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Travelling salesman problem

The travelling salesman problem asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science. The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP. In the theory of computational complexity, the decision version of the TSP belongs to the class of NP-complete problems. Thus, it is possible that the worst-case running time for any algorithm for the TSP increases superpolynomially with the number of cities; the problem was first formulated in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Though the problem is computationally difficult, many heuristics and exact algorithms are known, so that some instances with tens of thousands of cities can be solved and problems with millions of cities can be approximated within a small fraction of 1%.

The TSP has several applications in its purest formulation, such as planning and the manufacture of microchips. Modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the concept city represents, for example, soldering points, or DNA fragments, the concept distance represents travelling times or cost, or a similarity measure between DNA fragments; the TSP appears in astronomy, as astronomers observing many sources will want to minimize the time spent moving the telescope between the sources. In many applications, additional constraints such as limited resources or time windows may be imposed; the origins of the travelling salesman problem are unclear. A handbook for travelling salesmen from 1832 mentions the problem and includes example tours through Germany and Switzerland, but contains no mathematical treatment; the travelling salesman problem was mathematically formulated in the 1800s by the Irish mathematician W. R. Hamilton and by the British mathematician Thomas Kirkman.

Hamilton’s Icosian Game was a recreational puzzle based on finding a Hamiltonian cycle. The general form of the TSP appears to have been first studied by mathematicians during the 1930s in Vienna and at Harvard, notably by Karl Menger, who defines the problem, considers the obvious brute-force algorithm, observes the non-optimality of the nearest neighbour heuristic: We denote by messenger problem the task to find, for finitely many points whose pairwise distances are known, the shortest route connecting the points. Of course, this problem is solvable by finitely many trials. Rules which would push the number of trials below the number of permutations of the given points, are not known; the rule that one first should go from the starting point to the closest point to the point closest to this, etc. in general does not yield the shortest route. It was first considered mathematically in the 1930s by Merrill M. Flood, looking to solve a school bus routing problem. Hassler Whitney at Princeton University introduced the name travelling salesman problem soon afterward.

In the 1950s and 1960s, the problem became popular in scientific circles in Europe and the USA after the RAND Corporation in Santa Monica offered prizes for steps in solving the problem. Notable contributions were made by George Dantzig, Delbert Ray Fulkerson and Selmer M. Johnson from the RAND Corporation, who expressed the problem as an integer linear program and developed the cutting plane method for its solution, they wrote what is considered the seminal paper on the subject in which with these new methods they solved an instance with 49 cities to optimality by constructing a tour and proving that no other tour could be shorter. Dantzig and Johnson, speculated that given a near optimal solution we may be able to find optimality or prove optimality by adding a small number of extra inequalities, they used this idea to solve their initial 49 city problem using a string model. They found. While this paper did not give an algorithmic approach to TSP problems, the ideas that lay within it were indispensable to creating exact solution methods for the TSP, though it would take 15 years to find an algorithmic approach in creating these cuts.

As well as cutting plane methods, Dantzig and Johnson used branch and bound algorithms for the first time. In the following decades, the problem was studied by many researchers from mathematics, computer science, chemistry and other sciences. In the 1960s however a new approach was created, that instead of seeking optimal solutions, one would produce a solution whose length is provably bounded by a multiple of the optimal length, in doing so create lower bounds for the problem. One method of doing this was to create a minimum spanning tree of the graph and double all its edges, which produces the bound that the length of an optimal tour is at most twice the weight of a minimum spanning tree. Christofides made a big advance in this approach of giving an approach for which we know the worst-case scenario. Christofides algorithm given in 1976, at worst is 1.5 times longer than the optimal solution. As the algorithm was so simple and quick, many hoped it would give way to a near optimal solution method.

This remains th

Seabird breeding behavior

The term seabird is used for many families of birds in several orders that spend the majority of their lives at sea. Seabirds make up some, if not all, of the families in the following orders: Procellariiformes, Sphenisciformes and Charadriiformes. Many seabirds remain at sea for several consecutive years at a time, without seeing land. Breeding is the central purpose for seabirds to visit land; the breeding period is extremely protracted in many seabirds and may last over a year in some of the larger albatrosses. Seabirds nest in single or mixed-species colonies of varying densities on offshore islands devoid of terrestrial predators. However, seabirds exhibit many unusual breeding behaviors during all stages of the reproductive cycle that are not extensively reported outside of the primary scientific literature; the courtship stage of breeding is when pair bonds are formed and occurs before copulation and continues through the copulatory and chick-rearing stages of the breeding phenology. The sequence and variety of courting behaviors vary among species, but they begin with territorial defense, followed by mate-attraction displays, selection of a nest site.

