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In linear algebra, the trace of a square matrix A is defined to be the sum of elements on the main diagonal of A. The trace of a matrix is the sum of its eigenvalues, it is invariant with respect to a change of basis; this characterization can be used to define the trace of a linear operator in general. The trace is only defined for a square matrix; the trace is related to the derivative of the determinant. The trace of an n × n square matrix A is defined as tr ⁡ = ∑ i = 1 n a i i = a 11 + a 22 + ⋯ + a n n where aii denotes the entry on the ith row as well as ith column of A. Let A be a matrix, with A = = Then tr ⁡ = ∑ i = 1 3 a i i = a 11 + a 22 + a 33 = − 1 + 5 + = − 1 The trace is a linear mapping; that is, tr ⁡ = tr ⁡ + tr ⁡ tr ⁡ = c tr ⁡ for all square matrices A and B, all scalars c. A matrix and its transpose have the same trace: tr ⁡ = tr ⁡; this follows from the fact that transposing a square matrix does not affect elements along the main diagonal. The trace of a square matrix, the product of two matrices can be rewritten as the sum of entry-wise products of their elements.

More if A and B are two m × n matrices, then: tr ⁡ = tr ⁡ = tr ⁡ = tr ⁡ = ∑ i, j A i j B i j. This means that the trace of a product of equal-sized matrices functions to a dot product of vectors. For this reason, generalizations of vector operations to matrices involve a trace of matrix products. For real matrices A and B, the trace of a product can be written in the following forms: The matrices in a trace of a product can be switched without changing the result: If A is an m × n matrix and B is an n × m matrix Additionally, for real column matrices a ∈ R n and b ∈ R n, the trace of the outer product is equivalent to the inner product: More the trace is invariant under cyclic permutations, that is, This is known as the cyclic property. Arbitrary permutations are not allowed: in general, tr

Dame Janet Valerie Finch DBE, DL, FAcSS is a British sociologist and academic administrator. She was Vice-Chancellor and Professor of Social Relations at Keele University, has held a number of other public appointments in the UK, she holds an honorary position at the Morgan Centre for the Study of Relationships and Personal Life, based in the School of Social Sciences, University of Manchester. She is part of Flooved advisory board. Finch was educated at Merchant Taylors' Girls' School, Crosby and Bedford College, London where she obtained BA in Sociology, she obtained a PhD in Sociology from the University of Bradford in 1975. She was made an Honorary Fellow of RHC in 1999. Finch has published extensively on family relationships, her research interests focusing on inter-generational family relationships. In September 1995, Finch was appointed Vice-Chancellor at Keele University, having been a Pro-Vice-Chancellor at Lancaster University. At Keele, she has overseen substantial alterations of the University and the increasing development of a private Science and Business Park.

She attracted controversy in February 2007, when it was announced that she would receive a pay raise of 31.7%, the largest raise for any Vice-Chancellor in England that year. In a statement given to the THES, the university responded that the salary awarded would be frozen for a three-year term, was awarded following comparisons with other institutions, taking into account Finch's "outstanding performance during a period of cultural and procedural change and development", she retired from her position in 2010. She returned the following year to be made a Doctor of Letters. Finch is one of four main panel chairs in the Research Excellence Framework, based at the University of Manchester, a non-executive director of the Identity and Passport Service and the chair of the Council of Ombudsman Service Ltd. In 2011 the government selected Finch to be the chair of the UK Statistics Authority, but during a pre-appointment hearing with MPs on the Public Administration Select Committee differences became apparent on how the independence of the chair should be exercised, Finch decided to withdraw from the application process.

In 2012-13 Finch chaired the government's Working Group on Expanding Access to Published Research Findings, which produced a report recommending a policy direction for open access. She was independent co-chair of the Council for Technology. Janet Finch was named a CBE in the 1999 New Year's Honours List for services to social science, a DBE in the 2008 Birthday Honours List, for services to social science and higher education. Married to the Job: Wives' Incorporation in Men's Work Education as Social Policy Research and Policy: the Uses of Qualitative Methods in Social and Educational Research Family Obligations and Social Change Negotiating Family Responsibilities Wills and Families Passing On: Kinship and Inheritance in England Carol Smart David Morgan Keele University University of Manchester Date of birth per Debrett's Citation presented at the dinner to mark Professor Dame Janet Finch's retirement from Keele University Janet Finch at "Pioneers of Qualitative Research" from the Economic and Social Data Service

