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Binary function

In mathematics, a binary function is a function that takes two inputs. Stated, a function f is binary if there exists sets X, Y, Z such that f: X × Y → Z where X × Y is the Cartesian product of X and Y. Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where belongs to the subset if and only if f = z. Conversely, a subset R defines a binary function if and only if for any x in X and y in Y, there exists a unique z in Z such that belongs to R. We define f to be this z. Alternatively, a binary function may be interpreted as a function from X × Y to Z; when thought of this way, one writes f instead of f. Division of whole numbers can be thought of as a function. Another example is that of inner products, or more functions of the form ↦ x T M y where x,y are real-valued vectors of appropriate size and M is a matrix. If M is a positive definite matrix, this yields an inner product. Functions whose domain is a subset of R 2 are also called functions of two variables if their domain does not form a rectangle and thus the cartesian product of two sets.

In turn, one can derive ordinary functions of one variable from a binary function. Given any element x of X, there is a function f f, from Y to Z, given by f x: = f. Given any element y of Y, there is a function f y, or f, from X to Z, given by f y:= f. In computer science, this identification between a function from X × Y to Z and a function from X to ZY, where ZY is the set of all functions from Y to Z, is called currying; the various concepts relating to functions can be generalised to binary functions. For example, the division example above is surjective because every rational number may be expressed as a quotient of an integer and a natural number; this example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables because f = f. One can consider partial binary functions, which may be defined only for certain values of the inputs. For example, the division example above may be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero.

But this function is undefined. A binary operation is a binary function where the sets X, Y, Z are all equal. In linear algebra, a bilinear transformation is a binary function where the sets X, Y, Z are all vector spaces and the derived functions f x and fy are all linear transformations. A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear. However, the bilinear transformation can be interpreted as a single linear transformation from the tensor product X ⊗ Y to Z; the concept of binary function generalises to ternary function, quaternary function, or more to n-ary function for any natural number n. A 0-ary function to Z is given by an element of Z. One can define an A-ary function where A is any set. In category theory, n-ary functions generalise to n-ary morphisms in a multicategory; the interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category.

The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above

Aaron Slegers

Aaron Allan Slegers is an American professional baseball pitcher in the Tampa Bay Rays organization. He played college baseball at Indiana University Bloomington for the Indiana Hoosiers, he has played in Major League Baseball for the Minnesota Twins and the Rays. Slegers attended Notre Dame Preparatory High School in Arizona, he pitched for the school's baseball team, spending most of his junior year on the junior varsity team. He underwent a growth spurt between his junior and senior years, increasing from 6 feet 2 inches to 6 feet 9 inches, resulting arm pain as his bones grew faster than his tendons limited him to only two innings pitched in his senior year, he enrolled at Indiana University Bloomington, where he made the Indiana Hoosiers baseball team as a walk-on. He threw one inning; the next year, he had a stress fracture in his right tibia. His injuries limited him to throwing 8 1⁄3 total innings in his sophomore years. In 2013, his junior year, Slegers helped Indiana win the Big Ten Conference's regular-season championship and the conference tournament, Indiana appeared in the 2013 NCAA Division I Baseball Tournament.

He was named the Big Ten Conference Baseball Pitcher of the Year after going 9-2 with a 2.04 ERA. The Minnesota Twins selected Slegers in the fifth round of the 2013 Major League Baseball draft, he signed with the Twins. He made his professional debut that year with the Elizabethton Twins of the Rookie-level Appalachian League, compiling a 0.47 ERA in 19 innings pitched. He began the 2014 season with the Cedar Rapids Kernels of the Class A Midwest League, was promoted to the Fort Myers Miracle of the Class A-Advanced Florida State League in July. In 23 total starts between both clubs, he pitched to a combined 9-8 record with a 4.53 ERA. He returned to Fort Myers in 2015, after pitching to an 8-6 record and a 2.87 ERA in 19 starts, he was promoted to the Chattanooga Lookouts of the Class AA Southern League in August. In six starts for the Lookouts, he was 1-4 with a 4.91 ERA. He compiled a 10-7 record with a 3.41 ERA in 25 starts. The Twins invited him to spring training as a non-roster player in 2017.

He began the season with the Rochester Red Wings of the Class AAA International League. On August 17, 2017, Slegers made his major league debut for the Twins, he was optioned back to Rochester the next day, was recalled on September 4. In 24 starts for Rochester he was 15-4 with a 3.40 ERA, in four games for Minnesota, he was 0-1 with a 6.46 ERA. Slegers began 2018 with Rochester and was recalled by the Twins on April 26. Slegers was designated for assignment on January 3, 2019. On January 11, 2019, Slegers was claimed off waivers by the Pittsburgh Pirates. Slegers was designated for assignment on March 28, 2019, after the contracts of Melky Cabrera, Francisco Liriano, J. B. Shuck were selected. On March 30, 2019, Slegers was traded to the Tampa Bay Rays in exchange for cash considerations. On May 16, 2019, Slegers was designated for assignment by the Rays without appearing in an MLB game for them. On August 21, the Rays selected his contract. On September 2, Slegers was designated for assignment.

