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Sharkovskii's theorem

In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolaiovych Sharkovskii, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3 it must have periodic points of every other period. For some interval I ⊂ R, suppose f: I → I is a continuous function. We say that the number x is a periodic point of period m if f m = x and having least period m if furthermore f k ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers: 3 5 7 9 11 … ⋅ 2 0 … 3 ⋅ 2 5 ⋅ 2 7 ⋅ 2 9 ⋅ 2 11 ⋅ 2 … ⋅ 2 1 … 3 ⋅ 2 2 5 ⋅ 2 2 7 ⋅ 2 2 9 ⋅ 2 2 11 ⋅ 2 2 … ⋅ 2 2 … 3 ⋅ 2 3 5 ⋅ 2 3 7 ⋅ 2 3 9 ⋅ 2 3 11 ⋅ 2 3 … ⋅ 2 3 … ⋮ … 2 n … 2 4 2 3 2 2 2 1 It consists of: the odd numbers in increasing order, 2 times the odds in increasing order, 4 times the odds in increasing order, 8 times the odds, etc. at the end we put the powers of two in decreasing order.

This ordering is a total order, but not a well-order. Sharkovskii's theorem states that if f has a periodic point of least period m, m precedes n in the above ordering f has a periodic point of least period n; as a consequence, we see that if f has only finitely many periodic points they must all have periods that are powers of two. Furthermore, if there is a periodic point of period three there are periodic points of all other periods. Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer-generated picture; the assumption of continuity is important, as the discontinuous piecewise linear function f: [ 0, 3 ) → [ 0, 3 ) defined as: f: x ↦ { x + 1 f o r 0 ≤ x < 2 x − 2 f o r 2 ≤ x < 3 for which every value has period 3, would otherwise be a counterexample.

Essential is the assumption of f being defined on an interval – otherwise f: x ↦ − 1, whi

Hyland Bay and Moyle Floodplain

The Hyland Bay and Moyle Floodplain comprises the floodplain of the lower reaches of the Moyle and Little Moyle Rivers, the adjoining mudflats of Hyland Bay, on the west coast of the Top End of the Northern Territory of Australia. The site lies about 200 kilometres south-west of Darwin and 30 km north-east of the Aboriginal community of Wadeye, it is an important site for waterbirds. The site has been identified as a 1,062 km2 Important Bird Area by BirdLife International because the floodplain supports up to 500,000 magpie geese and over 1% of the world population of pied herons; the intertidal mudflats of the bay support large numbers of waders, or shorebirds great knots. Other waterbirds recorded breeding in the area in large numbers include egrets, little pied cormorants, nankeen night herons and royal spoonbills

Oliver Hanrahan

Oliver Daniel Hanrahan is a professional Australian rules footballer with the Hawthorn Football Club in the Australian Football League. Hanrahan was recruited to Hawthorn with selection 14 in the 2017 AFL rookie draft. Hanrahan was born in Fitzroy, Victoria where he grew up in Brighton East. In his earlier days he was well known for his cricket abilities, being a talented all rounder for the St Kevin's College 1st XI as well as playing for the Melbourne cricket club. Hanrahan was drafted straight from school football in the APS competition after quitting the Sandringham Dragons due to cricket commitments, he played for St Kevin's College 1st XVIII. The small forward finished with 19 goals for the season, his goal in the dying seconds of the elimination final against Port Melbourne gave the win for the Hawks. Three weeks they were premiers. Hanrahan made his AFL debut in round 15, 2019, against West Coast at the MCG, he kicked a goal in his debut match. Hanrahan played well enough to hold his spot in the team for the rest of the season.

