In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism, an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself. In any category, the composition of any two endomorphisms of X is again an endomorphism of X, it follows that the set of all endomorphisms of X forms a monoid, the full transformation monoid, denoted End. An invertible endomorphism of X is called an automorphism; the set of all automorphisms is a subset of End with a group structure, called the automorphism group of X and denoted Aut. In the following diagram, the arrows denote implication: Any two endomorphisms of an abelian group, A, can be added together by the rule = f + g. Under this addition, with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring.

For example, the set of endomorphisms of ℤn is the ring of all n × n matrices with integer entries. The endomorphisms of a vector space or module form a ring, as do the endomorphisms of any object in a preadditive category; the endomorphisms of a nonabelian group generate an algebraic structure known as a near-ring. Every ring with one is the endomorphism ring of its regular module, so is a subring of an endomorphism ring of an abelian group. In any concrete category for vector spaces, endomorphisms are maps from a set into itself, may be interpreted as unary operators on that set, acting on the elements, allowing to define the notion of orbits of elements, etc. Depending on the additional structure defined for the category at hand, such operators can have properties like continuity, so on. More details should be found in the article about operator theory. An endofunction is a function. A homomorphic endofunction is an endomorphism. Let S be an arbitrary set. Among endofunctions on S one finds permutations of S and constant functions associating to every x in S the same element c in S.

Every permutation of S is bijective and invertible. If S has more than one element, a constant function on S has an image, a proper subset of its codomain, thus is not bijective; the function associating to each natural number n the floor of n/2 has its image equal to its codomain and is not invertible. Finite endofunctions are equivalent to directed pseudoforests. For sets of size n there are nn endofunctions on the set. Particular examples of bijective endofunctions are the involutions. Adjoint endomorphism Epimorphism Frobenius endomorphism Monomorphism Jacobson, Basic algebra, 1, Dover, ISBN 978-0-486-47189-1 Hazewinkel, Michiel, ed. "Endomorphism", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 "Endomorphism". PlanetMath

Lissendorf is an Ortsgemeinde – a municipality belonging to a Verbandsgemeinde, a kind of collective municipality – in the Vulkaneifel district in Rhineland-Palatinate, Germany. It belongs to the Verbandsgemeinde of Gerolstein; the municipality lies in the Vulkaneifel, a part of the Eifel known for its volcanic history and geological features, ongoing activity today, including gases that sometimes well up from the earth. Yearly precipitation in Lissendorf amounts to 838 mm, rather high, falling into the highest third of the precipitation chart for all Germany. At 70% of the German Weather Service’s weather stations, lower figures are recorded; the driest month is April. The most rainfall comes in December. In that month, precipitation is. Precipitation varies minimally. At only 5% of the weather stations are lower seasonal swings recorded. Lissendorf lies on the former Roman road between Cologne. Many sideroads favoured settlement in this area in Roman times. Roman graves that have been unearthed confirm this.

Lissendorf had its first documentary mention in Prüm Abbey’s directory of holdings, the Prümer Urbar, in 893. In the Middle Ages, a noble house named “von Lissendorf” was mentioned; this family never attained greatness outside this region, though. Beginning in the 14th century, the village found itself in the hands of the House of Manderscheid-Blankenheim. In the 16th century, the village belonged to the County of Gerolstein. In the time of French rule, Lissendorf became the seat of the cantonal administration. By the end of the 19th century, the local railway, the Eifelbahn between Trier and Cologne, had opened; the council is made up of 16 council members, who were elected by proportional representation at the municipal election held on 7 June 2009, the honorary mayor as chairman. The 16 seats are shared among three voters’ groups. Lissendorf’s mayor is Lothar Schun, his deputies are Paul Leuwer and Monika Deckers; the German blazon reads: In Grün das goldene Geweih mit Grind eines Achtenders.

The municipality’s arms might in English heraldic language be described thus: Vert a stag’s attires, each with four points, fixed to the scalp Or. The noble family von Lissendorf, mentioned above, bore arms with this charge; the stag's attires fixed to the scalp. The municipality chose vert and Or. Saint Dionysius’s Catholic Parish Church, Kirchstraße 10, Romanesque Revival aisleless church, 1886-1887, sandstone Gothic Revival memorial cross, latter half of the 19th century. Am Bungert 20 – ten-axis, one-floor Quereinhaus from 185, late 19th century. Bahnhofstraße – signal box “Lf”, oldest still preserved lever signal box using pulled cables on the Eifelbahn, quarrystone walls with overhanging wood construction on the upper level. Bahnhofstraße 46 – two-floor solid building. Bahnhofstraße/corner of Wiesentalstraße – red sandstone Crucifixion Bildstock possibly from 1613. Friedhofstraße – wayside chapel, plastered building from 1889. Friedhofstraße, graveyard – Gothic Revival red sandstone graveyard cross from 1876.