He elected free agency on October 1, but re-signed with the Rays on a minor league deal in the offseason. Slegers' father, Robert, is 7 feet tall, while Christie, is 5 feet 11 inches, his older sister, played volleyball at Lehigh University. Career statistics and player information from MLB, or ESPN, or Baseball-Reference, or Fangraphs, or Baseball-Reference, or Retrosheet

Murchison East railway station

Murchison East railway station is located on the Shepparton railway line in Victoria, Australia. It serves the town of Murchison, opened on 13 January 1880, it became a junction station in September 1890 when the line to Rushworth opened, branching off at the southern end of the station. That line was extended to Colbinabbin and Girgarre; the Rushworth line closed in 1987. The station has the longest passing loop on the line, 870 metres in length, but in 2013 the loop was booked out of service due to poor track condition. A number of other sidings that serve a grain silo complex are opposite the platform; the main timber station building had been fenced off and a new passenger shelter constructed alongside. On 4 June 2014, the station building was destroyed by fire. Flashing lights were provided at the nearby Goulburn Valley Highway level crossing in 1974. Boom barriers have since been added; the closed Wahring station was located between Murchison East and Nagambie, while the closed Arcadia and Toolamba railway stations were located between Murchison East and Mooroopna.

Murchison East has one platform. It is served by V/Line Shepparton line trains; some services terminate here instead of going to Shepparton. Platform 1: Shepparton line: services to Shepparton & Southern Cross Media related to Murchison East railway station at Wikimedia Commons Victorian Railway Stations gallery

Engystomops freibergi

Engystomops freibergi is a frog native to the Amazonian Brazil, southeastern Peru, Amazonian Bolivia. For a while, it was considered to be a synonym of Engystomops petersi, its sibling species, but its species status was resurrected in a study published in 1998; these two species have been mixed in studies, there are records from the Guianas that have not yet been allocated to either species. Divergence of these two species seems to have been driven by behavioural isolation related to male call characteristics more than geographic isolation. Engystomops freibergi are small frogs. Males measure 24–36 mm in snout–vent length and females 25–39 mm. Dorsal colouration is variable. Skin on dorsum is warty. Engystomops freibergi is a locally common species found in lowland Amazon rainforest; these frogs are nocturnal and found in the leaf litter in primary forest. They feed on termites; the breeding period coincides with the rainy season. Males call at night near slow-moving streams; the call consists of a prefix and a "whine" component, in some populations, a third "squawk" component.

Foam nests are deposited on the surface of ponds

Jonathan Creek (New Brunswick)

Jonathan Creek is a tributary of the Petitcodiac River in New Brunswick. The creek's watershed area is around 50 km²; the majority of Jonathan Creek flows through the city of Moncton joining Jones Lake. Because of its close proximity to commercial and residential areas, water run off and bank erosion have impacted the condition of Jonathan Creek. In 2003, city of Moncton pleaded guilty to environmental charges regarding leachate from an out of service landfill being found in Jonathan Creek. List of rivers of New Brunswick https://web.archive.org/web/20120308231933/http://www.petitcodiacwatershed.org/jonathan_creek https://web.archive.org/web/20131228065401/http://www.bassinpetitcodiac.org/jonathan_creek_restoration http://ebi.probeinternational.org/investigations/news-coverage-2/april-19-2007-2/

Stephen Walkom

Stephen Walkom serves as vice president and director of officiating for the National Hockey League. This is second tenure in that position, having served from 2005 to 2009. From 1990 to 2004, from 2009 to 2013, he worked as an on-ice referee for the league, he is married to Annie and together they have three children. They reside in the Pittsburgh suburb of Pennsylvania. Walkom worked as a referee, until retiring August 3, 2005 when he elevated to the management position. Walkom succeeded Andy Van Hellemond as director of officiating, who resigned under controversy in mid-July 2004, he has a bachelor's degree in commerce from Laurentian University. As a referee, he was hired in 1990 after a successful amateur career that included obtaining Level VI certification - the highest in Hockey Canada's Officiating Program - and refereeing in the Memorial Cup tournament. In the NHL, he officiated more than 600 regular season games, 84 Stanley Cup playoff games, the 2002 Winter Olympic Games, 2004 World Cup of Hockey, two Stanley Cup Finals.

Walkom was president of the National Hockey League Officials Association, the labor union that represents NHL referees and linesmen. He was the director and owner of the North American School of Officiating, a summer development camp for aspiring hockey officials in Guelph, Canada; the Sports Business Journal released Walkom's annual salary in July, 2010, to be US$488,736 per year. On August 25, 2009 he announced his decision to step down from his position as the NHL's Director of Officiating to allow him to attempt a comeback at on-ice officiating as a referee, his first game back as an NHL Referee was October 2009 when Florida hosted Buffalo. Since his return, he has officiated in two Stanley Cup Finals. On August 7, 2013, he returned to the position of the NHL's Director of Officiating. HockeyRefs.com's Profile of Stephen Walkom