He was elevated at the end of the season onto the main list. Statistics are correct to the end of 2019. VFL premiership player: 2018 Hawthorn best first year player: 2019 Oliver Hanrahan's profile on the official website of the Hawthorn Football Club Oliver Hanrahan's playing statistics from AFL Tables Oliver Hanrahan at

Sidney McCrory

Sidney Jackson McCrory was the Louisiana Commissioner of Agriculture and Forestry from 1956 to 1960 during the final term of his political ally, Governor Earl Kemp Long. He was a key organizer in 1960 for the John F. Kennedy/Lyndon B. Johnson ticket, which handily carried Louisiana's ten electoral votes. McCrory was one of six children born in the village of Hope Villa on Bayou Manchac in Ascension Parish near Baton Rouge to the former Estelle Buffington Buillon and Cecil C. McCrory, a cotton farmer and a graduate of Louisiana State University with degrees in both mechanical and electrical engineering. Cecil McCrory served as adjutant general of the Louisiana National Guard, which he worked to reorganize during the administration of Governor Ruffin G. Pleasant from 1916 to 1920. Cecil McCrory was the head of the national draft system during World War I. In 1927, he became county agent in Caldwell Parish south of Monroe and transferred to Caddo Parish, where in Shreveport, he was the agent for fifteen years before he returned to his farm at Hope Villa.

Sidney McCrory, whose middle name "Jackson" comes from his paternal grandfather, finished LSU with a degree in entomology. Before he was elected agriculture commissioner, he had been the state entomologist and considered himself well versed in the science of insects. McCrory married the former Nettie Fay Cooper, a schoolteacher in East Baton Rouge Parish, a native of Merryville in Beauregard Parish, the daughter of Mars LeRoy Cooper and Laura Elvira Cooper; the McCrorys had two daughters, Sandra M. Lang and husband and Sharon M. Balser. One of McCrory's sisters, Cherrie Claire, married J. L. Iles, who during World War II was stationed in the Solomon Islands, where he was for a time a roommate of John F. Kennedy; the two were PT boat commanders. Their friendship continued after the war when both attended for several years the annual reunion of their military group in New York City. Sidney McCrory hence endorsed Kennedy when he ran for president, escorted him around the state, made speeches on Kennedy's behalf.

According to Mrs. McCrory, Sidney McCrory worked to have Kennedy named king of the annual International Rice Festival in Crowley, where as a U. S. senator in 1960, he addressed a crowd estimated at 100,000. Promoting Kennedy in Louisiana was Judge Edmund Reggie of Crowley, whose daughter was the second wife of U. S. Senator Ted Kennedy, John Kennedy's younger brother. Sidney McCrory unseated the one-term commissioner, Dave L. Pearce of West Carroll Parish in northeastern Louisiana, in the primary election held on January 17, 1956. McCrory carried Earl Long's support though Long in 1952 had given Pearce a short-term appointment to the office, which Pearce won in a special election in the year. Pearce, a former member of both houses of the Louisiana Legislature, first ran for agriculture commissioner in 1948 on the intraparty ticket of former Governor Sam Houston Jones of Lake Charles. Jones was handily defeated by Long. Pearce lost in 1948 to W. E. Anderson of Tangipahoa Parish, who had succeeded veteran commissioner Harry D. Wilson of Tangipahoa Parish, whose tenure had extended from 1916 until Wilson's death in office in January 1948.

Anderson was died at the end of his current term. Long appointed Pearce to finish Anderson's term, Pearce won a special election and served in the term of the anti-Long Governor Robert F. Kennon of Minden in Webster Parish in northwestern Louisiana. William J. "Bill" Dodd, a veteran state officeholder and an astute observer of Louisiana politics in the mid-twentieth century, said that Earl Long "hated" Pearce—the two became estranged shortly after Pearce became commissioner—and put up the "egghead" McCrory to unseat Pearce in the 1956 primary. Dodd did not McCrory unseated Pearce that year. In his Peapatch Politics: The Earl Long Era in Louisiana Politics, Dodd noted with humor how Long became irritated with McCrory, invited on Long's intraparty ticket to harass and, we hoped, defeat Uncle Earl's old political enemy, Dave Pearce. All McCrory could talk about was pesticides and how to get rid of different kinds of crop-killing bugs, his main topic and claim to fame, which dominated all of his speeches, whether he was in cotton country, forestry areas, or the city of New Orleans, was his eradicating the pink boll worms from Louisiana cotton fields.