Grausweg 5 – two-floor plastered building. Hauptstraße 21 – small Quereinhaus from 1829. Kirchstraße 33 – corner estate, bungalow from 1896. Kirchstraße/corner of Burgstraße – wayside cross, sandstone niche cross from the 18th century. Roslay 2 – two-floor plastered building. Wayside cross, northwest of the village on the Burgberg, former Station on a Way of the Cross, red sandstone from 1872. Wayside cross, southwest of the village on the road to Steffeln, red sandstone shaft cross from 18th or early 19th century. Lissendorf railway station lies on the Eifelbahn, served by the following local passenger services: the Eifel-Express. In Lissendorf, the historical Dümpelfeld–Lissendorf railway, known as the Middle Ahrtalbahn branched off. For all local public transport, three tariff systems apply: the Verkehrsverbund Region Trier, the Verkehrsverbund Rhein-Sieg, for journeys crossing tariff zones, the NRW-Tarif. Municipality’s official webpage

Glenn Jones is an American guitarist, He is most recognized for his work in the experimental rock group Cul de Sac. At the age of fourteen Jones picked up playing the acoustic guitar, which he purchased after hearing Axis: Bold as Love. During the early seventies, Jones discovered American primitive guitar and became influenced by Robbie Basho and John Fahey, it wasn't until he was asked by Robin Amos to join Shut-Up. In 1989, he founded Cul de Sac with Amos in Massachusetts; this Is the Wind That Blows It Out: Solos for 6 & 12 String Guitar Against Which the Sea Continually Beats Barbecue Bob in Fishtown Even to Win is to Fail The Wanting My Garden State Welcomed Wherever I Go Fleeting The Giant Who Ate Himself and Other New Works for 6 & 12 String Guitar

Renato Guerino Turano is an Italian and American politician and businessman. He served in the Italian Senate from 2006 to 2008 as a representative of Italian citizens in North America and Central America and was re-elected to the same position in the 2013 Italian general election. Turano was born in Castrolibero, Calabria and moved to the United States with his family at age fifteen, he attended the University of Illinois at Chicago from 1962 to 1966 and returned in 1990 for a Master of Business Administration degree. Turano holds an Honorary Doctorate in Letters from the University of Wisconsin-Parkside. Turano's family purchased a small Chicago baking company in 1962, renamed it as the Turano Baking Company, built it into one of the largest artisan bread producers in North America. Turano worked for the company in his youth and became its president and chief executive officer in 1982, he served as chair of the American Bakers Associations in the 2000s and led a march of eighty bakers to Capitol Hill to lobby for sector relief during a commodity crisis affecting wheat.

In 2009, he was described as one of the twenty most influential baking managers in America. Turano has been active in the Italian-American community for several decades, he founded Chicago's Casa Italia, served on the National Italian American Foundation, from 1996 to 2006 was an American consulate to the Region of Calabria, representing Calabrian Americans at annual conferences in Italy. Turano received a special achievement award from the NIAF for public service in 2007. Turano was elected to the Italian Senate in the 2006 general election, the first in which persons with Italian citizenship living overseas were able to elect their own representatives. Turano was elected to represent voters in North America and Central America, he credited Canadian support as vital to his victory, his party, Romano Prodi's The Union, won the election, Turano helped consolidate its narrow majority in the Senate. A political moderate, he defended the Prodi administration's centrist course in a 2006 interview with the Chicago Tribune, saying that the government was shifting Italy from its recent "quasi-socialistic" history.

Turano was unsuccessful. Turano received more first preference votes than any other candidate in North and Central America, but the governing coalition's party list narrowly lost to Silvio Berlusconi's People of Freedom, he was re-elected in the 2013 general election, again as a candidate of the Democratic Party. Partito Democratico page Senate biography

Beauvais is a city and commune in northern France. It serves in the Hauts-de-France region. Beauvais is located 75 kilometres from Paris; the residents of the city are called Beauvaisiens. The municipality of Beauvais has a population of 56,020 as of 2016, population estimate from the Insee, ranks as the most populous city in the Oise department, the third most-populous city in Picardy. Together with its suburbs and satellite towns, the metropolitan area of Beauvais has a population of 128,020. Beauvais was known to the Romans by the Gallo-Roman name of Caesaromagus; the post-Renaissance Latin rendering is Bellovacum from the Belgic tribe the Bellovaci, whose capital it was. In the ninth century it became a countship, which about 1013 passed to the bishops of Beauvais, who became peers of France from the twelfth century. At the coronations of kings the Bishop of Beauvais wore the royal mantle and went, with the Bishop of Langres, to raise the king from his throne to present him to the people. De Bello Gallico II 13 reports that as Julius Caesar was approaching a fortified town called Bratuspantium in the land of the Bellovaci, its inhabitants surrendered to him when he was about 5 Roman miles away.

