's writing lacked subtlety or provided excessive detail. He suggested diluting Birdie Coonan's mistrust of Eve so the audience would not recognize Eve as a villainess until much later in the story. Zanuck reduced the screenplay by about 50 pages and chose the title All About Eve from the opening scene in which Addison DeWitt says that he will soon tell "more of Eve ... All about Eve, in fact."

Casting
Among the actresses originally considered to play Margo Channing were Mankiewicz's original inspiration Susan Hayward, who was rejected by Zanuck as "too young", Marlene Dietrich, dismissed as "too German" and Gertrude Lawrence, who was ruled out when her lawyer insisted that she not have to drink or smoke in the film and that the script would be rewritten to allow her to sing a torch song. Zanuck favored Barbara Stanwyck, but she was not available. Tallulah Bankhead was considered, as was Joan Crawford, who was working on the film The Damned Don't Cry.

The role went to Claudette Colbert, but she withdrew after an injury shortly before filming began. Mankiewicz briefly considered Ingrid Bergman before offering the role to Bette Davis. Davis, who had recently ended an 18-year association with Warner Bros. after several poorly received films, accepted the role, saying later that the script was among the best that she had ever read. Margo had been originally conceived as genteel and knowingly humorous, but with the casting of Davis, Mankiewicz revised the character to introduce abrasive qualities. Mankiewicz praised Davis for her professionalism and for the caliber of her performance.

Anne Baxter had spent a decade in supporting roles and had won the 1946 Academy Award for Best Supporting Actress for The Razor's Edge. She won the role of Eve after Jeanne Crain, the first choice, became pregnant. Crain was at the height of her popularity and had established a career playing likable heroines; Zanuck believed that she lacked the "bitch virtuosity" required by the part and that audiences would not accept her as a deceitful character.

Mankiewicz greatly admired Thelma Ritter and wrote the character of Birdie Coonan for her after working with her on A Letter to Three Wives in 1949. As Coonan is the only character immediately suspicious of Eve Harrington, Mankiewicz was confident that Ritter would contribute a shrewd characterization that cast doubt on Eve and provided a counterpoint to the more theatrical personalities of the other characters. Marilyn Monroe, relatively unknown at the time, was cast as Miss Casswell, referred to by DeWitt as a "graduate of the Copacabana School of Dramatic Art."

Monroe won the part after a lobbying campaign by her agent, despite Zanuck's initial antipathy and belief that she was better suited to comedy. The inexperienced Monroe was cowed by Davis, and it took 11 takes to complete the scene in the theater lobby; when Davis barked at her, Monroe left the set to vomit. Smaller roles were filled by Gregory Ratoff as the producer Max Fabian, Barbara Bates as Phoebe and Walter Hampden as the host of the award ceremony. Hampden was the president of the prestigious Players Club in New York, a club for actors that gives a lifetime achievement award.

Box office
The film earned $3.1 million in receipts in the United States during its release, more than double its original budget of $1.4 million. To date the film has a cumulative gross of $8.4 million, more than five times its production costs.

Critical response
All About Eve received widespread critical acclaim upon its release on October 13, 1950, at a New York City premiere. The film's competitor, Sunset Boulevard, released the same year, drew similar praise, and the two were often favorably compared. Film critic Bosley Crowther of The New York Times loved the picture, stating that "a fine Darryl Zanuck production, excellent music and an air of ultra-class complete this superior satire." Variety called it "a literate, adult film" with "exceedingly well-cast performances," while Harrison's Reports called it "a fascinating, continually absorbing story about Broadway theatrical people, given a mature treatment and penetrated with realistic dialogue and flashes of slick, sardonic humor." John McCarten of The New Yorker called it "a thoroughly entertaining movie."

Writing in 2000, film critic Roger Ebert of the Chicago Sun Times praised the film, saying of Bette Davis that "veteran actress Margo Channing in All About Eve was her greatest role." Boxoffice.com stated that it "is a classic of the American cinema – to this day the quintessential depiction of ruthless ambition in the entertainment industry, with legendary performances from Bette Davis, Anne Baxter and George Sanders anchoring one of the very best films from one of Hollywood's very best Golden Era filmmakers: Joseph L. Mankiewicz."

As of 2021 review aggregator website Rotten Tomatoes, All About Eve holds an approval rating of 99% based on 107 reviews, with an average rating of 9.30/10. The site's critics consensus reads: "Smart, sophisticated, and devastatingly funny, All About Eve is a Hollywood classic that only improves with age." In 2019 Metacritic assigned a weighted average score of 98 out of 100, based on 15 critics, indicating "universal acclaim".

Thematic content
Critics and academics have delineated various themes in the film. Rebecca Flint Marx, in her Allmovie review, notes the antagonism (active hostility or opposition) that existed between Broadway and Hollywood at the time, stating that the "script summoned into existence a whole array of painfully recognizable theatre types, from the aging, egomaniacal grand dame to the outwardly docile, inwardly scheming ingenue to the powerful critic who reeks of malignant charm." Abel Green, writing in Variety said, "The snide references to picture people, the plug for San Francisco ("an oasis of civilization in the California desert") and the like are purposeful and manifest an intelligent reflex from a group of hyper-talented people towards the picture business."

Roger Ebert, in his review in The Great Movies, says Eve Harrington is "a universal type", and focuses on the aging actress plot line, comparing the film to Sunset Boulevard. Similarly, Marc Lee's 2006 review of the film for The Daily Telegraph describes a subtext "into the darker corners of show business, exposing its inherent ageism, especially when it comes to female stars." Kathleen Woodward's 1999 book, Figuring Age: Women, Bodies, Generations (Theories of Contemporary Culture), also discusses themes that appeared in many of the "aging actress" films of the 1950s and 1960s, including All About Eve. She reasons that Margo has three options: "To continue to work, she can perform the role of a young woman, one she no longer seems that interested in. She can take up the position of the angry bitch, the drama queen who holds court (the deliberate camp that Susan Sontag finds in this film). Or she can accept her culture's gendered discourse of aging which figures her as in her moment of fading. Margo ultimately chooses the latter option, accepting her position as one of loss."

