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1829_49 | The Underground is a multicultural student-run media site devoted to telling the untold stories within the Penn State community. The publication seeks to foster the multicultural student voice through creating an open forum of discussion and promoting diversity and community involvement. The media site was founded in 2015.
The LION 90.7 FM (WKPS-FM) was founded in 1995 as a replacement for Penn State's original student radio station WDFM. The LION broadcasts from the ground floor of the HUB–Robeson Center, serving the Penn State and State College communities with alternative music and talk programming, including live coverage of home Penn State football games. |
1829_50 | CommRadio is operated by the Penn State College of Communications. It was founded in the spring of 2003 as an internet-based audio laboratory and co-curricular training environment for aspiring student broadcasters. It airs both sports coverage and news. Other programming includes student talk shows, political coverage, AP syndicated news, and soft rock music. In recent years, ComRadio broadcasters have won numerous state awards for their on-air work.
La Vie (the Life), the university's annual student yearbook, has been in production documenting student life continuously since 1890. La Vie 1987, edited by David Beagin, won a College Gold Crown for Yearbooks award from the Columbia Scholastic Press Association. |
1829_51 | Kalliope is an undergraduate literary journal produced by students and sponsored by the university's English Department. It is published in the spring. Kalliope includes works of fiction, nonfiction, poetry, and visual art. In addition, Klio, an online publication, provides students with literary pieces in the fall semester.
Valley is Penn State's student-run life and style magazine. It was founded in 2007.
The student-run humor magazine is Phroth, which publishes two to four issues each year. Its roots date back to 1909 when it was called Froth. Several Froth writers and editors have gone on to win fame: Julius J. Epstein wrote the screenplay for the film Casablanca (1942) and won three Academy Awards; Jimmy Dugan wrote for the Saturday Evening Post, National Geographic, and The New York Times; and Ronald Bonn was a producer with NBC Nightly News and CBS Evening News. |
1829_52 | In addition, Penn State's newspaper readership program provides free copies of USA Today, The New York Times, as well as local and regional newspapers depending on the campus location (for example, the Centre Daily Times in University Park). This program, initiated by then-President Graham Spanier in 1997, has since been instituted on several other universities across the country.
Athletics
Penn State's mascot is the Nittany Lion, a representation of a type of mountain lion that once roamed what is now University Park. The school's official colors, now blue and white, were originally black and dark pink. Penn State participates in the NCAA Division I FBS and in the Big Ten Conference for most sports.
Two sports participate in different conferences: men's volleyball in the Eastern Intercollegiate Volleyball Association (EIVA) and women's hockey in College Hockey America (CHA). The fencing teams operate as independents. |
1829_53 | Athletic teams at Penn State have won 79 national collegiate team championships (51 NCAA, 2 consensus Division I football titles, 6 AIAW, 3 USWLA, 1 WIBC, and 4 national titles in boxing, 11 in men's soccer and one in wrestling in years prior to NCAA sponsorship). The 51 NCAA Championships ranks fifth all time in NCAA Division I, and is the most of any Big Ten school.
Since joining the Big Ten in 1991, Penn State teams have won 103 conference regular season and tournament titles.
Penn State has one of the most successful overall athletic programs in the country, as evidenced by its rankings in the NACDA Director's Cup, a list compiled by the National Association of Collegiate Directors of Athletics that charts institutions' overall success in college sports. From the Cup's inception in the 1993–1994 season, the Nittany Lions have finished in the top 25 every year. |
1829_54 | Despite widespread success in the overall athletic program, the school is best known for its football team and draws a very large following. Penn State's Beaver Stadium has the second largest seating capacity of any stadium in the nation, with an official capacity of 106,572 slightly behind Michigan Stadium with an official capacity of 107,601. For decades, the football team was led by coach Joe Paterno. Paterno was in a close competition with Bobby Bowden, the head coach for Florida State, for the most wins ever in Division I-A (now the FBS) history. This competition effectively ended with Paterno still leading following Bowden's retirement after the 2010 Gator Bowl. In 2007, he was inducted into the College Football Hall of Fame. Paterno amassed 409 victories over his career, the most in NCAA Division I history. Paterno died on January 22, 2012, at the age of 85. Paterno was posthumously honored by Penn State during the September 17, 2016 football game that marked the 50th |
1829_55 | anniversary of his first game as head coach. |
1829_56 | The school's wrestling team has also become noticed. Under Cael Sanderson, the Nittany Lions won eight national titles in nine years, from 2011 to 2019.
The university opened a new Penn State All-Sports Museum in February 2002. This two-level museum is located inside Beaver Stadium.
In addition to the school-funded athletics, club sports also play a major role in the university, with over 68 club sports organizations meeting regularly. Many club teams compete nationally in their respective sports. The Penn State Ski Team, which competes as part of the United States Collegiate Ski and Snowboard Association (USCSA) in the Allegheny Conference, as well as the Penn State Swim Club, which competes in the American Swimming Association – University League (ASAU), are just a few examples. Some other clubs include baseball, squash, karate, crew, and sailing. |
1829_57 | Penn State's most well-known athletic cheer is "We are...Penn State." Typically, the students and cheerleaders shout, "We are," followed by a "Penn State" response from the rest of the fans. By tradition, this is done three times and followed by "Thank you..." "... You're welcome!"
Notable people
The list of eminent past and present individuals associated with Penn State—as alumni, faculty, and athletic staff—can be found in the list of Pennsylvania State University people.
Alumni association |
1829_58 | Established in 1870, nine years after the university's first commencement exercises, the Penn State Alumni Association has the stated mission "to connect alumni to the University and each other, provide valuable benefits to members and support the University's mission of teaching, research, and service." The Alumni Association supports a number of educational and extracurricular missions of Penn State through financial support and is the network that connects alumni through over 280 "alumni groups", many of which are designated based on geographical, academic, or professional affiliation. |
1829_59 | As of July 1, 2010, the Alumni Association counts 496,969 members within the United States, with an additional 16,180 in countries around the globe. About half the United States alumni reside in Pennsylvania, primarily in the urban areas of Philadelphia (and the surrounding counties), the Pittsburgh Area and in the Centre County region surrounding State College, although alumni can be found in every region of the country and abroad. About 34 percent of United States alumni and 21 percent of international alumni are members of the Alumni Association. With membership totaling 176,426 as of FY2016, the Penn State Alumni Association is the largest dues-paying alumni association in the world, a distinction it has held since 1995. |
1829_60 | Since 2001, the university, along with all schools in the Big Ten, has participated in the "Big Ten Challenge" website, which is a "competitive" clearinghouse of alumni donation statistics for member schools. Results are tracked to determine a percentage of each school's alumni from the previous decade who gave to their alma mater each calendar year (during the 2005–2006 year, alumni donations from 1996 to 2005 were tallied). With the exception of 2005–2006, when Penn State fell to second behind Northwestern University, Penn State has won the challenge each year since its inception.
See also
Palmer Museum of Art
List of colleges and universities in Pennsylvania
References
External links
Penn State Athletics website
1855 establishments in Pennsylvania
Commonwealth System of Higher Education
Educational institutions established in 1855
Forestry education
Land-grant universities and colleges
State College, Pennsylvania
Universities and colleges in Centre County, Pennsylvania |
1830_0 | is a eroge visual novel series created by Frontwing composed of the classic entries and one spinoff, titled MegaChu. Each of the entries have also spawned trimmed down erotic OVA releases, which enjoyed mild popularity in the west. This series is created by Kūchū Yōsai and produced by Noboru Yamaguchi.
The game series also won a Bishōjo Game Award on September 5, 2008, as part of the best game series ever. The first two games were rereleased in 2008 due to a problem with DirectX 10 and the video playback software, causing Front Wing to standardize all three games released that year with the same engine, which removed compatibility with Windows 98 and Me. Makai Tenshi Djibril 4, the beginning of a new saga based at a school seemingly founded by Luvriel for Devil Angels, was released on April 23, 2010, and used, for the first time, graphics displayed in native 576i HD. |
1830_1 | For the 2011 game, instead of Djibril The Devil Angel 5, the game is called , and features all six of the main characters from Djibril Episode 1 to 4 as if they are from the Sengoku period of Japan, the last game produced by Noboru Yamaguchi before his death.