Uncle Earl went crazy when had to listen to... McCrory kill enough pink boll worms to fill the Atlantic Ocean. Pearce staged a comeback and defeated McCrory in the primary held on December 5, 1959, when anti-Long sentiment was running in the state. McCrory was eliminated from the runoff election. Instead Pearce defeated a third candidate, George W. Shannon, the choice of gubernatorial candidate deLesseps Story Morrison, defeated in the runoff election by Jimmie Davis. McCrory ran again in 1963, but Pearce was renominated and unopposed in the general election held on March 3, 1964. At the time the office was called "Commissioner of Agriculture and Immigration". Pearce prevailed in 1967 and 1971, he had no Republican opposition during any of those elections. In the 1971 contest, Pearce's last successful one, he referred to himself in an advertisement as "Louisiana Top Salesman... Progressive, Experienced Administrator."


Multipotentiality is an educational and psychological term referring to the ability and preference of a person one of strong intellectual or artistic curiosity, to excel in two or more different fields. It can refer to an individual whose interests span multiple fields or areas, rather than being strong in just one; such traits are called multipotentialities, while "multipotentialites" has been suggested as a name for those with this trait. By contrast, those whose interests lie within a single field are called "specialists." An early instance of the term in the record comes from relevant research in giftedness. In 1972, R. H. Frederickson et al. defined a multipotentialed person as someone who, “when provided with appropriate environments, can select and develop a number of competencies to a high level.” In 1999, "multipotentiality" appears in Laurie Diane Shute's doctoral dissertation, titled "An investigation of multipotentiality among university honors students." In 2010, multipotentiality appears again in Tamara Fisher's article in Education week.

She defines it thus: Multipotentiality is the state of having many exceptional talents, any one or more of which could make for a great career for that person. Around 2010 or Emilie Wapnick coined the term "multipotentialite" to establish a shared identity for the community, she defines it this way: A multipotentialite is a person who has many different interests and creative pursuits in life. Multipotentialites have no "one true calling". Being a multipotentialite is our destiny. We have many paths and we pursue all of them, either sequentially or simultaneously. Multipotentialites thrive on learning and mastering new skills. We are excellent at bringing disparate ideas together in creative ways; this makes us incredible innovators and problem solvers. When it comes to new interests that emerge, our insatiable curiosity leads us to absorb everything we can get our hands on; as a result, we tend to be a wealth of information. While the term "multipotentialite" is used interchangeably with "polymath" or "Renaissance Person", the terms are not identical.

One need not be an expert in any particular field to be a multipotentialite. Indeed, Isis Jade makes a clear distinction between multipotentiality and polymaths. Multipotentiality refers to one's potential in multiple fields owing to his/her diverse interests and attempts. Polymaths, on the other hand, are distinguished by their expertise in several fields. In this sense, multipotentialites can be viewed as potential polymaths. Other terms used to refer to multipotentialites are "scanners", "slashers", "generalist", "multipassionate", "RP2", "multipods", among others. With the advent of the industrial age, cultural norms have shifted in favor of specialization. Indeed, in the modern day, the more narrow the specialization, the higher the pay and respect accorded, for example: PhD graduates, specialized lawyers and engineers; the aphorism Jack of all trades, master of none emphasizes this. Older emphasis towards generalism and multiple potentials such as Renaissance humanism and the Renaissance man were replaced.

However, the convergence economy, Internet age, the rise of the Creative Class, other modern developments are bringing about a return of a more positive opinion for generalists and multipotentialites. In Specialization, Polymaths And The Pareto Principle In A Convergence Economy, Jake Chapman writes: Economists tell us that the history of human labor is one of continually increasing specialization. In the days of the hunter-gatherer, every member of the tribe would have been expected to command some degree of proficiency with each task; as we progressed along the economic continuum from hunter-gatherer through agrarian and industrial and now into post-industrial economies, the labor force has become more fragmented, with workers having more and more specialized skill sets.... Specialization has been a path to prosperity. Although specialization has certain economic advantages, in the era of technological convergence, well-educated generalists will be those who are the most valuable, it is time for a renaissance of the “Renaissance Man.”...

