Heidelberg University the Ruprecht Karl University of Heidelberg, is a public research university in Heidelberg, Baden-Württemberg, Germany. Founded in 1386 on instruction of Pope Urban VI, Heidelberg is Germany's oldest university and one of the world's oldest surviving universities, it was the third university established in the Holy Roman Empire. Heidelberg has been a coeducational institution since 1899; the university consists of twelve faculties and offers degree programmes at undergraduate and postdoctoral levels in some 100 disciplines. Heidelberg comprises three major campuses: the humanities are predominantly located in Heidelberg's Old Town, the natural sciences and medicine in the Neuenheimer Feld quarter, the social sciences within the inner-city suburb Bergheim; the language of instruction is German, while a considerable number of graduate degrees are offered in English as well as some in French. As of 2017, 29 Nobel Prize winners have been affiliated with the university. Modern scientific psychiatry, psychopharmacology, psychiatric genetics, environmental physics, modern sociology were introduced as scientific disciplines by Heidelberg faculty.

1,000 doctorates are completed every year, with more than one third of the doctoral students coming from abroad. International students from some 130 countries account for more than 20 percent of the entire student body. Heidelberg is a German Excellence University, part of the U15, as well as a founding member of the League of European Research Universities and the Coimbra Group; the university's noted alumni include eleven domestic and foreign Heads of State or Heads of Government. In international comparison Heidelberg University occupies top positions in rankings and enjoys a high academic reputation; the Great Schism of 1378 made it possible for Heidelberg, a small city and capital of the Electorate of the Palatinate, to gain its own university. The Great Schism was initiated by the election of two popes after the death of Pope Gregory XI in the same year. One successor the other in Rome; the German secular and spiritual leaders voiced their support for the successor in Rome, which had far-reaching consequences for the German students and teachers in Paris: they lost their stipends and had to leave.

Rupert I recognized the opportunity and initiated talks with the Curia, which led to a Papal Bull for foundation of a university. After having received, on 23 October 1385, permission from pope Urban VI to create a school of general studies, the final decision to found the university was taken on 26 June 1386 at the behest of Rupert I, Count Palatine of the Rhine; as specified in the papal charter, the university was modelled after University of Paris and included four faculties: philosophy, theology and medicine. On 18 October 1386 a special Pontifical High Mass in the Heiliggeistkirche was the ceremony that established the university. On 19 October 1386 the first lecture was held. In November 1386, Marsilius of Inghen was elected first rector of the university; the rector seal motto was semper apertus—i.e. "the book of learning is always open." The university grew and in March 1390, 185 students were enrolled at the university. Between 1414 and 1418, theology and jurisprudence professors of the university took part in the Council of Constance and acted as counselors for Louis III, who attended this council as representative of the emperor and chief magistrate of the realm.

This resulted in establishing a good reputation for its professors. Due to the influence of Marsilius, the university taught the nominalism or via moderna. In 1412, both realism and the teachings of John Wycliffe were forbidden at the university but around 1454, the university decided that realism or via antique would be taught, thus introducing two parallel ways; the transition from scholastic to humanistic culture was effected by the chancellor and bishop Johann von Dalberg in the late 15th century. Humanism was represented at Heidelberg University by the founder of the older German Humanistic School Rudolph Agricola, Conrad Celtes, Jakob Wimpfeling, Johann Reuchlin. Æneas Silvius Piccolomini was chancellor of the university in his capacity of provost of Worms, always favored it with his friendship and good-will as Pope Pius II. In 1482, Pope Sixtus IV permitted laymen and married men to be appointed professors in the ordinary of medicine through a papal dispensation. In 1553, Pope Julius III sanctioned the allotment of ecclesiastical benefice to secular professors.

Martin Luther's disputation at Heidelberg in April 1518 made a lasting impact, his adherents among the masters and scholars soon became leading Reformationists in Southwest Germany. With the Electorate of the Palatinate turn to the Reformed faith, Otto Henry, Elector Palatine, converted the university into a calvinistic institution. In 1563, the Heidelberg Catechism was created under collaboration of members of the university's divinity school; as the 16th century was passing, the late humanism stepped beside Calvinism as a predominant school of thought. It developed into a cultural and academic center. However, with the beginning of the Thirty Years' War in 1618, the intellectual and fiscal wealth of the university declined. In 1622, the then-world-famous Bibliotheca Palatina was stolen f

Medoids are representative objects of a data set or a cluster with a data set whose average dissimilarity to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most used on data when a mean or centroid cannot be defined, such as graphs, they are used in contexts where the centroid is not representative of the dataset like in images and 3-D trajectories and gene expression. These are of interest while wanting to find a representative using some distance other than squared euclidean distance. For some data sets. A common application of the medoid is the k-medoids clustering algorithm, similar to the k-means algorithm but works when a mean or centroid is not definable; this algorithm works as follows. First, a set of medoids is chosen at random. Second, the distances to the other points are computed. Third, data are clustered according to the medoid. Fourth, the medoid set is optimized via an iterative process.

