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Convex hull

In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Formally, the convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. Convex hulls of open sets are open, convex hulls of compact sets are compact; every convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, every antimatroid can be represented by applying this closure operator to finite sets of points; the algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time O for two or three dimensional point sets, in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions.

Convex hulls have been studied for simple polygons, Brownian motion, space curves, epigraphs of functions. Convex hulls have wide applications in mathematics, combinatorial optimization, geometric modeling, ethology. Related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, convex skull. A set of points in a Euclidean space is defined to be convex if it contains the line segments connecting each pair of its points; the convex hull of a given set X may be defined as The minimal convex set containing X The intersection of all convex sets containing X The set of all convex combinations of points in X The union of all simplices with vertices in X For bounded sets in the Euclidean plane, not all on one line, the boundary of the convex hull is the simple closed curve with minimum perimeter containing X. One may imagine stretching a rubber band so that it surrounds the entire set S and releasing it, allowing it to contract; this formulation does not generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a spanning tree of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull.

However, in higher dimensions, variants of the obstacle problem of finding a minimum-energy surface above a given shape can have the convex hull as their solution. For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of the objects; the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces. It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing X, for every X? However, the second definition, the intersection of all convex sets containing X, is well-defined, it is a subset of every other convex set Y that contains X, because Y is included among the sets being intersected. Thus, it is the unique minimal convex set containing X. Therefore, the first two definitions are equivalent; each convex set containing X must contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it contains the intersection of all convex sets containing X, therefore the second and third definitions are equivalent.

In fact, according to Carathéodory's theorem, if X is a subset of a d -dimensional vector space, every convex combination of finitely many points from X is a convex combination of at most d + 1 points in X. The set of convex combinations of a -tuple of points is a simplex. Therefore, every convex combination of points of X belongs to a simplex whose vertices belong to X, the third and fourth definitions are equivalent. In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points, downward-facing points, e

José María Estudillo

José María Estudillo, was an early settler of San Diego and was a governing official during San Diego's Mexican period. Born in Andalusia, Captain Estudillo was Commandant of the Presidio of San Diego from October 23, 1820 to September 1821 and again from 1827 to his death in 1830. Estudillo married Gertrudis Horcasitas. In 1827 Estudillo's son, José Antonio Estudillo, built a large L-shaped adobe house for his father on land granted by Governor José María de Echeandía; the adobe was enlarged and became U-shaped. The house is still standing, known as Casa de Estudillo, is one of the oldest surviving buildings in California, it is located in Old Town San Diego State Historic Park, on the southeast side of the Old Town San Diego plaza, is designated a National Historic Landmark in its own right. José Antonio Estudillo was the grantee of Rancho Janal. Estudillo's other children were José Joaquin Estudillo, grantee of Rancho San Leandro, on the eastern shore of the San Francisco Bay. In December 1823 he was diarist with Brevet Captain José Romero when they were sent to find a route from Sonora to Alta California.

California Pioneer Register and Index, 1542—1848 "The Estudillo Family", The Journal of San Diego History 15:1 by Sister Catherine McShane "Rancho Guajome", The Journal of San Diego History 41:4 by Iris H. W. Engstrand and Mary F. Ward José Romero papers. Archival material. Abstract: "Report, 16 January 1824, to Antonio Narbona from Palm Springs, on his activities in Alta California, on the expedition undertaken with José María Estudillo to locate a trail to the Colorado River, on the conditions that forced them to return to the Cahuilla Indian ranchería." University of California Library, Berkeley. OCLC 122576202 "José Maria Estudillo", from Smythe's History of San Diego, p. 169

Navtol

Navtol is a small village of sarisab-pahi west panchayat of Pandaul block in Madhubani district of Bihar State in India. It is located 1.5 kilometres north of National Highway-57, Ganguli chawk. The village Navtol "Sarisabpahi" is situated at 16–17 kilometers south-east from district headquarters, in Darbhanga commissionary of state Bihar, it is an important Tola of revenue village Sarisab alise Sarisab-pahi. Now there are two Panchayats, Sarisab-pahi to the Sarisab-pahi to the West. Other Tolas are Bitthotol. Total area of Navtol now is 1.5 km2. The population of this village is 6500–7000. Hindi and English are spoken and written in this area, Maithili being the main language of this village, and scripts of this village are Devnagari and Mithilakshar. Durga pooja celebrated in Navtol, every year in "Ashwin" since 1840, it is near jhanjharpur railway station. This place is famous for durgapuja. There is a great sarpanch called Baldeo Jha, he has done a lot of work for their people and village. According to Skanda Purana, Siddharth Kshetra is situated to 2 Yojan in Aagney kon to Kapil Muni ashram Madhubani.