The Renaissance thinkers recognized both the potential of individuals as well as the enormous value to being well-rounded. Somewhere along the way the idea of someone who dabbled in many fields lost its cultural appeal and we began to praise those who sought deep subject matter expertise. We now live in a world where distinctions between separate industries are breaking down and the real opportunities for growth are where those industries intersect. Harnessing these 21st-century opportunities will require people who are “jacks of all trades, masters of none,” or more master polymaths. Organizations such as startups that require adaptability and holding multiple roles can employ several multipotentialites and have one specialist as a resource. In Specialization, Polymaths And The Pareto Principle In A Convergence Economy, Chapman said: In the modern world, where a common job might require someone to be a social-media expert, public speaker and data analyst, the polymath wins and the deep subject-matter expert is relegated to a back corner to be used as a resource for others.

As an investor, if I were going to pick the perfect team, it would be a group of rock-star polymaths with a single subject matter expert as a resource. Stretch Magazine discusses the role of multipotentialites in organizations and how they will believe they will be more in demand in the future. Historical context, current conventional wisdom, comparative advantage, USP, among others contribute to the wide acceptance of spec

Jenkins Orphanage

The Jenkins Orphanage was established in 1891 by Rev. Daniel Joseph Jenkins in Charleston, South Carolina. Jenkins was a businessman and Baptist minister who encountered street children and decided to organize an orphanage for young African Americans; the original site of this orphanage was at 660 King Street, but the number of orphans on roll outgrew the facilities. In 1893, the orphanage moved to the Old Marine Hospital at 20 Franklin Street; this National Historic Landmark, designed by Robert Mills, served as home of the orphanage until 1937. Its present-day location is in South Carolina; the orphanage took in donations of musical instruments. Not being a musician, Jenkins hired two local Charleston musicians — P. M. "Hatsie" Logan and Francis Eugene Mikell — to tutor the children in music. Upon its establishment, it became the only black instrumental group organized in South Carolina; the band's debut was on the streets of Charleston with the permission of the mayor, police chief, Chamber of Commerce.

The Jenkins Orphanage Band, wearing discarded Citadel uniforms, performed throughout the United States and toured England raising money for the support of the orphanage. It played in inaugural parades of Presidents Theodore William Taft, it appeared at the St. Louis Exposition and the Anglo-American Exposition in 1914, it toured the United States from coast to coast, played in Paris, Rome and Vienna. As many as five bands were on tour during the 1920s; the band ceased to exist in the 1980s. In 2003, a 10-minute Fox Movietone News newsreel feature about the band, filmed on November 22, 1928, was entered into the United States National Film Registry as being worthy of preservation. William "Cat" Anderson, Jabbo Smith, Tom Delaney, Freddie Green are but a few of the alumni from the Jenkins Orphanage band who made it to the big time; the orphanage was responsible for providing numerous children with another chance, which many took advantage of by going on to lead productive lives, including teachers and attorneys.

Luther M. Reames Jr. Esq. is a vice-president with a major bank and an example of a Jenkins Orphanage success story. John Chilton A Jazz Nursery: The Story of the Jenkins' Orphanage Bands of Charleston, South Carolina, 60 p. London, U. K.: Bloomsbury, ISBN 095012902X. Jenkins Institute For Children Avery Research Center University of South Carolina Archives Time Magazine The Charleston Jazz Initiative South Carolina Music Hall of Fame Charleston Jazz by Jack McCray Fox Movietone News: Jenkins Orphanage Band on IMDb Excerpt of Fox Movietone News: Jenkins Orphanage Band at the South Carolina University Libraries Moving Image Research Collections