Note that a medoid is not equivalent to a median, a geometric median, or centroid. A median is only defined on 1-dimensional data, it only minimizes dissimilarity to other points for metrics induced by a norm. A geometric median is defined in any dimension, but is not a point from within the original dataset. Let x 1, x 2, ⋯, x n be a set of n points in a space with a distance function d. Medoid is defined as x medoid = argmin y ∈ ∑ i = 1 n d. From the definition above, it is clear that the medoid can be computed after computing all pairwise distances between points in the ensemble; this would take O distance evaluations. In the worst case, one can not compute the medoid with fewer distance evaluations. However, there are many approaches that allow us to compute medoids either or in sub-quadratic time under different statistical models. If the points lie on the real line, computing the medoid reduces to computing the median which can be done in O by Quick-select algorithm of Hoare. However, in higher dimensional real spaces, no linear-time algorithm is known.

RAND is an algorithm that estimates the average distance of each point to all the other points by sampling a random subset of other points. It takes a total of O distance computations to approximate the medoid within a factor of with high probability, where Δ is the maximum distance between two points in the ensemble. Note that RAND is an approximation algorithm, moreover Δ may not be known apriori. RAND was leveraged by TOPRANK which uses the estimates obtained by RAND to focus on a small subset of candidate points, evaluates the average distance of these points and picks the minimum of those. TOPRANK needs O distance computations to find the exact medoid with high probability under a distributional assumption on the average distances. Trimed presents an algorithm to find the medoid with O distance evaluations under a distributional assumption on the points; the algorithm uses the triangle inequality to cut down the search space. Meddit leverages a connection of the medoid computation with multi-armed bandits and uses a Upper-Confidence-bound type of algorithm to get an algorithm which takes O distance evaluations under statistical assumptions on the points.

Correlated Sequential Halving leverages multi-armed bandit techniques, improving upon Meddit. By exploiting the correlation structure in the problem, the algorithm is able to provably yield drastic improvement in both number of distance computations needed and wall clock time. An implementation of RAND, TOPRANK, trimed can be found here. An implementation of Meddit can be found here. An implementation of Correlated Sequential Halving can be found here. K-medoids k-means algorithm Centroid

Thomas Whittemore was an American scholar and archaeologist who founded the Byzantine Institute of America. His close personal relationship with Mustafa Kemal Ataturk and the first president of the Turkish Republic, enabled him to gain permission from the Turkish government to start the preservation of the Hagia Sophia mosaics in 1931. Thomas Whittemore was born in the Cambridgeport neighborhood of Cambridge, Massachusetts, on January 2, 1871, he earned a Bachelor of Arts degree in English Literature from Tufts College in 1894. He taught English Composition at Tufts for a year and studied at Harvard Graduate School of Arts and Sciences, he taught courses in the fine arts at New York University and Columbia University. From 1911 until his death Whittemore served as American representative on the Egyptian Exploration Fund. Whittemore worked in various capacities to provide relief to Russian refugees during World War I and following the Russian Revolution, he spent 8 months in Russia in 1915-16 and reported on conditions there when he returned to New York to organized shipments of supplies.

He was a member of the U. S.-based Russian Relief Commission and a committee for war relief organized by Grand Duchess Tatiana Nikolaovna. In 1930, Whittemore founded the Byzantine Institute of America, whose mission was to "conserve, restore and document" the monuments and artworks of the Byzantine world. In 1931, Whittemore traveled with the Institute to Istanbul with the permission of Mustafa Kemal Atatürk to oversee the removal of plaster covering the Byzantine mosaics in Hagia Sophia. Of the radical and sudden transformation of Hagia Sophia from an active mosque to a secular museum in 1931 he wrote: "Santa Sophia was a mosque the day that I talked to him; the next morning, when I went to the mosque, there was a sign on the door written in Ataturk's own hand. It said:'The museum is closed for repairs'"In 1934, Harvard University appointed him keeper of Byzantine coins and seals at the Fogg Art Museum for a year, he accepted a presidential appointment to represent the United States at the Byzantine Conference in Sofia in September of that year.

His work was reported in the United States. In 1942, the New York Times noted his return to Istanbul for his "ninth year in uncovering Byzantine mosaics in the St. Sophia Museum". Beginning in 1948, he sponsored a program for the restoration of the mosaics in the Chora Church in Istanbul. Brown University awarded him an honorary degree of Doctor of Laws in 1950. Asteroid 931 Whittemora, discovered by French astronomer François Gonnessiat at Algiers Observatory in March 1920, was named in his honor; the naming was mentioned in The Names of the Minor Planets by Paul Herget in 1955. On June 8, 1950, he suffered a heart attack while visiting the U. S. Department of State in Washington, D. C, he was buried in Mount Auburn Cemetery in Massachusetts. "Who Was Thomas Whittemore", at Dumbarton Oaks "Whittemore, Thomas", Dictionary of Art Historians Laurian Douthett, "The Leading Protagonist: Thomas Whittemore", May 10, 2013 Thomas Whittemore papers, ca. 1875-1966, at Dumbarton Oaks