Kapil Ashram is now known as Kapileshwar Sthan in Madhubani district. This place was the site of education center, it was the head of the center of Mithila Nyayvishesh. The ancient name of Sarisabpahi was Siddharth Kshetra according to Brihad Vishnupuran and Skand Puran

1000 Stars

1000 Stars is the debut solo album by Australian singer and former Rogue Traders lead singer Natalie Bassingthwaighte. It was released through Sony Music Australia as a digital download on 20 February 2009, followed by a physical release on 21 February 2009. Upon its release, 1000 Stars debuted at number one on the ARIA Albums Chart and was certified gold by the Australian Recording Industry Association for shipments of more than 35,000 units; the album spawned two top-ten singles, "Alive" and "Someday Soon", which were both certified platinum. In 2006, Bassingthwaighte had signed a recording contract with Sony Music Australia to embark on a solo career, she wrote and recorded 1000 Stars over three months in London, Los Angeles and Sweden with several songwriters and producers, including Paul Barry, Steve Anderson, Jimmy Harry and Ina Wroldsen, among others. The album was released digitally on 20 February 2009 and physically on 21 February 2009; the digital edition on iTunes includes the bonus track "Star".

"Alive" was released on 14 October 2008. It peaked at number eight on the ARIA Singles Chart and was certified platinum by the Australian Recording Industry Association for sales exceeding 70,000 copies; the second single "Someday Soon" was released on 8 December 2008. Upon its release, "Someday Soon" peaked at number seven on the ARIA Singles Chart and spent eight consecutive weeks in the top ten, it was certified platinum. The album's title track "1000 Stars" was released as the third single on 23 April 2009, peaked at number 30."Not for You" was released as the fourth single from 1000 Stars on 23 July 2009, but failed to impact the charts. "Love Like This" was released on 29 January 2010 as the album's fifth and final single. It was released to raise awareness of the AIDS Council Of New South Wales' "Wear It With Pride" campaign leading up to the 2010 Mardi Gras parade. All of the single's proceeds went towards the ongoing support of the LGBT community. Unlike the fourth single, "Love Like This" managed to chart at number 88 on the ARIA Singles Chart.

For the issue dated 2 March 2009, 1000 Stars debuted at number one on the ARIA Albums Chart with first-week sales of 9,000 copies. In its second week, the album sold 6,982 copies, it spent four consecutive weeks in the top ten and was certified gold by the Australian Recording Industry Association for shipments of more than 35,000 units. Notes ^a signifies an additional producer "Supersensual" incorporates elements of "Heart of Glass", written by Debbie Harry and Chris Stein. Adapted from the liner notes of 1000 Stars. Locations Recorded at Los Feliz. Technical credits List of number-one albums of 2009

Listed buildings in Whitehaven

Whitehaven is a town and civil parish in the Borough of Copeland, England. It contains over 170 buildings. Of these, one is listed at Grade I, the highest of the three grades, six are at Grade II*, the middle grade, the others are at Grade II, the lowest grade. Whitehaven is a natural port, the harbour developed in the 17th century for the export of coal from the local mines. During the 18th century the harbour expanded and the town was laid out in a grid plan with a building such as a church at the end to provide a vista. During the 20th century the amount of work done by the port declined, the export of coal ended in the 1980s. During this time a number of the town's houses and other buildings were demolished. Most of the listed buildings are houses and shops of various sizes, many of them in Georgian style. Listed buildings remaining from the industrial past include structures in and around the harbour, warehouses, a former flax mill, colliery buildings now used as a museum. Other listed buildings include churches, civic buildings and public houses, banks.

Air shaft caps providing ventilation for a railway tunnel, a market hall

Bear Dance

Bear Dance is a Native American ceremonial dance that occurs in the spring. It is a ten-day event to strengthen social ties within the community, encourage courtship, mark the end of puberty for girls. For the Utes, it is a ten-day event of dancing, games, horse racing, gambling, it is one of the oldest Ute ceremonies. The bear symbolizes leadership and wisdom. A group of men have played musical rasps for the dance. Reason For The Bear Dance The bear dance is performed by the Ute Indians after the first sound of thunder is heard as spring comes; this tradition began in the fifteenth century taught to humans by bears. The primal ancestor of the Ute Indians are believed by themselves to be bears; the reason for this dance was to help wake up the hibernating bears in winter, the Indians from being inside during the cold season. Along with waking up for winter finding a new mate for the new season is another reason this dance is performed by bears, humans. Men and children are involved in this yearly dance.

For this dance the men are to prepare everything for the performance. During the dance women invite men to dance in order to find a mate, dance together