In early 2020, Frontwing announced a sixth game, in conjunction with DMM Games, starring a new cast of angels, titled , with a cyberspace theme, with the only characters confirmed returning from the first four games being Luvriel and Rika Manabe. This is also the only game in the series with a partially censored version. |
1830_2 | Plot
At the beginning of summer break, A young man named Naoto Jinno asked Rika Manabe, his girlfriend, to go out with him, planning to propose to her, only for his attempt to do so be interrupted by Asmodeus and Luvriel, commanders of a holy conflict, with Naoto and Rika helping Luvriel recover after she is defeated. As a result, Rika becomes a Devil Angel, an angelic warrior who fights using Amore, gathered through the act of making love, something Naoto is fairly happy to do.
In the second game, and its related anime, Hikari Jinno, his sister who was studying overseas, comes to visit, at the same time as Rococo, Asmodeus's sister, decides to attack Rika in revenge for Asmodeus's defeat, turning her into a demonic version of a Devil Angel, causing Hikari to end up a Devil Angel, with Naoto slightly more reluctantly, at first, generating Amore within her, and they manage to purify Rika and defeat Rococo. |
1830_3 | In the third game, Naoto finds himself helping test out the New Angelic Gaia Interface, in the form of a blue haired girl, when a new antagonist, pretending to be a photography student, lures Rika and Hikari into a trap that sees Hikari corrupted and Rika unable to help. Due to NAGI's digital nature, she is able to regenerate from damage and take greater risks, which is needed when the demons deploy the first completely robotic devil...
The fourth game has a different concept, with Luvriel having established a school for potential Devil Angels, recruits Momo Sakura, Aoi Ayonokouji and Yuzuha Hoshikawa, also known as Spica, Althaea and Junos, to defend it from a pair of non-identical twin devils, Meimei and Maimai. |
1830_4 | The fifth game was an anniversary title, featuring the casts of Makai Tenshi DJibril 3 and DJibril 4 meeting when Luvriel goes to see the Jinnos, only for them to be thrown into a parallel version of Sengoku Japan, with the various Devil Angels taking on personalities that were based on famous Japanese heroes as they try to figure out how to get back to the present.
The sixth, and currently final, game has Luvriel having re-established her school, this time placing the campus within cyberspace, with the assistance of Rika Manabe. However, where there are angels, there will always be demons...
Characters |
1830_5 | The main protagonist of the series. Ever since he met Rika when they were children, he became great friends with her. Until that incident that Rika had, Naoto wants to spend some more time with Rika and wanted to have a nice relationship with her, until Asmo came along and started trying to kidnap her. When Luvriel was knocked out in a stalemate in the battle with him, Naoto and Rika take her to his home to help her. As they realize that Luvriel was really an Angel, she told them the facts behind the incident and that she was too injured to continue fighting the demon boy. Rika wanted to be an Angel just like Luvriel, but it turns out that she has to have sex (and thus lose her virginity) to store up Amore power to shapeshift into an Angel, so she decided to lose her virginity to Naoto. Later in the second series, he becomes more caring than perverted than before. Ever since Rika was kidnapped, he felt emotional feelings in himself and has been saddened when Rika was kidnapped and |
1830_6 | missed her. This series later introduces his little stepsister, Hikari. She usually gets jealous but is sweet to her brother since they had a genetic sexual attraction to each other, which causes Hikari to worry. However, Naoto lets her sister help save Rika for him by shapeshifting into a Devil Angel by having sex with him and fights for his love. |
1830_7 | A quiet and sweet student who has been a friend of Naoto since childhood. During her childhood, Naoto doesn't seem to be getting along with Rika pretty well back then, but when he got close to her, they were really close friends. As she accepts to be his girlfriend, she cried of joy. After hearing from Luvriel about the battles between the angels and the demons but that Luvriel was too injured to fight, Naoto wants to join in the fight. Unfortunately, for Naoto, angels are supposed to be girls, so Rika accepts to be a part of the army. But when Luvriel told Rika that she has to have sex to acquire Amore power, she felt shocked and worried. She was later kidnapped at the beginning of the second anime (Djibril The Second Coming Book 1) and has shapeshifted into a darker form of herself by the new demon character Miss Rococo. |
1830_8 | Introduced in the second game, Hikari was visiting her brother, and witnesses the defeat of the original Djibril. Partially due to being exposed to the secret of Djibril, and partially because she was the only person available to replace Rika, she ends up recruited as the second Djibril, with the suffix of Aries, in order to rescue Rika from the clutches of Misty Mei and Rococo. Later in the third game, Similar to the kidnapping of Rika, she was captured by demons, shapeshifts into her darker self, and plays a major role of the game.
The newest of the Devil Angels known as Djibril, Nagi is in fact a computer program called N.A.G.I., or the New Angelic Gaia Interface. Part of a new effort in Heaven to use less humans and angels in their battles, due to, effectively, demand outstripping supply, she is recruited to help with the main mission after some kind of strange spell has been cast on Rika, and Hikari was captured... |
1830_9 | The only character not retired after Makai Tenshi Djibril 3, Luvriel is a low-ranking angel who appears as Asmo is attacking Rika, attempting to stop him, but failing, and deciding instead to move in with Naoto, help him support Rika, and later Hikari and Nagi, although she seems to spend more time spying on them while they're having sex. Also, despite Luvriel's childish appearance, and occasional behavior, she is in fact 10,009 years old
Whereas the opening of Makai Tenshi Djibril 2 uses the romanization Luvriel, in the openings of Makai Tenshi Djibril 4 and Sengoku Tenshi Djibril the spelling Loveriel is used. |
1830_10 | Devil Angels
A major plot point of each of the three games is the metamorphosis of the female lead into one of the Devil Angels, under the title of Djibril, with later games, and the associated anime including the sub-plot of cleansing a former lead character. This is done using a halo, normally supplied by Luvriel, but alternative metamorphosis methods do appear in the various games. |
1830_11 | The form of a Devil Angel differs with the person, with each having both an angelic aspect and a demonic aspect, depending on how they gain the power to shapeshift. These usually are signified by a change in hair color, with angelic Djibrils, all having hair that is some shade of light blue, and the addition of a leotard-style outfit. The only exclusion to this is the New Angelic Gaia Interface, which only has the metamorphosis being that N.A.G.I. just gained a leotard. In Makai Tenshi Djibril 4, however, this is changed completely to a form of a jumpsuit, with hair colors not changing, and, by Dennou Tenshi Djibril, all Devil Angels are created in the same way as N.A.G.I. |
1830_12 | In the games, the dark aspect is indicated by the colors of their uniforms changing to red and black, as well as changing their eye color, as well as other additions dependent on the level of corruption. Later games used the specific title Djibril for them so they wouldn't be confused with the historical angel Gabriel. |
1830_13 | Known Devil Angels include -
Rika Manabe (Djibril) The first of the three Devil Angels, as well as the only one not related, by blood or choice, to Naoto. She is Luvriel's replacement in the first game, due to a mutually assured destruction strategy gone wrong. She was captured in the second game by a giant wormlike creature, over the course of the game, put through a perverse version of her training in the first game, and in the third game, she was weakened by the use of a spell cast through the corrupted hand of Dark Aries. |
1830_14 | Hikari Jinno (Djibril Aries/Dark Aries) The second of the three Devil Angels, Aries was recruited to replace Rika when she is defeated in battle and taken as spoils. She also spends most of the third game in a second, corrupted, persona of Dark Aries, which has a unique power to control minds. She is stronger-willed than Rika, managing to break the controlling influence of Misty Mei in the third game multiple times, and, at least in the anime, summoning her own 'Super Nova' style attack despite being almost completely controlled.