Hermann Köchly was a German philologist and educational reformer. He studied at Leipzig, taught at the Saalfeld Progymnasium and at the Dresden Kreuzschule. In February 1849, Köchly was elected to the lower house of the Kingdom of Saxony, but that same year was forced to flee to Brussels on account of his participation in the May insurrection, he was appointed professor of classical philology at Zürich in 1851, at Heidelberg in 1864. He attached himself to the Progressive Party. Ueber das Princip des Gymnasialunterrichts der Gegenwart Zur Gymnasialreform The scheme set forth in these pamphlets stressed the natural sciences, and, in Latin and Greek, urged emphasis on content rather than on grammar and style, the gradual abolition of speaking and writing those languages; the plan was adopted in Saxony immediately. Critical essays on Quintus Smyrnæus Hesiod, in collaboration with Gottfried Kinkel An edition of Aratus, Maximi et aliorum astrologica An edition of the text of Apostelesmata Dionysiaca of Nonnos Seven dissertations on De Iliadis carminibus De diversis Hesiodeæ Theogoniæ partibus Iliadis Carmina XVI Three dissertations on De Odysseæ carminibus Opuscula epica IV.

Geschichte des Griechischen Kriegswesens Griechische Kriegsschriftsteller, vol. 1, vol. 2 part 1, vol. 2 part 2. Einleitung in Cäsars Kommentarien über den gallischen Krieg Onosandri de imperatoris officio Liber An edition of Arrian's Anabasis Editions of Euripides and Iphigenia in Taurien An edition of Medea He did translations of Caesar, etc. A collection of his smaller works is found in his Opuscula academica, Akademische Vorträge und Reden and Opuscula philologica. Hug, Hermann Köchly Böckel, Hermann Köchly, ein Bild seines Lebens und seiner Persönlichkeit Arnold Hug, "Köchly, Hermann", Allgemeine Deutsche Biographie, 16, Leipzig: Duncker & Humblot, pp. 410–414 Peter Wirth, "Köchly, Hermann", Neue Deutsche Biographie, 12, Berlin: Duncker & Humblot, p. 294 Attribution This article incorporates text from a publication now in the public domain: "Köckly, Hermann". Encyclopedia Americana. 1920. This article incorporates text from a publication now in the public domain: Gilman, D. C.. "Köchly, Hermann".

New International Encyclopedia. New York: Dodd, Mead

Creston is a census-designated place in San Luis Obispo County, about 10 miles east of Atascadero. Creston was founded in 1884 on the Rancho Huerhuero Mexican land grant. Creston was home to Cardiff Stud Farm, a place once owned by Canadian-American television personality Alex Trebek, where a number of retired American thoroughbred racehorses were taken care of, including those below. Cardiff Stud Farm was sold in 2008 and is now an event center called Windfall Farms. An entity known as Cardiff Stud Farms still exists however, owned by Stephen Sahadi, but is situated in Atascadero, CA. Flying Paster Golden Act Skywalker Itsallgreektome Writer and Scientology founder L. Ron Hubbard died at his ranch near Creston. In 2014, Creston was made an American Viticultural Area called the Creston District AVA. Creston is near the following highways: California State Route 58 California State Route 46 California State Route 41 California State Route 229 The 2010 United States Census reported that Creston had a population of 94.

The population density was 165.3 people per square mile. The racial makeup of Creston was 89 White, 0 African American, 2 Native American, 1 Asian, 0 Pacific Islander, 0 from other races, 2 from two or more races. Hispanic or Latino of any race were 6 persons; the Census reported that 94 people lived in households, 0 lived in non-institutionalized group quarters, 0 were institutionalized. There were 36 households, out of which 15 had children under the age of 18 living in them, 20 were opposite-sex married couples living together, 3 had a female householder with no husband present, 3 had a male householder with no wife present. There were 3 unmarried opposite-sex partnerships, 1 same-sex married couples or partnerships. 8 households were made up of individuals and 2 had someone living alone, 65 years of age or older. The average household size was 2.61. There were 26 families; the population was spread out with 22 people under the age of 18, 4 people aged 18 to 24, 28 people aged 25 to 44, 27 people aged 45 to 64, 13 people who were 65 years of age or older.