All About Eve has long been a favored film among gay audiences, likely due to its campy overtones (in part due to the casting of Davis) and its general sophistication. Davis, who long had a strong gay fan base, expressed support for gay men in her 1972 interview with The Advocate.

Later recognition and rankings
In 1990, All About Eve was selected for preservation in the United States National Film Registry by the Library of Congress as being "culturally, historically, or aesthetically significant." The Academy Film Archive preserved All About Eve in 2000. The film received in 1997 a placement on the Producers Guild of America Hall of Fame. The film has been selected by the American Film Institute for many of their 100 Years lists.

When AFI named Bette Davis #2 on its list of the greatest female American screen legends, All About Eve was the film selected to highlight Davis' legendary career. The Writers Guild of America has ranked the film's screenplay as the fifth greatest ever written.

Sarah Siddons Award
The film opens with the image of a fictitious award trophy, described by DeWitt as the "highest honor our theater knows: the Sarah Siddons Award for Distinguished Achievement." The statuette is modeled after the famous painting of Siddons costumed as the tragic Muse by Joshua Reynolds, a copy of which hangs in the entrance of Margo's apartment and often visible during the party scene. In 1952, a small group of distinguished Chicago theater-goers began to give an award with that name, which was sculpted to look like the one used in the film. It has been given annually, with past honorees including Bette Davis and Celeste Holm.

Adaptations
The first radio adaptation was a one-hour broadcast on Lux Radio Theatre on CBS Radio on October 1, 1951, with Bette Davis, Gary Merrill and Anne Baxter reprising their original roles. Lux Radio Theatre did a follow-up adaptation on November 23, 1954, this time on NBC radio with Ann Blyth and Claire Trevor playing the lead roles, with Trevor replacing Ida Lupino when she became ill and was unable to attend the broadcast.

A radio version of All About Eve starring Tallulah Bankhead as Margo Channing was presented on NBC's The Big Show by the Theatre Guild of the Air on November 16, 1952. Bankhead and many contemporary critics felt that the characterization of Margo Channing was patterned on her, a long-rumored charge denied by both Mankiewicz and Davis, but attested by costume designer Edith Head. Additionally, Bankhead's rivalry with her understudy (Lizabeth Scott) during the production of The Skin of Our Teeth is cited as an alternative hypothesis for the origin of Mary Orr's The Wisdom of Eve, the original short story that formed the basis for the film. Bette Davis played three roles on film that Tallulah Bankhead had originated – Dark Victory, Jezebel and The Little Foxes, much to Bankhead's chagrin. Bankhead and Davis were considered to be somewhat similar in style. Several decades later Davis called Channing "the essence of a Tallulah Bankhead kind of actress" in an interview with Barbara Walters. The production is notable in that Mary Orr, of The Wisdom of Eve, played the role of Karen Richards. The cast also featured Alan Hewitt as Addison DeWitt (who narrated), Beatrice Pearson as Eve Harrington, Don Briggs as Lloyd Richards, Kevin McCarthy as Bill Samson, Florence Robinson as Birdie Coonan, and Stefan Schnabel as Max Fabian.

In 1970, All About Eve was the inspiration for the stage musical Applause, with book by Betty Comden and Adolph Green, lyrics by Lee Adams, and music by Charles Strouse. The original production starred Lauren Bacall as Margo Channing, and it won the Tony Award for Best Musical that season. It ran for four previews and 896 performances at the Palace Theatre on Broadway. After Bacall left the production, she was replaced by Anne Baxter in the role of Margo Channing.

In 2019, a stage adaptation of All About Eve premiered at the Noël Coward Theatre in London, directed by Ivo van Hove and starring Gillian Anderson as Margo Channing, Julian Ovenden as Bill, and Lily James as Eve Harrington.