New Angelic Gaia Interface (Djibril Zero)Djibril Zero is a last-resort Djibril, in that a N.A.G.I. has the capability to regenerate damage, and even incapacitation, within reason, can be recovered from as long as the interface can be restarted properly. She has only been used once in this capacity, during the third game, when Jibril Aries is abducted, and used to further incapacitate the original Djibril. |
1830_15 | Momo Sakura, Aoi Ayonokouji and Yuzuha Hoshikawa (Spica, Althaea and Junos) A trio of students at Luvriel's first attempt at a school for angels, who manage to become Devil Angels. Notably, from this point on, the appearance of Devil Angels varies wildly.
DJibril Regulus, Muse and Alnair Three of the students at Luvriel's second attempt at a school. Notably, there are over a dozen other confirmed Devil Angels, but these are the only ones explicitly named.
Meimi Otonashi (Misty Mei) While not a Devil Angel in the sense that she was recruited by Asmo originally, Misty Mei does gain power in the same ways as all three Djibrils. Numerous attempts have been made to befriend Misty Mei, but she is never able to achieve her actual Djibril appearance or identity, appearing, in all three original games, as a gothic lolita with black bat wings. Notably, She was pivotal in some cases to the capture and controlling of the first two Devil Angels. |
1830_16 | Super X-32 A false angel created in the third game and anime, intended to be a counter for the Djibril series. Rather than having a human component, Super X-32 is a gynoid with several weapons and features intended to render the need for agents like Dark Aries and Misty Mei largely redundant. In the anime, she is considered to be N.A.G.I.'s mirror duplicate, an artificial devil program to fight the artificial angel program, while in the games, she is only encountered once during N.A.G.I.'s ending path, and is not actually fought.
Powers
"Amore" (Magical Energy): Metamorphoses Rika, Hikari or Nagi into Djbril from having sex with a person of the opposite sex for whom they care for. Negative Amore, created by tentacle and/or demonic sex causes a Devil Angel to turn dark. Notably, Super X-32 can absorb and use positively charged Amore according to the anime, making it clear the Amore is virtually identical, but processed differently by a Devil Angel, however it is made. |
1830_17 | "Tenshi no Shiki" (Angelic Ceremony): By performing particular sexual acts under the same conditions as when they accumulate Amore certain powers are stored up. Each power has two different forms: A Heaven Form and a Hell Form, which are achieved by a different act, but give the same effect. Despite the names of the forms, they do not visibly cause a Devil Angel to have a greater or lesser chance of turning Dark over time. These were phased out with later revisions of the Djibril metamorphosis process.
"Cellular Regeneration": A unique power of Djibril Zero, She is able to regenerate battle damage like the loss of limbs without too much effort, but she is not able to avoid some forms of damage, due to limits in her programming. This also allows her to shapeshift parts of her body to create extra weapons to offset the loss of some human-type Djibrils. |
1830_18 | "Angel Super Nova": A Super Form, used once when their main powers fail them, this powers them up, allowing them to break free from whatever danger they are in. It is empowered by the wishes of someone who they especially care about.
In Series 1, Djibril used this to break up the perverted Angelic Ceremony and allowed herself to destroy Rirouge.
In Series 2, Djibril Aries used a variant of the technique to both defeat Dark Djibril and to break free of the dark trance.
As far as can be told, Djibril Zero, Super X-32 and Misty Mei cannot use any version of this ability, the former due to being artificially created, the latter due to never appearing in any of the three games as an angel.
"Zero Cannon": An ability shown for Djibril Zero in the anime, it allows her to attack demons at more extreme ranges, by firing a beam of positively charged Amore, with the side effect that it cannot be used to defeat Devil Angels, including artificial ones like Super X-32.
Theme songs |
1830_19 | In the series, the opening theme songs differ from one another. Game theme songs were later inserted in the anime series at the end of credits, when the episode ends, in lieu of title credits. These were supplemented in the latest two games by ending themes, with both games having their opening sequences released online well in advance of the game's release.
In the original three games, The themes, where appropriate, are used in instrumental form as a leitmotif for each of the main female leads, and a remixed instrumental form is used as the menu theme for all four games. The lyrics, composer, and vocals of the theme songs were written by the Japanese pop rock band, Funta (Under the name of U), with the help of the anime J-pop company, GWAVE.
Themes in Episode 1
Opening:
Themes in Episode 2
Opening: Little my star
Themes in Episode 3
Opening: Kuru Kuru Lovely Day (クルクルlovely day)
Ending:
Themes in Episode 4
Opening: the first the last
Ending: 新生時〜この恋ときみとあたし〜 |
1830_20 | Kuru Kuru Lovely Day was released as an audio CD as part of the promotional materials for Episode 3. Other themes include "Control Is Impossible", A fast tune used in the three games and two anime to indicate a comical or crazy moment, which seems, in a lot of cases, to involve directly Luvriel, becoming almost a leitmotif in itself.
References
External links
Makai Tenshi Djibril - PC game at Front Wing
Makai Tenshi Djibril -Episode 2- - PC game at Front Wing
Makai Tenshi Djibril 3 - PC game at Front Wing
Makai Tenshi Djibril 4 - PC game at Front Wing
Sengoku Tenshi Djibril - PC game at Front Wing
2004 anime OVAs
2004 video games
2007 anime OVAs
2009 anime OVAs
Bishōjo games
Eroge
Fictional angels
Fictional demons and devils
Frontwing games
Hentai anime and manga
Japan-exclusive video games
Video games developed in Japan
Visual novels
Windows games
Windows-only games |
1831_0 | In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, is a factorization of the integer , and is a factorization of the polynomial .
Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any can be trivially written as whenever is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. |
1831_1 | Factorization was first considered by ancient Greek mathematicians in the case of integers. They proved the fundamental theorem of arithmetic, which asserts that every positive integer may be factored into a product of prime numbers, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique up to the order of the factors. Although integer factorization is a sort of inverse to multiplication, it is much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. |
1831_2 | Polynomial factorization has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its roots to finding the roots of the factors. Polynomials with coefficients in the integers or in a field possess the unique factorization property, a version of the fundamental theorem of arithmetic with prime numbers replaced by irreducible polynomials. In particular, a univariate polynomial with complex coefficients admits a unique (up to ordering) factorization into linear polynomials: this is a version of the fundamental theorem of algebra. In this case, the factorization can be done with root-finding algorithms. The case of polynomials with integer coefficients is fundamental for computer algebra. There are efficient computer algorithms for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see factorization of polynomials). |
1831_3 | A commutative ring possessing the unique factorization property is called a unique factorization domain. There are number systems, such as certain rings of algebraic integers, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of Dedekind domains: ideals factor uniquely into prime ideals.
Factorization may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds of matrix factorizations. For example, every matrix has a unique LUP factorization as a product of a lower triangular matrix with all diagonal entries equal to one, an upper triangular matrix , and a permutation matrix ; this is a matrix formulation of Gaussian elimination.
Integers |
1831_4 | By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the order of the factors) factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one.
For computing the factorization of an integer , one needs an algorithm for finding a divisor of or deciding that is prime. When such a divisor is found, the repeated application of this algorithm to the factors and gives eventually the complete factorization of .
For finding a divisor of , if any, it suffices to test all values of such that and . In fact, if is a divisor of such that , then is a divisor of such that . |
1831_5 | If one tests the values of in increasing order, the first divisor that is found is necessarily a prime number, and the cofactor cannot have any divisor smaller than . For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of that is not smaller than and not greater than .
There is no need to test all values of for applying the method. In principle, it suffices to test only prime divisors. This needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes. As the method of factorization does essentially the same work as the sieve of Eratosthenes, it is generally more efficient to test for a divisor only those numbers for which it is not immediately clear whether they are prime or not. Typically, one may proceed by testing 2, 3, 5, and the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. |
1831_6 | This method works well for factoring small integers, but is inefficient for larger integers. For example, Pierre de Fermat was unable to discover that the 6th Fermat number
is not a prime number. In fact, applying the above method would require more than , for a number that has 10 decimal digits.
