Ancona is a city and a seaport in the Marche region in central Italy, with a population of around 101,997 as of 2015. Ancona is the capital of the province of Ancona and of the region; the city is located 280 km northeast of Rome, on the Adriatic Sea, between the slopes of the two extremities of the promontory of Monte Conero, Monte Astagno and Monte Guasco. Ancona is one of the main ports on the Adriatic Sea for passenger traffic, is the main economic and demographic centre of the region. Ancona was founded by Greek settlers from Syracuse in about 387 BC, who gave it its name: Ancona stems from the Greek word Ἀγκών, meaning "elbow". Greek merchants established a Tyrian purple dye factory here. In Roman times it kept its own coinage with the punning device of the bent arm holding a palm branch, the head of Aphrodite on the reverse, continued the use of the Greek language; when it became a Roman town is uncertain. It was occupied as a naval station in the Illyrian War of 178 BC. Julius Caesar took possession of it after crossing the Rubicon.
Its harbour was of considerable importance in imperial times, as the nearest to Dalmatia, was enlarged by Trajan, who constructed the north quay with his Syrian architect Apollodorus of Damascus. At the beginning of it stands the marble triumphal arch, the Arch of Trajan with a single archway, without bas-reliefs, erected in his honour in 115 by the Senate and Roman people. Ancona was successively attacked by the Goths and Saracens between the 3rd and 5th centuries, but recovered its strength and importance, it was one of the cities of the Pentapolis of the Exarchate of Ravenna, a lordship of the Byzantine Empire, in the 7th and 8th centuries. In 840, Saracen raiders burned the city. After Charlemagne's conquest of northern Italy, it became the capital of the Marca di Ancona, whence the name of the modern region. After 1000, Ancona became independent turning into an important maritime republic clashing against the nearby power of Venice. An oligarchic republic, Ancona was ruled by six Elders, elected by the three terzieri into which the city was divided: S. Pietro and Capodimonte.
It had a coin of its own, the agontano, a series of laws known as Statuti del mare e del Terzenale and Statuti della Dogana. Ancona was allied with the Republic of Ragusa and the Byzantine Empire. In 1137, 1167 and 1174 it was strong enough to push back the forces of the Holy Roman Empire. Anconitan ships took part in the Crusades, their navigators included Cyriac of Ancona. In the struggle between the Popes and the Holy Roman Emperors that troubled Italy from the 12th century onwards, Ancona sided with the Guelphs. Differently from other cities of northern Italy, Ancona never became a seignory; the sole exception was the rule of the Malatesta, who took the city in 1348 taking advantage of the black death and of a fire that had destroyed many of its important buildings. The Malatesta were ousted in 1383. In 1532 it definitively lost its freedom and became part of the Papal States, under Pope Clement VII. Symbol of the papal authority was the massive Citadel. Together with Rome, Avignon in southern France, Ancona was the sole city in the Papal States in which the Jews were allowed to stay after 1569, living in the ghetto built after 1555.
In 1733 Pope Clement XII extended the quay, an inferior imitation of Trajan's arch was set up. The southern quay was built in 1880, the harbour was protected by forts on the heights. From 1797 onwards, when the French took it, it appears in history as an important fortress. Ancona, as well as Venice, became a important destination for merchants from the Ottoman Empire during the 16th century; the Greeks formed the largest of the communities of foreign merchants. They were refugees from former Byzantine or Venetian territories that were occupied by the Ottomans in the late 15th and 16th centuries; the first Greek community was established in Ancona early in the 16th century. Ancona entered the Kingdom of Italy when Christophe Léon Louis Juchault de Lamoricière surrendered here on 29 September 1860, eleven days after his defeat at Castelfidardo. On 23 May 1915, Italy joined the Entente Powers. In 1915, following Italy's entry, the battleship division of the Austro-Hungarian Navy carried out extensive bombardments causing great damage to all installations and killing several dozen people.
Ancona was one of the most important Italian ports on the Adriatic Sea during the Great War. During World War II, the city was taken by the Polish 2nd Corps against Nazi German forces, as Free Polish forces were serving as part of the British Army. Poles were tasked with capture of the city on 16 June 1944 and accomplished the task a month on 18 July 1944 in what is known as the battle of Ancona; the attack was part of an Allied operation to gain access to a seaport closer to the Gothic Line in order to shorten their lines of communication for the advance into northern Italy. Jews began to live in Ancona in 967 A. D. In 1270, a Jewish resident of Ancona, Jacob of Ancona, travelled to China, four years before Marco Polo and documented his impressions in a book called "The City of Lights". From 1300 and on, the Jewish community of Ancona grew most due to the city importance and it being a center of trade with the Levant. In that year, Jewish poet Immanuel the Roman
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2X² of all binary relations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition of binary relations R and S, with the converse of R as the converse relation. Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminated in the algebraic logic of Ernst Schröder; the equational form of relation algebra treated here was developed by Alfred Tarski and his students, starting in the 1940s. Tarski and Givant applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself be conducted without variables. A relation algebra is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, negation x−, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, the relational constant I, such that these operations and constants satisfy certain equations constituting an axiomatization of a calculus of relations.