In popular culture

 * The plot of the film has been used numerous times, frequently as an outright homage to the film, with one notable example being a 1974 episode of The Mary Tyler Moore Show titled "A New Sue Ann". In the episode, the character of Sue Ann Nivens (Betty White), hostess of a popular local cooking show, hires a young, pretty and very eager fan (Linda Kelsey) as her apprentice and assistant, but the neophyte quickly begins to sabotage her mentor, in an attempt to replace her as host of the show. Sue Ann, however, unlike Margo Channing, prevails in the end, countering the young woman's attempts to steal her success and sending her on her way.
 * Brazilian telenovela Celebridade was loosely based on All About Eve plot; it features Malu Mader as the successful businesswoman and former model Maria Clara Diniz, who hires Laura Prudente da Costa (Cláudia Abreu) for a job in her company after she saves her life and claims to be her greatest fan — but she is, in fact, an imposter who wants not only to take everything from the other woman, but to become a new Maria Clara.
 * The English rock band All About Eve took their name from the film.
 * Steven Soderbergh's 2012 film Magic Mike is a loose re-working of the All About Eve plot and includes subtle references in homage to the original. The lead actor Channing Tatum name checks Bette Davis's character Margo Channing, and instead of Eve, the ambitious young upstart is named Adam, played by Alex Pettyfer. Like Eve, Adam gets his stage debut filling in for an absent star, and his subsequent ruthless rise to glory at the expense of others mirrors that of Eve.
 * A 2008 episode of The Simpsons titled "All About Lisa" is influenced by this film. In the episode, Lisa Simpson becomes Krusty the Clown's assistant, eventually taking his place on television and receiving an entertainment award.
 * Pedro Almodóvar's 1999 Academy Award-winning Spanish-language film Todo sobre mi madre (All About My Mother) has elements similar to those found in All About Eve. The title of the film itself is an homage to the 1950 film.
 * In the fifth season of The L Word, a fan becomes Jenny Schecter's assistant while she is directing a movie; later the fan blackmails the movie studio into letting her direct and she proceeds to take over Jenny's life.
 * In the second season of Glee, Kurt Hummel calls his fellow glee club member Santana Lopez "a Latina Eve Harrington", after learning she is blackmailing a closeted jock into becoming her "beard" and running mate for Prom Queen and King.
 * In the first season of Will & Grace, Grace becomes dependent on a maid to give her a confidence boost during a design competition. This prompts her drunken assistant Karen to suspect a plot and she confronts the maid, exclaiming "I've seen All About Eve. Poooor Eve!"
 * In the pilot episode of Political Animals, when Susan suspects Georgia, a fellow reporter, has a crush on her boyfriend and is attempting to outshine her at the newspaper, she says, "If Eve Harrington were an actual person today, she would look like Georgia. She would bake cupcakes, and she would have a blog."
 * In the third season of Gilligan's Island, the episode "All About Eva" concerns the character of Eva Grubb coming on the island and taking over Ginger's persona, with both roles played by actress Tina Louise.
 * In the fifth season of Quantum Leap, the plot of the episode "Goodbye Norma Jean" mirrors that of All About Eve. In it, Sam Beckett leaps into Marilyn Monroe's chauffeur and finds himself pitted against an aspiring actress who is trying to steal Monroe's part for the film The Misfits. Sam succeeds in stopping Monroe's rival, and she rightfully takes her place as Clark Gable's leading lady.
 * In Alex Holdridge's 2007 film In Search of a Midnight Kiss, Vivian, an amateur actress played by Sara Simmonds, says to Wilson, played by Scoot McNairy: "I just don't think people out here have the raw ambition that I do." And he replies: "You are a real All About Eve."
 * All About Steve is a 2009 American comedy film.
 * A season 6 episode of Designing Women titled, "All About Odes to Atlanta", sees Carlene, Mary Jo, and Julia entering a song contest and drawing the attention of a young "fan" who seems all too eager to help them, much to Allison's distrust.
 * The season 20 episode of Family Guy titled, "All About Alana", is a homage to the film and sees Lois Griffin allowing one of her piano students Alana Fitzgerald to move in with the family, who begins slowly taking over Lois' life.
 * The storyline was partially adapted in 2023 and 2024 by All Elite Wrestling, in which Mariah May was first portrayed as a super-fan, and later, protegé of "Timeless" Toni Storm – who was herself portraying gimmick based on Sunset Boulevard – before May's transition to a more villainous character whose plan all along was to supplant Storm.

title: All About Eve


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MIL-STD-188

MIL-STD-188 is a series of U.S. military standards relating to telecommunications.

Purpose
Faced with "past technical deficiencies in telecommunications systems and equipment and software…that were traced to basic inadequacies in the application of telecommunication standards and to the lack of a well defined…program for their review, control and implementation", the U.S. Department of Defense looked to develop a series of standards that would alleviate the problem.

By 1988, the U.S. Department of Defense (DoD) issued Instruction 4630.8 (reissued in 1992, 2002, 2004) stating its policy that "all forces for joint and combined operations be supported through compatible, interoperable, and integrated Command, Control, Communications, and Intelligence systems. …[and that all such] systems developed for use by U.S. forces are considered to be for joint use." To achieve this the director of the Defense Information Systems Agency (DISA) is charged with "developing information technology standards to achieve interoperability and compatibility…[and ensure that all] systems and equipment shall conform to technical and procedural standards for interface, interoperability, and compatibility".

The MIL-STD-188 standards were created to "address telecommunication design parameters based on proven technologies." To ensure interoperability, DISA made these standards mandatory for use in all new DoD systems and equipment, or major upgrades.

"The mandatory use of these standards will aid significantly in achieving standardization and result in improvements in availability, maintainability, reliability, and supportability. This, in turn, will enhance lifecycle configuration management and logistic support with subsequent reductions in life cycle costs."

Evolution
When first developed, Military Standard 188 covered technical standards for tactical and long-haul communications, but as it was revised (MIL-STD-188A, MIL-STD-188B) it became a document applicable to tactical communications only (MIL-STD-188C 24 Nov 1969 ). The Defense Information Systems Agency published circulars which announced both standards and engineering criteria relating to the long-haul Defense Communications System and to the technical support of the Worldwide Military Command and Control System. In line with a decision by the Joint Chiefs of Staff, these standards are published in the MIL-STD-188 series of documents. This series is subdivided into "a MIL-STD-188-100 series covering common standards for tactical and long-haul communications, a MIL-STD-188-200 series covering standards for tactical communications only, and a MIL-STD-188-300 series covering standards for long-haul communications only."

The MIL-STD-188 series standards are encompassed by the DoD's Joint Technical Architecture.

Deviations and waivers
For any manufacturer seeking to deviate from the MIL–STD-188 series standards (prior to the manufacture of an item) they must request to do so with the Joint Steering Committee (JSC) which is constituted under the Defense Communications Agency. For any DoD Agency to get a waiver to receive an item that deviates from the standards they also must apply to the JSC.

Relation to other systems of standards
According to DoD documents, "The MIL-STD-188 series may be based on, or make reference to, Joint Technical Architecture, American National Standards Institute (ANSI) standards, International Telecommunication Union - Telecommunication Standardization Sector (ITU-T) recommendations, North Atlantic Treaty Organization (NATO) Standardization Agreements (STANAG), and other standards wherever applicable."

Current development emphasis
Currently the DoD is placing its emphasis "on the development of common standards for tactical and long-haul communications (the MIL-STD-188-100 series)."

Documents
Note: The following list of documents are those that are presently active. Documents with three digit numbers followed by a letter of the alphabet indicate that they are revisions of an older version of that document.

MIL-STD-188-100 series
According to the DoD the MIL-STD-188-100 series contains "technical standards and design objectives which are common to both the long haul and tactical communications systems."