There are more efficient factoring algorithms. However they remain relatively inefficient, as, with the present state of the art, one cannot factorize, even with the more powerful computers, a number of 500 decimal digits that is the product of two randomly chosen prime numbers. This ensures the security of the RSA cryptosystem, which is widely used for secure internet communication. |
1831_7 | Example
For factoring into primes:
Start with division by 2: the number is even, and . Continue with 693, and 2 as a first divisor candidate.
693 is odd (2 is not a divisor), but is a multiple of 3: one has and . Continue with 231, and 3 as a first divisor candidate.
231 is also a multiple of 3: one has , and thus . Continue with 77, and 3 as a first divisor candidate.
77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has , and thus . This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.
As , one has finished. Thus 11 is prime, and the prime factorization is
. |
1831_8 | Expressions
Manipulating expressions is the basis of algebra. Factorization is one of the most important methods for expression manipulation for several reasons. If one can put an equation in a factored form , then the problem of solving the equation splits into two independent (and generally easier) problems and . When an expression can be factored, the factors are often much simpler, and may thus offer some insight on the problem. For example,
having 16 multiplications, 4 subtractions and 3 additions, may be factored into the much simpler expression
with only two multiplications and three subtractions. Moreover, the factored form immediately gives roots x = a,b,c as the roots of the polynomial.
On the other hand, factorization is not always possible, and when it is possible, the factors are not always simpler. For example, can be factored into two irreducible factors and .
Various methods have been developed for finding factorizations; some are described below. |
1831_9 | Solving algebraic equations may be viewed as a problem of polynomial factorization. In fact, the fundamental theorem of algebra can be stated as follows: every polynomial in of degree with complex coefficients may be factorized into linear factors for , where the s are the roots of the polynomial. Even though the structure of the factorization is known in these cases, the
s generally cannot be computed in terms of radicals (nth roots), by the Abel–Ruffini theorem. In most cases, the best that can be done is computing approximate values of the roots with a root-finding algorithm.
History of factorization of expressions
The systematic use of algebraic manipulations for simplifying expressions (more specifically equations)) may be dated to 9th century, with al-Khwarizmi's book The Compendious Book on Calculation by Completion and Balancing, which is titled with two such types of manipulation. |
1831_10 | However, even for solving quadratic equations, the factoring method was not used before Harriot's work published in 1631, ten years after his death. In his book Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas, Harriot drew, tables for addition, subtraction, multiplication and division of monomials, binomials, and trinomials. Then, in a second section, he set up the equation , and showed that this matches the form of multiplication he had previously provided, giving the factorization .
General methods
The following methods apply to any expression that is a sum, or that may be transformed into a sum. Therefore, they are most often applied to polynomials, though they also may be applied when the terms of the sum are not monomials, that is, the terms of the sum are a product of variables and constants. |
1831_11 | Common factor
It may occur that all terms of a sum are products and that some factors are common to all terms. In this case, the distributive law allows factoring out this common factor. If there are several such common factors, it is worth to divide out the greatest such common factor. Also, if there are integer coefficients, one may factor out the greatest common divisor of these coefficients.
For example,
since 2 is the greatest common divisor of 6, 8, and 10, and divides all terms.
Grouping
Grouping terms may allow using other methods for getting a factorization.
For example, to factor
one may remark that the first two terms have a common factor , and the last two terms have the common factor . Thus
Then a simple inspection shows the common factor , leading to the factorization
In general, this works for sums of 4 terms that have been obtained as the product of two binomials. Although not frequently, this may work also for more complicated examples. |
1831_12 | Adding and subtracting terms
Sometimes, some term grouping lets appear a part of a recognizable pattern. It is then useful to add terms for completing the pattern, and subtract them for not changing the value of the expression.
A typical use of this is the completing the square method for getting quadratic formula.
Another example is the factorization of If one introduces the non-real square root of –1, commonly denoted , then one has a difference of squares
However, one may also want a factorization with real number coefficients. By adding and subtracting and grouping three terms together, one may recognize the square of a binomial:
Subtracting and adding also yields the factorization: |
1831_13 | These factorizations work not only over the complex numbers, but also over any field, where either –1, 2 or –2 is a square. In a finite field, the product of two non-squares is a square; this implies that the polynomial which is irreducible over the integers, is reducible modulo every prime number. For example,
since
since
since
Recognizable patterns
Many identities provide an equality between a sum and a product. The above methods may be used for letting the sum side of some identity appear in an expression, which may therefore be replaced by a product.
Below are identities whose left-hand sides are commonly used as patterns (this means that the variables and that appear in these identities may represent any subexpression of the expression that has to be factorized.
Difference of two squares
For example,
Sum/difference of two cubes
Difference of two fourth powers |
1831_14 | Sum/difference of two th powers
In the following identities, the factors may often be further factorized:
Difference, even exponent
Difference, even or odd exponent
This is an example showing that the factors may be much larger than the sum that is factorized.
Sum, odd exponent
(obtained by changing by in the preceding formula)
Sum, even exponent
If the exponent is a power of two then the expression cannot, in general, be factorized without introducing complex numbers (if and contain complex numbers, this may be not the case). If n has an odd divisor, that is if with odd, one may use the preceding formula (in "Sum, odd exponent") applied to
Trinomials and cubic formulas
Binomial expansions
The binomial theorem supplies patterns that can easily be recognized from the integers that appear in them
In low degree:
More generally, the coefficients of the expanded forms of and are the binomial coefficients, that appear in the th row of Pascal's triangle. |
1831_15 | Roots of unity
The th roots of unity are the complex numbers each of which is a root of the polynomial They are thus the numbers
for
It follows that for any two expressions and , one has:
If and are real expressions, and one wants real factors, one has to replace every pair of complex conjugate factors by its product. As the complex conjugate of is and
one has the following real factorizations (one passes from one to the other by changing into or , and applying the usual trigonometric formulas:
The cosines that appear in these factorizations are algebraic numbers, and may be expressed in terms of radicals (this is possible because their Galois group is cyclic); however, these radical expressions are too complicated to be used, except for low values of . For example, |
1831_16 | Often one wants a factorization with rational coefficients. Such a factorization involves cyclotomic polynomials. To express rational factorizations of sums and differences or powers, we need a notation for the homogenization of a polynomial: if its homogenization is the bivariate polynomial Then, one has
where the products are taken over all divisors of , or all divisors of that do not divide , and is the th cyclotomic polynomial.
For example,
since the divisors of 6 are 1, 2, 3, 6, and the divisors of 12 that do not divide 6 are 4 and 12.
Polynomials
For polynomials, factorization is strongly related with the problem of solving algebraic equations. An algebraic equation has the form
where is a polynomial in with
A solution of this equation (also called a root of the polynomial) is a value of such that |
1831_17 | If is a factorization of as a product of two polynomials, then the roots of are the union of the roots of and the roots of . Thus solving is reduced to the simpler problems of solving and .
Conversely, the factor theorem asserts that, if is a root of , then may be factored as
where is the quotient of Euclidean division of by the linear (degree one) factor .
If the coefficients of are real or complex numbers, the fundamental theorem of algebra asserts that has a real or complex root. Using the factor theorem recursively, it results that
where are the real or complex roots of , with some of them possibly repeated. This complete factorization is unique up to the order of the factors. |
1831_18 | If the coefficients of are real, one generally wants a factorization where factors have real coefficients. In this case, the complete factorization may have some quadratic (degree two) factors. This factorization may easily be deduced from the above complete factorization. In fact, if is a non-real root of , then its complex conjugate is also a root of . So, the product
is a factor of with real coefficients. Repeating this for all non-real factors gives a factorization with linear or quadratic real factors.