A relation algebra is to a system of binary relations on a set containing the empty and identity relations and closed under these five operations as a group is to a system of permutations of a set containing the identity permutation and closed under composition and inverse. However, the first order theory of relation algebras is not complete for such systems of binary relations. Following Jónsson and Tsinakis it is convenient to define additional operations x◁y = x•y˘, dually, x▷y = x˘•y. Jónsson and Tsinakis showed that I◁x = x▷I, that both were equal to x˘. Hence a relation algebra can well be defined as an algebraic structure; the advantage of this signature over the usual one is that a relation algebra can be defined in full as a residuated Boolean algebra for which I◁x is an involution, that is, I◁ = x. The latter condition can be thought of as the relational counterpart of the equation 1/ = x for ordinary arithmetic reciprocal, some authors use reciprocal as a synonym for converse. Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras.
Hence the latter form the variety RA of relation algebras. Expanding the above definition as equations yields the following finite axiomatization; the axioms B1-B10 below are adapted from Givant, were first set out by Tarski in 1948. L is a Boolean algebra under binary disjunction, ∨, unary complementation −: B1: A ∨ B = B ∨ A B2: A ∨ = ∨ C B3: − ∨ − = AThis axiomatization of Boolean algebra is due to Huntington. Note that the meet of the implied Boolean algebra is not the • operator, nor is the 1 of the Boolean algebra the I constant. L is a monoid under binary composition and nullary identity I: B4: A• = •C B5: A•I = AUnary converse ˘ is an involution with respect to composition: B6: A˘˘ = A B7: ˘ = B˘•A˘Axiom B6 defines conversion as an involution, whereas B7 expresses the antidistributive property of conversion relative to composition. Converse and composition distribute over disjunction: B8: ˘ = A˘∨B˘ B9: •C = ∨B10 is Tarski's equational form of the fact, discovered by Augustus De Morgan, that A•B ≤ C− ↔ A˘•C ≤ B− ↔ C•B˘ ≤ A−. B10: ∨B− = B−These axioms are ZFC theorems.
After each of the following axioms is shown the number of the corresponding theorem in Chapter 3 of Suppes, an exposition of ZFC: B4 27, B5 45, B6 14, B7 26, B8 16, B9 23. The following table shows how many of the usual properties of binary relations can be expressed as succinct RA equalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B; the most complete set of results of this nature is Chapter C of Carnap, where the notation is rather distant from that of this entry. Chapter 3.2 of Suppes contains fewer results, presented as ZFC theorems and using a notation that more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry, or in an equational manner; the metamathematics of RA are discussed at length in Tarski and Givant, more in Givant. RA consists of equations manipulated using nothing more than uniform replacement and the substitution of equals for equals. Both rules are wholly familiar from abstract algebra generally.
Hence RA proofs are carried out in a manner familiar to all mathematicians, unlike the case in mathematical logic generally. RA can express any first-order logic formulas containing no more than three variables; this fragment of FOL suffices to express Peano arithmetic and all axiomatic set theories proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics, while dispensing with FOL and its connectives, quantifiers and modus ponens; because RA can express Peano arithmetic and set theory, Gödel's incompleteness theorems apply to it. (N. B; the Boolean algebra
In human anatomy, the deltoid tuberosity is a rough, triangular area on the anterolateral surface of the middle of the humerus to which the deltoid muscle attaches. It has been reported as prominent in less than 10% of cases; the deltoid tuberosity develops through endochondral ossification in a two-phase process. The initiating signal is tendon-dependent, whilst the growth phase is muscle-dependent. In mammals, the humerus displays a wide morphological variation; the size and orientation of its functionally important features, including the deltoid tubercle, greater tubercle, medial epicondyle, are pivotal to an animal's style of locomotion and habitat. In cursorial animals such as the pronghorn, the deltoid tubercle is located about a quarter of the way down the shaft, which allows for rapid but weak limb flexion and extension. In natatorial animals such as the North American river otter, the tubercle is located nearly halfway down the shaft, which allows for powerful limb flexion and extension.