The current articles in this series include:

Digital Interface Circuits (can interoperate with RS-422 and RS-423, but not identical)
 * 100 - Common Long Haul and Tactical Communication System Technical Standards
 * 105 - Interoperability and Performance Standards for the All Digital Tactical-To-Strategic Gateway
 * 110D - Interoperability And Performance Standards For Data Modems
 * 111A - Interoperability And Performance Standards For Fiber Optic Communications Systems
 * 112 - Subsystem Design And Engineering Standards For Common Long Haul Tactical Cable And Wire Communications
 * 113 - Interoperability And Performance Standards For Analog- To- Digital Conversion Techniques
 * 114A - Electrical Characteristics of PC teléfono apo 78
 * 115 - Interoperability And Performance Standards For Communications Timing And Synchronization Subsystems
 * 116 - Interoperability Standards For Information And Record Traffic Exchange Mode
 * 120 - Military Communication System Standards Terms And Definitions
 * 124B - Grounding, Bonding And Shielding For Common Long Haul/Tactical Communication Systems Including Ground Based Communications-Electronics Facilities And Equipments
 * 125A - High-Altitude Electromagnetic Pulse (HEMP) Protection For Ground-Based Facilities Performing Critical, Time-Urgent Missions
 * 136A - Satellite Data Link Standard (SDLS) For EHF Medium Data Rate (MDR) Uplinks And Downlinks
 * 140A - Equipment Technical Design Standards For Common Long Haul/Tactical Radio Communications In The Low Frequency Band And Lower Frequency Bands
 * 141D - Interoperability and Performance Standards for Medium and High Frequency Radio Systems (includes standards for automatic link establishment)
 * 145 - Interoperability And Performance Standards For Digital Line of sight Microwave Radio Equipment
 * 146 - Interoperability And Performance Standards For Satellite Communications
 * 148B - Interoperability Standard for Anti Jam (AJ) Communications In The High Frequency (2-30 MHz) Band
 * 154A - Subsystem, Equipment, And Interface Standards For Common Long Haul And Tactical Telecommunications Control Facilities
 * 161D - Interoperability and Performance Standards for Digital Facsimile Equipment
 * 164A - Interoperability and Performance Standards for C-Band, X-Band, and Ku-Band SHF Satellite Communications Earth Terminals
 * 165A - Interoperability and Performance Standards for SHF Satellite Communications PSK Modems (Fdma Operation)
 * 166 - Interface Standard, Interoperability and Performance Standard for SHF SATCOM Link Control
 * 167 - Interface Standard, Message Format for SHF SATCOM Link Control
 * 168 - Interface Standard, Interoperability and Performance Standards for SHF Satellite Communications Multiplexers and Demultiplexers
 * 171 - Interoperability Standards For Information And Record Traffic Exchange Mode I
 * 172 - Interoperability Standards For Information And Record Traffic Exchange Mode Ii
 * 173 - Interoperability Standards For Information And Record Traffic Exchange Mode V
 * 174 - Interoperability Standards For Information And Record Traffic Exchange Mode Vi
 * 176 - Standardized Profile For Asynchronous Transfer Mode (ATM)
 * 181B - Advanced Narrowband Digital Voice Terminal (ANDVT) and 56 kbit/s data
 * 181C - Interoperability Standard For Access To 5-kHz And 25-kHz UHF Satellite Communications Channels
 * 182B - Interoperability Standard For UHF SATCOM Demand Assigned Multiple Access (DAMA) Orderwire Messages And Protocols
 * 183B - Interoperability Standard For Multiple-Access 5-kHz And 25-kHz UHF Satellite Communications Channels
 * 184 - Interoperability and Performance Standard for the Data Control Waveform
 * 185 - Interoperability UHF Milsatcom DAMA Control System
 * 186 - Interoperability Standard For UHF SATCOM Short-Delay Report-Broadcast Service
 * 187 - Interoperability Standard for Mobile User Objective System (MUOS) Terminals
 * 190 - Methods for Communication System Measure
 * 194 - Integrated Services Digital Network Profile
 * 196 - Bi-Level Image Compression for the National Imagery Transmission Format Standard
 * 197A - Adaptive Recursive Interpolated Differential Pulse Code Modulation (ARIDPCM) Compression Algorithm For The National Imagery Transmission Format Standard
 * 198A - Joint Photographic Experts Group (JPEG) Image Compression for the National Imagery Transmission Format Standard
 * 199 - Vector Quantization Decompression for the National Imagery Transmission Format Standard
 * 203-1A - Interoperability and Performance Standards for Tactical Digital Information Link (Tadil) A (Link 11)

MIL-STD-188-200 series
According to the DoD the MIL-STD-188-200 series "contains current tactical communications, technical standards and design objectives…[this series includes] appropriate unclassified design objectives and tactical communications systems technical standards…[and] Appropriate communications-electronics systems standards and design objectives developed under joint projects…[which are] integrated in the tactical communications standards."

The current articles in this segment include:

 * 200 - System Design and Engineering Standards For Tactical Communications
 * 202 - Interoperability And Performance Standards For Tactical Digital Transmission Groups (Coaxial Cable) For Tactical Communications
 * 203 - Interoperability And Performance Standards For Tactical Digital Information Link (TADIL) A
 * 212 - Subsystem Design And Engineering Standards For Tactical Digital Information Link (TADIL) B
 * 216A - Interoperability Standards For Data Adapter Control Mode
 * 220D - Digital Message Transfer Device Subsystems (Combat-net radio)
 * 241 - RF Interface Requirements for VHF Frequency Hopping Tactical Radio Systems
 * 242 - Interoperability & Performance Standards For Tactical Single Channel Very High Frequency (VHF) Radio Equipment
 * 243 - Interoperability And Performance Standards For Tactical Single Channel Ultra High Frequency (UHF) Radio Communications
 * 256 - Interoperability And Performance Standards For Digital Signaling And Supervision Of Tactical Communications Systems
 * 260 - Design And Engineering Standards For Tactical Terminal Subsystems

MIL-STD-188-300 series
According to the DoD the MIL-STD-188–300 Series contains "communications system standards and design objectives applicable to the field of long haul and point-to–point communications in support of the Defense Communications System (DCS) and the National Military Command System (NMCS), and also to provide the necessary interface with non-DCS equipment."