For computing these real or complex factorizations, one needs the roots of the polynomial, which may not be computed exactly, and only approximated using root-finding algorithms. |
1831_19 | In practice, most algebraic equations of interest have integer or rational coefficients, and one may want a factorization with factors of the same kind. The fundamental theorem of arithmetic may be generalized to this case, stating that polynomials with integer or rational coefficients have the unique factorization property. More precisely, every polynomial with rational coefficients may be factorized in a product
where is a rational number and are non-constant polynomials with integer coefficients that are irreducible and primitive; this means that none of the may be written as the product two polynomials (with integer coefficients) that are neither 1 nor –1 (integers are considered as polynomials of degree zero). Moreover, this factorization is unique up to the order of the factors and the signs of the factors. |
1831_20 | There are efficient algorithms for computing this factorization, which are implemented in most computer algebra systems. See Factorization of polynomials. Unfortunately, these algorithms are too complicated to use for paper-and-pencil computations. Besides the heuristics above, only a few methods are suitable for hand computations, which generally work only for polynomials of low degree, with few nonzero coefficients. The main such methods are described in next subsections.
Primitive-part & content factorization
Every polynomial with rational coefficients, may be factorized, in a unique way, as the product of a rational number and a polynomial with integer coefficients, which is primitive (that is, the greatest common divisor of the coefficients is 1), and has a positive leading coefficient (coefficient of the term of the highest degree). For example: |
1831_21 | In this factorization, the rational number is called the content, and the primitive polynomial is the primitive part. The computation of this factorization may be done as follows: firstly, reduce all coefficients to a common denominator, for getting the quotient by an integer of a polynomial with integer coefficients. Then one divides out the greater common divisor of the coefficients of this polynomial for getting the primitive part, the content being Finally, if needed, one changes the signs of and all coefficients of the primitive part.
This factorization may produce a result that is larger than the original polynomial (typically when there are many coprime denominators), but, even when this is the case, the primitive part is generally easier to manipulate for further factorization.
Using the factor theorem
The factor theorem states that, if is a root of a polynomial
meaning , then there is a factorization
where |
1831_22 | with . Then polynomial long division or synthetic division give:
This may be useful when one knows or can guess a root of the polynomial.
For example, for one may easily see that the sum of its coefficients is 0, so is a root. As , and one has
Rational roots
For polynomials with rational number coefficients, one may search for roots which are rational numbers. Primitive part-content factorization (see above) reduces the problem of searching for rational roots to the case of polynomials with integer coefficients having no non-trivial common divisor.
If is a rational root of such a polynomial
the factor theorem shows that one has a factorization
where both factors have integer coefficients (the fact that has integer coefficients results from the above formula for the quotient of by ). |
1831_23 | Comparing the coefficients of degree and the constant coefficients in the above equality shows that, if is a rational root in reduced form, then is a divisor of and is a divisor of Therefore, there is a finite number of possibilities for and , which can be systematically examined.
For example, if the polynomial
has a rational root with , then must divide 6; that is and must divide 2, that is Moreover, if , all terms of the polynomial are negative, and, therefore, a root cannot be negative. That is, one must have
A direct computation shows that only is a root, so there can be no other rational root. Applying the factor theorem leads finally to the factorization
Quadratic ac method
The above method may be adapted for quadratic polynomials, leading to the ac method of factorization.
Consider the quadratic polynomial |
1831_24 | with integer coefficients. If it has a rational root, its denominator must divide evenly and it may be written as a possibly reducible fraction By Vieta's formulas, the other root is
with
Thus the second root is also rational, and Vieta's second formula gives
that is
Checking all pairs of integers whose product is gives the rational roots, if any.
In summary, if has rational roots there are integers and such and (a finite number of cases to test), and the roots are and In other words, one has the factorization
For example, let consider the quadratic polynomial
Inspection of the factors of leads to , giving the two roots
and the factorization
Using formulas for polynomial roots
Any univariate quadratic polynomial can be factored using the quadratic formula:
where and are the two roots of the polynomial. |
1831_25 | If are all real, the factors are real if and only if the discriminant is non-negative. Otherwise, the quadratic polynomial cannot be factorized into non-constant real factors.
The quadratic formula is valid when the coefficients belong to any field of characteristic different from two, and, in particular, for coefficients in a finite field with an odd number of elements.
There are also formulas for roots of cubic and quartic polynomials, which are, in general, too complicated for practical use. The Abel–Ruffini theorem shows that there are no general root formulas in terms of radicals for polynomials of degree five or higher.
Using relations between roots
It may occur that one knows some relationship between the roots of a polynomial and its coefficients. Using this knowledge may help factoring the polynomial and finding its roots. Galois theory is based on a systematic study of the relations between roots and coefficients, that include Vieta's formulas. |
1831_26 | Here, we consider the simpler case where two roots
and of a polynomial satisfy the relation
where is a polynomial.
This implies that is a common root of and Its is therefore a root of the greatest common divisor of these two polynomials. It follows that this greatest common divisor is a non constant factor of Euclidean algorithm for polynomials allows computing this greatest common factor.
For example, if one know or guess that:
has two roots that sum to zero, one may apply Euclidean algorithm to and The first division step consists in adding to giving the remainder of
Then, dividing by gives zero as a new remainder, and as a quotient, leading to the complete factorization
Unique factorization domains |
1831_27 | The integers and the polynomials over a field share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a unit, ±1 in the case of integers) and a product of irreducible elements (prime numbers, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. Integral domains which share this property are called unique factorization domains (UFD).
Greatest common divisors exist in UFDs, and conversely, every integral domain in which greatest common divisors exist is an UFD. Every principal ideal domain is an UFD.
A Euclidean domain is an integral domain on which is defined a Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. |
1831_28 | In a Euclidean domain, Euclidean division allows defining a Euclidean algorithm for computing greatest common divisors. However this does not imply the existence of a factorization algorithm. There is an explicit example of a field such that there cannot exist any factorization algorithm in the Euclidean domain of the univariate polynomials over .
Ideals
In algebraic number theory, the study of Diophantine equations led mathematicians, during 19th century, to introduce generalizations of the integers called algebraic integers. The first ring of algebraic integers that have been considered were Gaussian integers and Eisenstein integers, which share with usual integers the property of being principal ideal domains, and have thus the unique factorization property.
Unfortunately, it soon appeared that most rings of algebraic integers are not principal and do not have unique factorization. The simplest example is in which
and all these factors are irreducible. |
1831_29 | This lack of unique factorization is a major difficulty for solving Diophantine equations. For example, many wrong proofs of Fermat's Last Theorem (probably including Fermat's "truly marvelous proof of this, which this margin is too narrow to contain") were based on the implicit supposition of unique factorization.
This difficulty was resolved by Dedekind, who proved that the rings of algebraic integers have unique factorization of ideals: in these rings, every ideal is a product of prime ideals, and this factorization is unique up the order of the factors. The integral domains that have this unique factorization property are now called Dedekind domains. They have many nice properties that make them fundamental in algebraic number theory.
Matrices |
1831_30 | Matrix rings are non-commutative and have no unique factorization: there are, in general, many ways of writing a matrix as a product of matrices. Thus, the factorization problem consists of finding factors of specified types. For example, the LU decomposition gives a matrix as the product of a lower triangular matrix by an upper triangular matrix. As this is not always possible, one generally considers the "LUP decomposition" having a permutation matrix as its third factor.
See Matrix decomposition for the most common types of matrix factorizations.
A logical matrix represents a binary relation, and matrix multiplication corresponds to composition of relations. Decomposition of a relation through factorization serves to profile the nature of the relation, such as a difunctional relation.
See also
Euler's factorization method for integers
Fermat's factorization method for integers
Monoid factorisation
Multiplicative partition
Table of Gaussian integer factorizations
Notes |
1831_31 | References
External links
Wolfram Alpha can factorize too.