Seabirds are long-lived monogamous, birds that mate for life. This makes selecting a mate important with lifelong implications for the reproductive success of both individuals in the pair. Seabirds are one of the only avian families; these dances are complex and can include displays and vocalizations that vary between families and orders. Albatrosses are well known for their intricate mating dances. All species of albatross have some form of ritualized dance, with many species displaying similar forms. Albatrosses’ complex visual and vocal dances are considered some of the most developed mating displays in any long-lived animal. Both members of the pair use these dances as a proxy for mate quality and it is believed to be a important aspect of mate choice in this family. For black-footed and Laysan albatrosses there are ten described parts to their mating dance which can be given in various sequences. Several parts include “billing” where one individual touches the others bill and “sky pointing” where the bird rises on the tips of its toes, stretches its neck and points its bill upward.

In the wandering albatross, sky pointing is accompanied with “sky calling” where the displaying individual spreads its wings, revealing his massive 12 foot wingspan while pointing and vocalizing skyward. The mating dance may last for several minutes, it has been noted that many albatross species dance upon reuniting with their partner every year. Boobies are another group of seabirds known for their mating displays. Brown, red-footed and blue-footed boobies have at least nine described parts to their mating display. Sky pointing in boobies is similar to albatrosses. Parading is a well-known display in boobies as well. In blue-footed and red-footed boobies, parading includes lifting and flaunting their brightly colored feet at their prospective partner. Frigatebirds are known for breeding system. Unlike other seabirds, frigatebirds have a lek-breeding system where displaying males aggregate in groups of up to 30 individuals with prospecting females flying overhead. However, unlike classic leks, the pair builds a nest on the male’s display site.

The male participates in nest defense and chick-rearing. The main display that male frigatebirds use to attract females is a “gular presentation” where the male inflates his bright red throat pouch, points his head upwards and opens his wings, it has been shown experimentally that there is no correlation between energy expended by males during courtship display and mate selection by females. Once the pair bond is formed, courtship feeding occurs in some species. Courtship feeding is; the male feeds the female, but in certain species where the sex roles are reversed, the female may feed the male. Several reasons proposed as to why courtship feeding occurs is: 1) to help strengthen the pair bond 2) to reduce aggression between males and females and 3) to provide additional nutrition to the females during the egg-laying stage. Courtship feeding is seen in many tern species. In common terns, courtship feeding begins right at the start of pair formation with male terns carrying a fish around the breeding colony, displaying it to prospective mates.

The direct benefits hypothesis may explain. Homosexual behavior has been well documented in over 500 species of non-human animals ranging from insects to lizards to mammals. In birds, same-sex pairing has been shown in many

Hawthorn East, Victoria

Hawthorn East is an inner suburb of Melbourne, Australia, 7 km east of Melbourne's Central Business District. Its local government area is the City of Boroondara. At the 2016 Census, Hawthorn East had a population of 14,321; the suburb is bounded by Barkers Road to the north, Burke Road to the east, Toorak Road and the Monash Freeway to the south and Auburn Road to the west. Hawthorn East is the home of a number of head offices for some of Australia's largest companies, including Coles and Bunnings. Hawthorn East was established in the 1880s and many of the historical buildings and houses are still well-preserved, it is located in the local government area of the City of Boroondara and is between two shopping strips, located on Glenferrie Road and Burke Road. In the 2016 Census, there were 14,321 people in Hawthorn East. 63.8% of people were born in Australia. The next most common countries of birth were China 5.2%, India 3.5%, England 3.1%, New Zealand 1.8% and Malaysia 1.6%. 70.0% of people spoke only English at home.

Other languages spoken at home included Mandarin 6.5%, Greek 2.2%, Cantonese 1.6%, Italian 1.2% and Hindi 1.2%. The most common responses for religion were No Religion 40.0% and Catholic 19.4%. Fritsch Holzer Park is a large open space, popular for recreational activities and named after Augustus Fritsch and the Holzer brothers, who formed the Upper Hawthorn Brick Company on this site in 1883; the former Hawthorn Council purchased the site in 1972 and used it as a landfill site until 1986 a temporary waste transfer station until 1989. In 1995 a project was launched to reconstruct this area into a park. Anderson Park is another significant park in the area and offers panoramic views of the Melbourne CBD. Other parks of note include Victoria Road Reserve; the head office of Coles is located in Hawthorn East, adjacent to Toorak Rd. The Coles Myer group, acquired by Western Australia conglomerate Wesfarmers in 2007 was the original purpose for the site, the former Toorak Drive-In Theatre. Primary and secondary schools within Hawthorn East include Alia College, Auburn Primary School, Auburn South Primary School, Bialik College and Auburn High School.

Auburn Primary School was established in 1889. Hawthorn East is served by the following routes: 70, from Bourke St Docklands via Riversdale Rd to Wattle Park. 72, from Camberwell via Burke Road to Melbourne University. 75, from City via Riversdale Road Camberwell Road to Vermont South. The Belgrave and Alamein lines run through the suburb stopping at the following stations located nearby: Auburn Camberwell Glenferrie The Glen Waverley line runs past the southern boundary of the suburb and the closest station is: Tooronga William "Show me the Money" Terry- Civil rights leader, mentor to Benjamin Nolan. Benjamin "The price is right" Nolan - police informant. City of Hawthorn - the former local government area of which Hawthorn East was a part