The current articles in this series include:

 * 300 - Standards For Long Haul Communications System Design Standards Applicable To The Defense Communications System
 * 310A - Subsystem Design And Engineering Standards For Technical Control Facilities
 * 311 - Technical Design Standards For Frequency Division Multiplexers
 * 313 - Subsystem Design And Engineering Standards And Equipment Technical Design Standards For Long-Haul Communications Transversing Microwave Line-of-Sight (LOS) Radio And Tropospheric Scatter Radio
 * 315 - Subsystem Design/Engineering & Equipment Technical Design Standards For Wire Systems
 * 331  - Interoperability and Performance Standard for Video Teleconferencing
 * 340  - Equipment Technical Design Standards for Voice Orderwire Multiplex
 * 341  - Non-Diversity Digital Data 2400 bits per second
 * 342  - Standards for Long Haul Communications Equipment Technical Design Standards for Voice Frequency Carrier Telegraph (FSK)
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title: MIL-STD-188


text:
Weser

The Weser is a river of Lower Saxony in north-west Germany. It begins at Hannoversch Münden through the confluence of the Werra and Fulda. It passes through the Hanseatic city of Bremen. Its mouth is 50 km further north against the ports of Bremerhaven and Nordenham. The latter is on the Butjadingen Peninsula. It then merges into the North Sea via two highly saline, estuarine mouths.

It connects to the canal network running east–west across the North German Plain.

The river, when combined with the Werra (a dialectal form of Weser), is 744 km long and thus, the longest river entirely situated within Germany (the Main, however, is the longest if the Weser and Werra are not combined). The Weser itself is 452 km long. The Werra rises in Thuringia, the German state south of the main projection (tongue) of Lower Saxony.

Etymology
"Weser" and "Werra" are the same words in different dialects. The difference reflects the old linguistic border between Central and Low German, passing through Hannoversch Münden.

The name likely derives from the Old Germanic *waisōn "flow, ooze". It is cognate with the Wear in England and Vistula (Polish Wisła, German Weichsel) in Poland, all of which are derived from the Proto-Indo-European root *weys- "to flow", which also gives rise to Old English/Old Frisian wāse "mud, ooze", Old Norse veisa "slime, stagnant pool", Dutch waas "haze; soggy land" (see Waasland), Old Saxon waso "wet ground, mire", Old High German wasal "rain" and French vase "mud, sludge".

Course
The Weser starts at the confluence of the Fulda and the Werra. It then runs down to the Porta Westfalica between two high hill ranges, the Wiehengebirge, west and the Weserbergland in the east.

Between Minden and the North Sea, humans have largely canalised the river up to a limit of 1,200-ton ships. Eight hydroelectric dams stand at the ends of adjacent weirstreams that make up the river. The navigation is linked west to the Dortmund–Ems Canal via the Coastal Canal. It is linked east at Bremerhaven to the Elbe.

A large reservoir, the Edersee, on the Eder, the main tributary of the Fulda, is used to allow enough water depth for shipping year-round. The dam, built in 1914, was bombed and severely damaged by British aircraft in May 1943, causing great destruction and about 70 deaths downstream. It was rebuilt within four months. The reservoir is a major summer resort area. Turbines driven by its sluices provide electricity.

The Weser enters the North Sea in the southernmost part of the German Bight. In the sea it splits into two arms – the riverbed at the end of the last ice age. These sea arms are called Alte Weser (old Weser) and Neue Weser (new Weser). They are the waterways for ships heading for the ports of Bremerhaven, Nordenham, and Bremen. The Alte Weser Lighthouse marks the northernmost point of the Weser. This replaced the Roter Sand Lighthouse in 1964.

Tributaries
The largest tributary of the Weser is the Aller, which joins south of Bremen. Tributaries of the Weser and the Werra (from source to mouth) are:

Modes of the list: List:
 * Listed upstream, but sides seen with the flow
 * Distances ("km …") from the hydrographical limit towards the sea
 * "II", "III"and "IV" mark distances of secondary/tertiary tributaries from the confluence with the Weser etc.
 * After the names, lengths and basin sizes are given.
 * Lengths with longer affluents are given behind the slash, lengths including an upper course with another name with "or"
 * km 19, right: Geeste (in Bremerhaven), 42.5 km, 338 km2
 * km 33, right: Lune, 43 km, 383 km2
 * km 35.9, right: Drepte, 37.6 km, 101 km2
 * km 52.8, left: Hunte, 189 km, 2.785 km2
 * II: km 125.7: Lake Dümmer
 * km 67.6, right: Lesum, 9.9 or 131.5, 2,188 km2
 * II: km 9.9, right Hamme, 48.5 km, 549 km2
 * ↑ main stream: Wümme, 118 / 120, 1,585 km2
 * km 72.5, left: Ochtum, 25.6 or 45 km, 917 km2
 * II: km 25.6: left Hache, 33 km, 118 km2
 * km 125.6, right: Aller, 260 km, 15,744 km2
 * II: km 63.6, left: Leine, 278 km, 5,617 km2, stronger than river Aller above
 * III: km 112.7, right: Innerste, 99.7 km, 1,264 km2
 * III: km 192.8, right: Rhume, 44 km, 1,193 km2, stronger than river Leine above
 * IV: km 15.6, right: Oder, 56 km, 385 km2, headwater of the strongest waterway of Aller system
 * II: km 97.3, right: Örtze, 62 / 70 km, 760 km2
 * II: km 140.7, left: Oker, 218 km, 1822 km2, stronger than river Aller above
 * km 184.6, right: Steinhuder Meerbach
 * ↑ km II: 29 lake Steinhuder Meer
 * km 188.7, left: Große Aue, 84.5 km, 1,522 km2
 * km 261.3, left: Werre, 71.9 km, 1485 km2
 * II: km 12.7, left: Else, 34.6 km, 416 km2, branch of the Hase, an affluent of Ems
 * km 287.7, left: Exter, 26.1 km, 109 km2
 * km 323.3, left: Emmer, 61.8 km, 535 km2
 * km 387.5, left: Nethe, 50.4 km, 460 km2
 * km 406.5, left: Diemel, 110.5 km, 1,762 km2
 * km 451.5, left: Fulda, 220.4 km, 6.947 km2
 * II: km 45.3, left: Eder, 176.1 km, 3,361 km2, headwater of the strongest waterway of Weser system
 * III: km 17.1, left: Schwalm, 97.1 km, 1.299 km2
 * ↑ III: km 49.4–70.5: Edersee reservoir
 * II: 120.1, right: Haune, 66.5 km, 500 km2