Arithmetic
Elementary algebra |
1832_0 | Below are the rosters of the minor league affiliates of the Pittsburgh Pirates:
Players
Stephen Alemais
Stephen Gavin Alemais (born April 12, 1995) is an American professional baseball shortstop in the Pittsburgh Pirates organization. |
1832_1 | Alemais was born in the Bronx, New York and attended All Hallows High School. He earned all-state honors in his junior and senior years, and all-district, all-city, and all-league honors in his sophomore, junior, and senior year. He currently holds the schools' single season hit record. Undrafted out of high school in the 2013 MLB draft, he enrolled at Tulane University where he played college baseball. In 2015, he played collegiate summer baseball with the Cotuit Kettleers of the Cape Cod Baseball League. In 2016, his junior year at Tulane, he slashed .311/.368/.401 with one home run, 28 RBIs, and 19 stolen bases in 53 games, earning American Athletic Conference First Team honors. After the season, he was selected in the third round of the 2016 Major League Baseball draft by the Pittsburgh Pirates with the 105th overall pick. |
1832_2 | Alemais signed with Pittsburgh and made his professional debut with the West Virginia Black Bears before being promoted to the West Virginia Power in August. In fifty games, he hit .249 with one home run and twenty RBIs. In 2017, he played for the Power and the Bradenton Marauders along with making a rehab appearance with the Gulf Coast League Pirates. In 67 total games, he batted .265 with four home runs and 34 RBIS. In 2018, he played for the Altoona Curve, slashing .279/.346/.346 with one home run, 34 RBIs, and 16 stolen bases in 120 games. He returned to Altoona to begin 2019.
Carter Bins
Carter Bins (born October 7, 1998) is an American professional baseball catcher in the Pittsburgh Pirates organization.
Bins attended Angelo Rodriguez High School in Fairfield, California. He hit .313/.427/.447 with 9 RBI in 67 at-bats in his senior year. He was First-Team All Monticello Empire League honors and was twice named Defensive Player of the Year. |
1832_3 | Bins was drafted in the 35th round of the 2016 draft by the Philadelphia Phillies but chose to attend Fresno State Bulldogs baseball. Bins started at catcher for Fresno State for his three years there, earning Academic All-Mountain West honors each year. In his career at Fresno State, Bins hit .289/.383/.465 with 19 home runs and 96 RBI in 679 plate appearances. Bins was regarded as an excellent defensive catcher in college and was considered among the best in the 2019 draft. He was drafted by the Seattle Mariners in the 11th round of the 2019 MLB draft. Bins signed with the Mariners for a $325,000 signing bonus
Bins debuted in the minors with the Everett AquaSox, then of the Class A Short Season Northwest League. He played 50 games for the AquaSox, hitting .208/.391/.357 with 7 home runs and 26 RBI.
On July 28, 2021, Bins was traded to the Pittsburgh Pirates along with Joaquin Tejada in exchange for Tyler Anderson.
Fresno State Bulldogs bio
Cody Bolton |
1832_4 | Carl Donovan Bolton (born June 19, 1998) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Bolton attended Tracy High School in Tracy, California. As a senior, he went 9–2 with a 1.13 ERA, striking out 97 batters in 68 innings. After his senior year, he was drafted by the Pittsburgh Pirates in the sixth round of the 2017 Major League Baseball draft. He signed with the Pirates, forgoing his commitment to play college baseball at the University of Michigan. |
1832_5 | After signing, Bolton made his professional debut with the Rookie-level Gulf Coast Pirates. In nine starts, he pitched to a 3.16 ERA. Bolton spent 2018 with the West Virginia Power of the Class A South Atlantic League, going 3–3 with a 3.65 ERA in nine starts, and began 2019 with the Bradenton Marauders of the Class A-Advanced Florida State League, where he was named Pitcher of the Week on May 13 as well as an All-Star. He was promoted to the Altoona Curve of the Class AA Eastern League in June. Over 21 starts between the two clubs, Bolton went 8–6 with a 3.28 ERA, striking out 102 over innings. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. He began the 2021 season on the injured list with a knee injury and underwent surgery in May, forcing him to miss the whole season.
Diego Castillo
Diego Alejandro Castillo (born October 28, 1997) is a Venezuelan professional baseball infielder in the Pittsburgh Pirates organization. |
1832_6 | The New York Yankees signed Castillo as an international free agent in December 2014. In July 2021, the Yankees traded Castillo along with Hoy Park to the Pittsburgh Pirates for Clay Holmes.
The Pirates added Castillo to their 40-man roster after the 2021 season.
Omar Cruz
Omar Cruz (born January 26, 1999) is a Mexican professional baseball pitcher in the Pittsburgh Pirates organization. |
1832_7 | Cruz signed with the San Diego Padres as an international free agent in 2017. He made his professional debut in 2018 with the Rookie-level Arizona League Padres and was promoted to the Tri-City Dust Devils of the Class A Short Season Northwest League in July. Over 11 games (ten starts) with the two clubs, he pitched to a 1-1 record with a 1.91 ERA, striking out 59 batters over innings. He returned to Tri-City to begin the 2019 season before he was promoted to the Fort Wayne TinCaps of the Class A Midwest League. He compiled a combined 2-3 record and 2.73 ERA over 12 starts, striking out 76 over 56 innings. |
1832_8 | On January 19, 2021, Cruz (alongside David Bednar, Drake Fellows, Hudson Head, and Endy Rodriguez) was traded to the Pittsburgh Pirates in a three team deal that also sent Joe Musgrove to the Padres and Joey Lucchesi to the New York Mets. To begin the 2021 season, he was assigned to the Greensboro Grasshoppers of the High-A East. After seven starts in which he went 3-3 with a 3.45 ERA and 38 strikeouts over innings, he was promoted to the Altoona Curve of the Double-A Northeast. Over 14 starts with Altoona, he went 3-4 with a 3.44 ERA over innings.
Drake Fellows
Drake Robert Fellows (born March 6, 1998) is an American professional baseball pitcher for the Pittsburgh Pirates organization. |
1832_9 | Fellows attended Joliet Catholic Academy in Joliet, Illinois and Vanderbilt University, where he played college baseball for the Vanderbilt Commodores. He was selected by the San Diego Padres in the sixth round of the 2019 Major League Baseball draft. He did not play in 2019 after signing, and did not play in 2020 due to the cancellation of the minor league season.
On January 19, 2021, Fellows was traded to the Pittsburgh Pirates in a three team trade that also sent David Bednar, Omar Cruz, Hudson Head and Endy Rodriguez to the Pirates, Joe Musgrove to the Padres and Joey Lucchesi to the New York Mets. He made his professional debut with the Florida Complex League Pirates and the Bradenton Marauders, but pitched only innings due to an elbow injury.
Matt Fraizer
Matthew Teran Fraizer (born January 12, 1998) is an American professional baseball outfielder in the Pittsburgh Pirates organization. |
1832_10 | Fraizer attended Clovis North High School in Clovis, California. He was drafted by the Oakland Athletics in the 38th round of the 2016 Major League Baseball Draft but did not sign and played college baseball at the University of Arizona. In 2018, he played collegiate summer baseball with the Orleans Firebirds of the Cape Cod Baseball League. He was selected by the Pittsburgh Pirates in the third round of the 2019 MLB draft and signed.
Fraizer made his professional debut with the West Virginia Black Bears, batting .221 over 43 games. He did not play a minor league game in 2020 due to the season being cancelled because of the COVID-19 pandemic. He started 2021 with the Greensboro Grasshoppers before being promoted to the Altoona Curve. Over 112 games between the two teams, he slashed .306/.388/.552 with 23 home runs, 68 RBIs, and 15 stolen bases.
Hudson Head
John Hudson Head (born April 8, 2001) is an American professional baseball outfielder for the Pittsburgh Pirates organization. |
1832_11 | Head attended Winston Churchill High School in San Antonio, Texas. The San Diego Padres selected him in the third round of the 2019 Major League Baseball draft. He signed with the Padres for a $3 million signing bonus, a record for a player taken in the third round.