The median age was 39.0 years. For every 100 females, there were 80.8 males. For every 100 females age 18 and over, there were 80.0 males. There were 39 housing units at an average density of 68.6 per square mile, of which 27 were owner-occupied, 9 were occupied by renters. The homeowner vacancy rate was 0%. 64 people lived in owner-occupied housing units and 30 people lived in rental housing units. History of Creston, in Creston News, a website

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The Crown Colony of Labuan was a British Crown colony on the northwestern shore of the island of Borneo established in 1848 after the acquisition of the island of Labuan from the Sultanate of Brunei in 1846. Apart from the main island, Labuan consists of six smaller islands. Labuan was expected by the British to be a second Singapore, but it did not fulfill its promise after the failure of its coal production that did not become fruitful, causing investors to withdraw their money, leaving all machinery equipment and Chinese workers that had entered the colony previously; the Chinese workers began involving themselves in other businesses with many becoming chief traders of the island's produce of edible bird's nest, pearl and camphor, with the main successful production being the coconut and sago. World War II brought the invasion of Japanese forces. Subsequently, Labuan became the place where the Japanese commander in Borneo surrendered to the Allied forces, with the territory placed under a military administration before merging into a new crown colony.

Since 1841, when James Brooke had established a solid presence in northwestern Borneo with the establishment of the Raj of Sarawak and began to assist in the suppression of piracy along the island coast, he had persistently promoted the island of Labuan to the British government. Brooke urged the British to establish a naval station, colony or protectorate along the northern coast to prevent other European powers from doing so which being responded by the Admiralty with the arrival of Admiral Drinkwater Bethune to look for a site for a naval station and to investigate Labuan in November 1844, along with Admiral Edward Belcher with his HMS Samarang to survey the island; the British Foreign Office appointed Brooke as a diplomat to Brunei in 1845 and asked him to co-operate with Bethune. At the same time, Lord Aberdeen, the British Foreign Minister at the time sent a letter to the Sultan of Brunei requesting the Sultan to not enter any treaties with other foreign powers while the island was under consideration as a British base.

On 24 February 1845, Admiral Bethune with his HMS Driver and several other political commissions left Hong Kong to survey the island more. The crews found that it was the most suitable for inhabitants than any other island in the coast of Borneo with its coal deposits; the British saw the potential the island could be the next Singapore. Brooke acquired the island for Britain through the Treaty of Labuan with the Sultan of Brunei, Omar Ali Saifuddin II on 18 December 1846. Admiral Rodney Mundy visited Brunei with his ship HMS Iris to keep the Sultan in line until the British government made a final decision to take the island and he took Pengiran Mumin to witness the island's accession to the British Crown on 24 December 1846. Brooke supervised the transferring process and by 1848, the island was made a crown colony and free port with him appointed as the first Governor. From 1890, Labuan came to be administered by the North Borneo Chartered Company before been reverted to British government rule in 1904.

By 30 October 1906, the British government proposed to extend the boundaries of the Straits Settlements to include Labuan. The proposal took effect from 1 January 1907, with the administration area being taken directly from Singapore, the capital of the Straits Settlements; as part of the World War II, the Japanese navy anchored at Labuan on 3 January 1942 without being met by any strong resistance. Most treasury notes on the island had been burned and destroyed by the British to prevent them from falling into Japanese hands; the remaining Japanese forces proceeded to Mempakul in the western coast of neighbouring North Borneo to strengthen their main forces there. Following the complete takeover of the rest of Borneo island, Labuan was ruled as part of the Empire of Japan and garrisoned by units of the Japanese 37th Army, which controlled northern Borneo; the island was renamed Maeda Island after Marquis Toshinari Maeda, the first commander of Japanese forces in northern Borneo. The Japanese planned to construct two airfields on the island with eleven others to be located in different parts of Borneo.

To achieve this, the Japanese brought one hundred thousand Javanese forced labourers from Java to work for them. The liberation of the whole of Borneo began on 10 June 1945 when the Allied forces under the command of General Douglas MacArthur and Lieutenant-General Leslie Morshead landed at Labuan with a convoy of 100 ships; the 9th Australian Division launched an attack, with its 24th Brigade landing two battalions at the island southeast protrudance and the north side of Victoria Harbour on Brown Beach while being supported by massive air and sea bombardments. The landings was witnessed by MacArthur on board the USS Boise when he decided to proceed further south from the southern Philippines to Labuan. Following the surrender of Japan on 15 August 1945, Lieutenant General Masao Baba, the last commander of the Japanese army in northern Borneo surrendered at the island's Layang-layang beach on 9 September 1945, he was brought to the 9th Division headquarters on the island to sign the surrender document in front of the commander of the 9th Division, Major General George Wootten.

The official surrender ceremony was held on the next day on 10 September at Surrender Point. The town of Victoria was rebuilt after the war; the island assumed its former name and was under British Military Administration along with the rest of the British territories in Borneo before joining the Crown Colony of North Borneo on 15 July