 * ↑ main stream above km 451.5: Werra, 299.6 km, 5.497 km2
 * km 566.5, right: Hörsel, 55.2 or 64.3, 784 km2
 * km 9.8, right: Nesse, 54.5 km, 426 km2
 * km 513.1, left: Ulster, 57.2 km, 421 km2
 * km 604.4, right: Schleuse, 34.2 km, 283 km2

Notable towns
Main towns along the Weser are (from the head of the river to its mouth): Hann. Münden, Beverungen, Höxter, Holzminden, Bodenwerder, Hamelin, Hessisch Oldendorf, Rinteln, Vlotho, Bad Oeynhausen, Porta Westfalica, Minden, Petershagen, Nienburg, Achim, Bremen, Brake, Nordenham, Bremerhaven.

Popular culture
The river features in the legend and folk tale the Pied Piper of Hamelin.

Etymology

 * Dieter Berger: Geographische Namen in Deutschland. Duden-Verlag, Mannheim 1999.
 * Hans Krahe: Sprache und Vorzeit. Quelle & Meyer, Heidelberg 1954. (Zur alteuropäischen Hydronomie.)
 * Julius Pokorny: Indogermanisches etymologisches Wörterbuch. Francke, Bern 1959.

Geology

 * Karsten Meinke: Die Entwicklung der Weser im Nordwestdeutschen Flachland während des jüngeren Pleistozäns. Diss., Göttingen 1992. Mit Bodenprofilen der Weserstädte.
 * Ludger Feldmann und Klaus-Dieter Meyer (Hrsg.): Quartär in Niedersachsen. Exkursionsführer zur Jubiläums-Hauptversammlung der Deutschen Quartärvereinigung in Hannover. DEUQUA-Exkursionsführer, Hannover 1998, page 89ff.
 * Hans Heinrich Seedorf und Hans-Heinrich Meyer: Landeskunde Niedersachsen. Natur und Kulturgeschichte eines Bundeslandes. Volume 1: Historische Grundlagen und naturräumliche Ausstattung. Wachtholz, Neumünster 1992, page 105ff.
 * Ludger Feldmann: Das Quartär zwischen Harz und Allertal mit einem Beitrag zur Landschaftsgeschichte im Tertiär. Papierflieger, Clausthal-Zellerfeld 2002, page 133ff and others.

Archaeology

 * Bremer Archäologische Blätter, Beiheft 2/2000 zur gleichnamigen Ausstellung im Focke-Museum: Siedler, Söldner und Piraten, Chauken und Sachsen im Bremer Raum, Der Landesarchäologe Bremen,.
 * Bremer Archäologische Blätter, Beiheft 3/2004 zur gleichnamigen Ausstellung im Focke-Museum: Gefundene Vergangenheit, Archäologie des Mittelalters in Bremen, Der Landesarchäologe Bremen, ISBN 3-7749-3233-6. (wg.Geschichte des Weserarms Balge)

History

 * Georg Bessell: Geschichte Bremerhavens. Morisse, Bremerhaven 1927, 1989.
 * Heinz Conradis: Der Kampf um die Weservertiefung in alter Zeit. In: Bremisches Jahrbuch. Bremen 41.1944.
 * J. W. A. Hunichs: Practische Anleitung zum Deich-, Siel- und Schlengenbau. Erster Theil, von den Sielen. Bremen 1770.
 * Die Kanalisierung der Mittelweser. Published by the Mittelweser AG, Carl Schünemann Verlag, Bremen 1960.
 * Kuratorium für Forschung im Küsteningenieurswesen: Die Küste. In: Archiv für Forschung und Technik an der Nord- und Ostsee. Boyens, Heide 51.1991.

Description

 * Karl Löbe: Das Weserbuch. Niemeyer, Hameln 1968.
 * Nils Aschenbeck, Wolf Dietmar Stock: Eine Flussfahrt von der Aller bis zur Nordsee. Atelier im Bauernhaus, Fischerhude 1998. ISBN 3-88132-350-3.

title: Weser


text:
Felix Klein

Felix Christian Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations between geometry and group theory. His 1872 Erlangen program classified geometries by their basic symmetry groups and was an influential synthesis of much of the mathematics of the time.

During his tenure at the University of Göttingen, Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His seminars covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction, and promoted mathematics education reform at all grade levels in Germany and abroad. He became the first president of the International Commission on Mathematical Instruction in 1908 at the Fourth International Congress of Mathematicians in Rome.

Life
Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.

Plücker died in 1868, leaving his book concerning the basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. In July 1870, at the beginning of the Franco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army before being appointed Privatdozent (lecturer) at Göttingen in early 1871.