Head made his professional debut with the Rookie-level Arizona League Padres, batting .283 with one home run and seven doubles over 32 games. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic.
On January 19, 2021, Head was traded to the Pittsburgh Pirates as part of a three team trade that also sent David Bednar, Omar Cruz, Drake Fellows and Endy Rodriguez to the Pirates, Joe Musgrove to the Padres and Joey Lucchesi to the New York Mets. He spent the 2021 season with the Bradenton Marauders of the Low-A Southeast, slashing .213/.362/.394 with 15 home runs, fifty RBIs, and 16 doubles over 101 games.
Jared Jones |
1832_12 | Jared Keith Jones (born August 6, 2001) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Jones attended La Mirada High School in La Mirada, California. He was drafted by the Pittsburgh Pirates in the second round of the 2020 Major League Baseball draft. He signed with the Pirates rather than play college baseball at the University of Texas at Austin.
Jones made his professional debut in 2021 with the Bradenton Marauders. Over 18 games (15 starts), he went 3-6 with a 4.64 ERA and 103 strikeouts over 66 innings.
Brennan Malone
Brennan Russell Malone (born September 8, 2000) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Malone attended Porter Ridge High School in Indian Trail, North Carolina before transferring to IMG Academy in Bradenton, Florida for his senior year. At IMG, he was recorded throwing as high as 97 miles per hour. He committed to play college baseball at the University of North Carolina. |
1832_13 | Malone was drafted by the Arizona Diamondbacks in the first round of the 2019 Major League Baseball draft, making him one of only three high school pitchers selected in the first round of the 2019 draft. He signed for $2.2 million. After signing, he was assigned to the Arizona League Diamondbacks, going 1–2 with a 5.14 ERA over seven innings. He also pitched in one game for the Hillsboro Hops at the end of the year.
On January 27, 2020, the Diamondbacks traded Malone and Liover Peguero to the Pittsburgh Pirates in exchange for Starling Marte and cash considerations. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. He missed a majority of the 2021 season due to a lat injury, and pitched only 14 innings for the year.
Mason Martin
Mason Neil Martin (born June 2, 1999) is an American professional baseball first baseman in the Pittsburgh Pirates organization. |
1832_14 | Martin attended Southridge High School in Kennewick, Washington, where he played football and baseball. In 2017, his senior year, he hit .507 with five home runs and ten stolen bases. After his senior year, he was drafted by the Pittsburgh Pirates in the 17th round of the 2017 Major League Baseball draft. He signed for $350,000, forgoing his commitment to play college baseball at Gonzaga University. |
1832_15 | After signing with the Pirates, Martin made his professional debut with the Rookie-level Gulf Coast League Pirates, slashing .307/.457/.630 with 11 home runs (breaking the GCL Pirates record) and 22 RBIs over 39 games, earning the title of Gulf Coast League Most Valuable Player. Martin began the 2018 season with the West Virginia Power of the Class A South Atlantic League, but was reassigned to the Bristol Pirates of the Rookie-level Appalachian League halfway through the year. Over 104 games between the two teams, he batted .220 with 14 home runs and 58 RBIs. In 2019, he began the year with the Greensboro Grasshoppers of the Class A South Atlantic League (with whom he was named an All-Star) before being promoted to the Bradenton Marauders of the Class A-Advanced Florida State League in July, with whom he finished the season. Over 131 games, Martin slashed .254/.351/.558 with 35 home runs and 129 RBIs. His 35 home runs were fourth in all of the minor leagues and his 129 RBIs were |
1832_16 | first. Following the season's end, he was named Pittsburgh's Minor League Player of the Year. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. Martin was assigned to the Altoona Curve of the Double-A Northeast for a majority the 2021 season, slashing .242/.318/.481 with 22 home runs, 75 RBIs, and 29 doubles over 112 games. Following the end of Altoona's season in mid-September, he was promoted to the Indianapolis Indians of the Triple-A East, appearing in eight games in which he hit three home runs to end the year. Martin led all Pittsburgh minor leaguers in home runs (25) and runs batted in (81), while also striking out 171 times over 439 at-bats. |
1832_17 | Cal Mitchell
Calvin David Mitchell (born March 8, 1999) is an American professional baseball outfielder in the Pittsburgh Pirates organization.
Mitchell graduated from Rancho Bernardo High School in San Diego, California, where he played four years of varsity baseball. As a junior in 2016, he batted .371 with 12 home runs and 41 RBIs. In 2017, as a senior, he batted .369 with 11 home runs and 34 RBIs. For his high school career, he batted .337 with 29 home runs and 120 RBIs. Mitchell committed to the University of San Diego to play college baseball for the San Diego Toreros baseball team. However, the Pittsburgh Pirates selected him in the second round (50th overall) of the 2017 Major League Baseball draft and he signed for $1.4 million, forgoing his commitment to USD. |
1832_18 | After signing, Mitchell made his professional debut with the Rookie-level Gulf Coast League Pirates. He spent the all of his first professional season there, batting .245 with two home runs and 20 RBIs in 43 games. He spent 2018 with the West Virginia Power of the Class A South Atlantic League. He was named the SAL Player of the Week for April 16–22 after hitting .467 with one home run, nine RBIs, and a 1.233 OPS, and also earned All-Star honors. Over 119 games for the Power, he hit .280 with ten home runs and 65 RBIs. He spent 2019 with the Bradenton Marauders of the Class A-Advanced Florida State League, with whom he was named an All-Star. Over 118 games, he slashed .251/.304/.406 with 15 home runs and 64 RBIs. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. Mitchell was assigned to the Altoona Curve of the Double-A Northeast for a majority of the 2021 season, slashing .280/.330/.429 with 12 home runs, 61 RBIs, and 19 doubles |
1832_19 | over 108 games. Following the end of Altoona's season in mid-September, he was promoted to the Indianapolis Indians of the Triple-A East with whom he played seven games. |
1832_20 | Kyle Nicolas
Kyle Nicolas (born February 22, 1999) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Nicolas grew up in Massillon, Ohio and attended Jackson High School, where he played baseball and basketball. He won state titles in both sports as a senior and was named the Federal League Co-Player of the Year in baseball after going 8-0 with a save and a 0.50 ERA on the mound while also batting .349 with 24 RBIs.
Nicolas played for the Ball State Cardinals for three seasons. In 2019, he played collegiate summer baseball with the Cotuit Kettleers of the Cape Cod Baseball League. As a junior, Nicolas went 0-1 with a 2.74 ERA in four starts before the season was cut short due to the coronavirus pandemic. |
1832_21 | Nicolas was selected 61st overall by the Miami Marlins in the 2020 Major League Baseball draft. After not playing in the minor leagues in 2020 following the cancelation of the season due to Covid-19, he began the 2021 season with the High-A Beloit Snappers. Nicolas was promoted to the Double-A Pensacola Blue Wahoos after posting 3-2 record with a 5.28 ERA and 86 strikeouts in innings with Beloit.
On November 29, 2021, Nicolas was traded along with Zach Thompson and Connor Scott to the Pittsburgh Pirates in exchange for Jacob Stallings.
Nicolas is the nephew of former Penn State and NFL quarterback Todd Blackledge.
Ball State Cardinals bio
Liover Peguero
Liover Peguero (born December 31, 2000) is a Dominican professional baseball shortstop in the Pittsburgh Pirates organization. |
1832_22 | Peguero signed with the Arizona Diamondbacks as an international free agent in July 2017. He spent his first professional season in 2018 with the Dominican Summer League Diamondbacks and Arizona League Diamondbacks, batting .259 with one home run and 21 RBIs over 41 games. In 2019, he played for the Missoula Osprey and Hillsboro Hops and slashed .326/.382/.485 with five home runs, 38 RBIS, and 11 stolen bases over sixty games.