Its name is Gaulish for "place", from * bratu-spantion. Some say. Others theorize that it is Bailleul sur Thérain. From 1004 to 1037, the Count of Beauvais was Odo Count of Blois. In a charter dated 1056/1060, Eudo of Brittany granted land "in pago Belvacensi" to the Abbey of Angers Saint-Aubin. In 1346 the town had to defend itself against the English, who again besieged it in 1433; the siege which it endured in 1472 at the hands of the Duke of Burgundy was rendered famous by the heroism of the town's women, under the leadership of Jeanne Hachette, whose memory is still celebrated by a procession on 27 June, during which women take precedence over men. An interesting hoard of coins from the High Middle Ages became known as the Beauvais Hoard, because some of the British and European coins found with the lot were from the French abbey located in Beauvais; the hoard, which contained a variety of rare and rare Anglo-Norman pennies and foreign coins, was reputed to have been found in or near Paris.

Beauvais was extensively damaged during World War I, again in World War II during the German advance on Paris in June 1940. Much of the older part of the city was all but destroyed, the cathedral badly damaged before being liberated by British forces on 30 August 1944. Beauvais lies at the foot of wooded hills on the left bank of the Thérain at its confluence with the Avelon, its ancient ramparts have been destroyed, it is now surrounded by boulevards, outside of which run branches of the Thérain. In addition, there are spacious promenades in the north-east of the town. Beauvais experiences an oceanic climate; the average annual temperature is the sunlight annual average of 1669 hours. Hills Bray are provided to the precipitation of Beauvais; the precipitation is 669 mm on average per year. However, the frequency of rainfall is high; the average number of days per year above the precipitation of a 1 mm is 116 days, or every third day. The fog is present, it is estimated at about 55 days a year; the department is affected by 41 days of average wind year it comes from the west to the south.

The city's cathedral, dedicated to Saint Peter, in some respects the most daring achievement of Gothic architecture, consists only of a transept and quire with apse and seven apse-chapels. The vaulting in the interior exceeds 150 feet in height; the cathedral underwent a major repair and restoration process in 2008. The small Romanesque church of the 10th century known as the Basse Oeuvre occupies the site destined for the nave. Begun in 1247, under Bishop William of Grès, an extra 5 metres were added to the height, to make it the tallest cathedral in Europe: the work was interrupted in 1284 by the collapse of the vaulting of the choir, a disaster that produced a temporary failure of nerve among the masons working in Gothic style; the transept was built from 1500 to 1548. In 1573 the fall of a too-ambitious central tower stopped work again, after which little addition was made, its façades that on the south, exhibit all the richness of the late Gothic style. The carved wooden doors of both the north and the south portals are masterpieces of Gothic and Renaissance workmanship.

The church possesses an elaborate astronomical clock and tapestries of the fifteenth and seventeenth centuries. To him is due some of the stained glass in St. Etienne, the second church of the town, an interesting example of the transition stage between the Romanesque and Gothic styles. During the Middle Ages, on 14 January, the Feast of Asses was celebrated in the Beauvais Cathedral, in commemoration of the Flight into Egypt. In the Place de l'Hôtel de Ville and in the old streets near the cathedral there are several houses dating from the 12th to the 16th centuries; the Hôtel de ville, c

Saint-Julien-de-Concelles is a commune in the Loire-Atlantique department in western France. Saint-Julien-de-Concelles is the home of the popular beurre blanc sauce; the sauce was first discovered in a small restaurant on the banks of the river Loire. The town is one of France's main growers of the mâche, its flat and sandy land is suited to growing vegetables, such as leeks, some flowers. Saint-Julien-de-Concelles grows the grapes on its hills that produce local Muscadet wine. A large lake, known as the Plan d'Eau du Chêne, is located near to the center of Saint-Julien-de-Concelles; the lake supports sailing, various sports and attracts walkers and nature lovers. The following cities are twinned with Saint-Julien-de-Concelles: Saarijärvi, Central Finland, Finland Trittau consisting of the following municipalities: Grande.