The University of Erlangen appointed Klein professor in 1872, when he was only 23 years old. For this, he was endorsed by Clebsch, who regarded him as likely to become the best mathematician of his time. Klein did not wish to remain in Erlangen, where there were very few students, and was pleased to be offered a professorship at the Technische Hochschule München in 1875. There he and Alexander von Brill taught advanced courses to many excellent students, including Adolf Hurwitz, Walther von Dyck, Karl Rohn, Carl Runge, Max Planck, Luigi Bianchi, and Gregorio Ricci-Curbastro.

In 1875, Klein married Anne Hegel, granddaughter of the philosopher Georg Wilhelm Friedrich Hegel.

After spending five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig University. His colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed and he battled with depression for the next two years. Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period.

Klein accepted a professorship at the University of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer of geometry. He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular, mechanics and potential theory.

The research facility Klein established at Göttingen served as model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruited David Hilbert from the University of Königsberg. This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932.

Under Klein's editorship, Mathematische Annalen became one of the best mathematical journals in the world. Founded by Clebsch, it grew under Klein's management, to rival, and eventually surpass Crelle's Journal, based at the University of Berlin. Klein established a small team of editors who met regularly, making decisions in a democratic spirit. The journal first specialized in complex analysis, algebraic geometry, and invariant theory. It also provided an important outlet for real analysis and the new group theory.

In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the World's Columbian Exposition. Due partly to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, by Grace Chisholm Young, an English student of Arthur Cayley's, whom Klein admired. In 1897, Klein became a foreign member of the Royal Netherlands Academy of Arts and Sciences.

Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending that analytic geometry, the rudiments of differential and integral calculus, and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians. Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of the Royal Society in 1885, and was awarded its Copley Medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for several further years.

Klein was one of ninety-three signatories of the Manifesto of the Ninety-Three, a document penned in support of the German invasion of Belgium in the early stages of World War I.

He died in Göttingen in 1925.

Work
Klein's dissertation, on line geometry and its applications to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors.

Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan.

Klein devised the "Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip, with one method of construction being the attachment of the edges of two Möbius strips.

During the 1890s, Klein began studying mathematical physics more intensively, writing on the gyroscope with Arnold Sommerfeld. During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften. This enterprise, which endured until 1935, provided an important standard reference of enduring value.

Erlangen program
In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric. This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.

Currently, the significance of Klein's contributions to geometry is evident. They have become so much part of mathematical thinking that it is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries.

Complex analysis
Klein saw his work on complex analysis as his major contribution to mathematics, specifically his work on:
 * The link between certain ideas of Riemann and invariant theory,
 * Number theory and abstract algebra;
 * Group theory;
 * Geometry in more than 3 dimensions and differential equations, especially equations he invented, satisfied by elliptic modular functions and automorphic functions.

Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate the plane. In 1879, he examined the action of PSL(2,7), considered as an image of the modular group, and obtained an explicit representation of a Riemann surface now termed the Klein quartic. He showed that it was a complex curve in projective space, that its equation was x3y + y3z + z3x = 0, and that its group of symmetries was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats complex analysis in a geometric way, connecting potential theory and conformal mappings. This work drew on notions from fluid dynamics.

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on methods of Charles Hermite and Leopold Kronecker, he produced similar results to those of Brioschi and later completely solved the problem by means of the icosahedral group. This work enabled him to write a series of papers on elliptic modular functions.

In his 1884 book on the icosahedron, Klein established a theory of automorphic functions, associating algebra and geometry. Poincaré had published an outline of his theory of automorphic functions in 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during an asthma attack at 2:30 A.M. on 23 March 1882.

Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years.

Selected works

 * 1882: Über Riemann's Theorie der Algebraischen Functionen und ihre Integrale
 * , also available from Cornell
 * 1884:Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade
 * English translation by G. G. Morrice (1888) Lectures on the Ikosahedron; and the Solution of Equations of the Fifth Degree via Internet Archive
 * 1886: Über hyperelliptische Sigmafunktionen. Erster Aufsatz, pp. 323–356, Mathematische Annalen Bd. 27,
 * 1888: Über hyperelliptische Sigmafunktionen. Zweiter Aufsatz, pp. 357–387, Math. Annalen, Bd. 32,
 * 1890: (with Robert Fricke) Vorlesungen über die Theorie der elliptischen Modulfunktionen (2 volumes) and 1892)
 * 1894: Über die hypergeometrische Funktion
 * 1894: Über lineare Differentialgleichungen der 2. Ordnung
 * 1894: Evanston Colloquium (1893) reported and published by Ziwet (New York, 1894)
 * 1895: Vorträge über ausgewählte Fragen der Elementargeometrie
 * 1897: English translation by W. W. Beman and D. E. Smith Famous Problems of Elementary Geometry via Internet Archive
 * 1897: (with Arnold Sommerfeld) Theorie des Kreisels (later volumes: 1898, 1903, 1910)
 * Zweiter Band. 1901.
 * 1897: Mathematical Theory of the Top (Princeton address, New York)
 * 1901:
 * 1908: Elementarmathematik vom höheren Standpunkte aus (Leipzig)
 * 1926: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (2 Bände), Julius Springer Verlag, Berlin & 1927. S. Felix Klein Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert
 * 1928: Vorlesungen über nichteuklidische Geometrie, Grundlehren der mathematischen Wissenschaften, Springer Verlag
 * 1933: Vorlesungen über die hypergeometrische Funktion, Grundlehren der mathematischen Wissenschaften, Springer Verlag
 * 1928: Vorlesungen über nichteuklidische Geometrie, Grundlehren der mathematischen Wissenschaften, Springer Verlag
 * 1933: Vorlesungen über die hypergeometrische Funktion, Grundlehren der mathematischen Wissenschaften, Springer Verlag

title: Felix Klein


text:
Group theory

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.

The early history of group theory dates from the 19th century. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups.

History
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term "group" and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry.

Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups (now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory.

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups.

Main classes of groups
The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations.