On January 27, 2020 the Diamondbacks traded Peguero and Brennan Malone to the Pittsburgh Pirates for Starling Marte. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. The Pirates invited him to their Spring Training in 2021. He spent the season with the Greensboro Grasshoppers, slashing .270/.332/.444 with 14 home runs, 45 RBIs, and 28 stolen bases over ninety games. Pittsburgh selected his contract and added him to their 40-man roster after the season.
Canaan Smith-Njigba |
1832_23 | Canaan Elijah Smith-Njigba (born April 30, 1999) is an American professional baseball left fielder in the Pittsburgh Pirates organization.
Smith-Njigba attended Rockwall-Heath High School in Heath, Texas, where he played baseball. He committed to play college baseball at the University of Arkansas. During his senior year, he garnered attention after he was intentionally walked 32 times in 24 games, or a rate of 1.67 per game. After his senior season, he was selected by the New York Yankees in the fourth round of the 2017 Major League Baseball draft. |
1832_24 | After signing with the Yankees, Smith-Njigba made his professional debut with the Rookie-level Gulf Coast League Yankees, slashing .289/.430/.422 with five home runs and 28 RBIs over 57 games. In 2018, he played with the Staten Island Yankees of the Class A Short Season New York–Penn League where he hit .191 with three home runs and 16 RBIs over 45 games, missing time due to injury. In 2019, he played for the Charleston RiverDogs of the Class A South Atlantic League, with whom he earned All-Star honors. Over 124 games, he batted .307/.405/.465 with 11 home runs, 74 RBIs, and 16 stolen bases. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. |
1832_25 | On January 24, 2021, the Yankees traded Smith-Njigba (alongside Miguel Yajure, Roansy Contreras and Maikol Escotto) to the Pittsburgh Pirates in exchange for Jameson Taillon. He was assigned to the Altoona Curve of the Double-A Northeast for a majority of the 2021 season. He was placed on the injured list briefly at the end of July, but activated in early August. He was placed back on the injured list in mid-August with a thigh injury, but was activated one month later. Over 66 games, he hit .274/.398/.406 with six home runs, forty RBIs, and 13 stolen bases. Following the end of Altoona's season in mid-September, he was promoted to the Indianapolis Indians of the Triple-A East with whom he appeared in seven games. He was selected to play in the Arizona Fall League for the Peoria Javelinas after the season. On November 19, 2021, the Pirates selected his contract and added him to their 40-man roster.
Smith-Njigba is the brother of Ohio State wide receiver Jaxon Smith-Njigba. |
1832_26 | Hunter Stratton
Hunter Stratton (born November 17, 1996) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Stratton attended Sullivan East High School in Bluff City, Tennessee, where he finished his high school career with 168 strikeouts and was inducted into their Hall of Fame. He played two seasons of college baseball at Walters State Community College, throwing two no-hitters during his sophomore year. Following the end of his sophomore year, he was selected by the Pittsburgh Pirates in the 16th round of the 2017 Major League Baseball draft. |
1832_27 | Stratton signed with the Pirates and made his professional debut with the Bristol Pirates of the Rookie-level Appalachian League, going 0-2 with a 4.81 ERA and 38 strikeouts over 43 innings. He spent the 2018 season with the West Virginia Power of the Class A South Atlantic League with whom he appeared in 22 games (making twenty starts) and went 6-5 with a 4.16 ERA and 82 strikeouts over innings. In 2019, he pitched for the Bradenton Marauders of the Class A-Advanced Florida State League where he pitched 72 innings and compiled a 5-4 record and 4.25 ERA. He did not play a minor league game in 2020 due the cancellation of the minor league season. Stratton began the 2021 season with the Altoona Curve of the Double-A Northeast and was promoted to the Indianapolis Indians of the Triple-A East during the season. Over 38 relief appearances between the two teams, he went 2-2 with a 2.39 ERA and seventy strikeouts over 49 innings. After the season, he played in the Dominican Winter League |
1832_28 | for the Gigantes del Cibao. |
1832_29 | Beau Sulser
Beau Grayson Sulser (born May 5, 1994) is an American professional baseball pitcher in the Pittsburgh Pirates organization.
Sulser played college baseball at Dartmouth from 2013 to 2017. In 2017, he was named Ivy League Pitcher of the Year.
He was drafted by the Pittsburgh Pirates in the tenth round of the 2017 Major League Baseball draft.
His brother Cole Sulser is also a professional baseball player.
Jack Suwinski
Jack William Suwinski (born July 29, 1998) is an American professional baseball outfielder in the Pittsburgh Pirates organization.
Suwinski attended Taft High School in Chicago, Illinois. He was drafted by the San Diego Padres in the 15th round of the 2016 Major League Baseball draft. On July 26, 2021 the Padres traded Suwinski, Tucupita Marcano, and Michell Miliano to the Pittsburgh Pirates for Adam Frazier.
After the 2021 season, the Pirates added Suwinski to their 40-man roster.
Tahnaj Thomas |
1832_30 | Tahnaj A'kheel Thomas (born June 16, 1999) is a Bahamian professional baseball pitcher in the Pittsburgh Pirates organization.
Thomas signed with the Cleveland Indians as an international free agent in 2016. He spent his first professional season in 2017 with the Dominican Summer League Indians and Arizona League Indians, going 0-5 with a 5.63 ERA over innings, and played 2018 with the Arizona League Indians where he posted a 4.58 ERA over innings. |
1832_31 | On November 14, 2018 the Indians traded Thomas, Erik González and Dante Mendoza to the Pittsburgh Pirates for Jordan Luplow and Max Moroff. Thomas spent his first season with the Pirates organization in 2019 with the Bristol Pirates and pitched to a 2-3 record with a 3.17 ERA and 59 strikeouts over innings. He did not play a minor league game in 2020 since the season was cancelled due to the COVID-19 pandemic. Thomas spent 2021 with the Greensboro Grasshoppers. Over 16 starts, Thomas went 3-3 with a 5.19 ERA and 62 strikeouts over innings.
Blake Weiman
Blake Gerald Weiman (born November 5, 1995) is an American professional baseball pitcher in the Pittsburgh Pirates organization. |
1832_32 | Weiman attended Columbine High School in Columbine, Colorado. He signed to play college baseball at the University of Kansas. During his high school career, he played in three Colorado Rockies Futures games. In 2014, as a senior, he went 7–0 with a 1.40 ERA and was named to the All-Colorado baseball team. Undrafted out of high school in the 2014 Major League Baseball draft, he enrolled at Kansas.
In 2015, as a freshman at Kansas, Weiman appeared in 21 games (seven starts), going 2–7 with a 6.75 ERA. As a sophomore in 2016, he became a full-time starter, appearing in 17 games in which he went 2–7 with a 6.82 ERA. In 2017, in Weiman's junior season, he moved to the bullpen where he greatly improved, pitching to a 5–1 record with a 2.80 ERA, striking out 55 batters in 45 relief innings pitched. After his junior year, he was drafted by the Pittsburgh Pirates in the eighth round of the 2017 Major League Baseball draft. |
1832_33 | Weiman signed with Pittsburgh and made his professional debut with the West Virginia Black Bears of the Class A Short Season New York–Penn League, going 4–3 with a 3.78 ERA in 21 relief appearances. He began the 2018 season with the West Virginia Power of the Class A South Atlantic League and was promoted to the Bradenton Marauders of the Class A-Advanced Florida State League and the Altoona Curve of the Class AA Eastern League during the year. In 67 relief innings pitched between the three clubs, he went 4–1 with a 2.42 ERA and 77 strikeouts. After the season, he played for the Surprise Saguaros of the Arizona Fall League and was named to the Fall Stars Game. Weiman was a non-roster invitee to 2019 spring training. He returned to Altoona to begin 2019 and was promoted to the Indianapolis Indians of the Class AAA International League in June after pitching to a 1.86 ERA over relief innings. Over eight relief appearances with Indianapolis, Weiman went 0–1 with a 4.63 ERA. He missed |
1832_34 | nearly all of the last two months of the season due to injury. |
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