Permutation groups
The first class of groups to undergo a systematic study was permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations) that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn; in general, any permutation group G is a subgroup of the symmetric group of X. An early construction due to Cayley exhibited any group as a permutation group, acting on itself (X = G) by means of the left regular representation.

In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥ 5 in radicals.

Matrix groups
The next important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the n-dimensional vector space Kn by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G.

Transformation groups
Permutation groups and matrix groups are special cases of transformation groups: groups that act on a certain space X preserving its inherent structure. In the case of permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: transformation groups frequently consist of all transformations that preserve a certain structure. The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous.

Abstract groups
Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two isomorphic groups are considered as the same group. A typical way of specifying an abstract group is through a presentation by generators and relations,

 * $$ G = \langle S|R\rangle. $$

A significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group G/H is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy.

The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under isomorphism, as well as the classes of group with a given such property: finite groups, periodic groups, simple groups, solvable groups, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school.

Groups with additional structure
An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion),

 * $$ m: G\times G\to G, (g,h)\mapsto gh, \quad i:G\to G, g\mapsto g^{-1}, $$

are compatible with this structure, that is, they are continuous, smooth or regular (in the sense of algebraic geometry) maps, then G is a topological group, a Lie group, or an algebraic group.

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transformation groups) are the mainstays of differential geometry and unitary representation theory. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, compact connected Lie groups have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group Γ can be realized as a lattice in a topological group G, the geometry and analysis pertaining to G yield important results about Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.

Finite group theory
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.

During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

Representation of groups
Saying that a group G acts on a set X means that every element of G defines a bijective map on the set X in a way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group homomorphism:

 * $$\rho:G \to \operatorname{GL}(V),$$

where GL(V) consists of the invertible linear transformations of V. In other words, to every group element g is assigned an automorphism ρ(g) such that ρ(g) ∘ ρ(h) = ρ(gh) for any h in G.

This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. On the one hand, it may yield new information about the group G: often, the group operation in G is abstractly given, but via ρ, it corresponds to the multiplication of matrices, which is very explicit. On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts (see Maschke's theorem). These parts, in turn, are much more easily manageable than the whole V (via Schur's lemma).

Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group's characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L2-space of periodic functions.

Lie theory
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse, page 3.

Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations (differential Galois theory), in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.

Combinatorial and geometric group theory
Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more compact way of defining a group is by generators and relations, also called the presentation of a group. Given any set F of generators $$\{g_i\}_{i\in I}$$, the free group generated by F surjects onto the group G. The kernel of this map is called the subgroup of relations, generated by some subset D. The presentation is usually denoted by $$\langle F \mid D\rangle.$$ For example, the group presentation $$\langle a,b\mid aba^{-1}b^{-1}\rangle$$ describes a group which is isomorphic to $$\mathbb{Z}\times\mathbb{Z}.$$ A string consisting of generator symbols and their inverses is called a word.

Combinatorial group theory studies groups from the perspective of generators and relations. It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. A fundamental theorem of this area is that every subgroup of a free group is free.

There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that there is in general no algorithm solving this task. Another, generally harder, algorithmically insoluble problem is the group isomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation $$\langle x,y \mid xyxyx = e \rangle,$$ is isomorphic to the additive group Z of integers, although this may not be immediately apparent. (Writing $$z=xy$$, one has $$G \cong \langle z,y \mid z^3 = y\rangle \cong \langle z\rangle.$$)

Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X, for example a compact manifold, then G is quasi-isometric (i.e. looks similar from a distance) to the space X.

Connection of groups and symmetry
Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example
 * If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups.
 * If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X.
 * If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example.
 * Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation $$x^2-3=0$$ has the two solutions $$\sqrt{3}$$ and $$-\sqrt{3}$$. In this case, the group that exchanges the two roots is the Galois group belonging to the equation.  Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.

The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative.

Frucht's theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object.

The saying of "preserving the structure" of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.

Applications of group theory
Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities.

Galois theory
Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S5, the symmetric group in 5 elements, is not solvable which implies that the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as class field theory.

Algebraic topology
Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group "counts" how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.

Algebraic geometry
Algebraic geometry likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the Hodge conjecture (in certain cases).) The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.

Algebraic number theory
Algebraic number theory makes uses of groups for some important applications. For example, Euler's product formula,

\begin{align} \sum_{n\geq 1}\frac{1}{n^s}& = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}, \\ \end{align} \!$$ captures the fact that any integer decomposes in a unique way into primes. The failure of this statement for more general rings gives rise to class groups and regular primes, which feature in Kummer's treatment of Fermat's Last Theorem.

Harmonic analysis
Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques.

Combinatorics
In combinatorics, the notion of permutation group and the concept of group action are often used to simplify the counting of a set of objects; see in particular Burnside's lemma.

Music
The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. Transformational theory models musical transformations as elements of a mathematical group.

Physics
In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether's theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the Standard Model, gauge theory, the Lorentz group, and the Poincaré group.

Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution.

Chemistry and materials science
In chemistry and materials science, point groups are used to classify regular polyhedra, and the symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct molecular orbitals.

Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule.

In chemistry, there are five important symmetry operations. They are identity operation (E), rotation operation or proper rotation (Cn), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (Sn). The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (Cn) consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/n, where n is an integer, about a rotation axis. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, n = 2, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the Cn axis having the largest value of n is the highest order rotation axis or principal axis. For example in boron trifluoride (BF3), the highest order of rotation axis is C3, so the principal axis of rotation is C3.

In the reflection operation (σ) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called σh (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (σv) or dihedral (σd).

Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, methane and other tetrahedral molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (Sn) requires rotation of  360°/n, followed by reflection through a plane perpendicular to the axis of rotation.

Cryptography
Very large groups of prime order constructed in elliptic curve cryptography serve for public-key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar's cipher, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite non-abelian groups such as a braid group.

title: Group theory