Aix-Marseille University is a public research university located in the region of Provence, southern France. It was founded in 1409 when Louis II of Anjou, Count of Provence, petitioned the Pisan Antipope Alexander V to establish the University of Provence; the university as it is today was formed by the merger of the University of Provence, the University of the Mediterranean and Paul Cézanne University. The merger became effective on 1 January 2012, resulting in the creation of the largest university in the French-speaking world, with about 74,000 students. AMU has the largest budget of any academic institution in the Francophone world, standing at €750 million; the university is organized around five main campuses situated in Marseille. Apart from its major campuses, AMU owns and operates facilities in Arles, Avignon, Digne-les-Bains, Gap, La Ciotat and Salon-de-Provence; the university is headquartered at the Marseille. AMU has produced many notable alumni in the fields of law, business, science and arts.
To date, there have been four Nobel Prize laureates amongst its alumni and faculty, as well as a two-time recipient of the Pulitzer Prize, four César Award winners, multiple heads of state or government, parliamentary speakers, government ministers and members of the constituent academies of the Institut de France. AMU has hundreds of research and teaching partnerships, including close collaboration with the French National Centre for Scientific Research and the French Alternative Energies and Atomic Energy Commission. AMU is a member of numerous academic organisations including the European University Association and the Mediterranean Universities Union; the institution developed out of the original University of Provence, founded on 9 December 1409 as a Studium generale by Louis II of Anjou, Count of Provence, recognized by papal bull issued by the Pisan Antipope Alexander V. However, there is evidence that teaching in Aix existed in some form from the beginning of the 12th century, since there were a doctor of theology in 1100, a doctor of law in 1200 and a professor of law in 1320 on the books.
The decision to establish the university was, in part, a response to the already-thriving University of Paris. As a result, in order to be sure of the viability of the new institution, Louis II compelled his Provençal students to study in Aix only. Thus, the letters patent for the university were granted, the government of the university was created; the Archbishop of Aix-en-Provence, Thomas de Puppio, was appointed as the first chancellor of the university for the rest of his life. After his death in 1420, a new chancellor was elected by the rector and licentiates – an uncommon arrangement not repeated at any other French university; the rector had to be an “ordinary student”, who had unrestricted civil and criminal jurisdiction in all cases where one party was a doctor or scholar of the university. Those displeased with the rector's decisions could appeal to a doctor legens. Eleven consiliarii provided assistance to the rector; these individuals represented all were elected from among the students.
The constitution was of a student-university, the instructors did not have great authority except in granting degrees. Mention should be made that a resident doctor or student who married was required to pay charivari to the university, the amount varying with the degree or status of the man, being increased if the bride was a widow. Refusal to submit to this statutable extortion was punished by the assemblage of students at the summons of the rector with frying-pans and horns at the house of the newly married couple. Continued recusancy was followed by the piling up of dirt in front of their door upon every Feast-day; these injunctions were justified on the ground that the money extorted was devoted to divine service. In 1486 Provence passed to the French crown; the university's continued existence was approved by Louis XII of France, Aix-en-Provence continued to be a significant provincial centre. It was, for instance, the seat of the Parliament of Aix-en-Provence from 1501 to 1789, no doubt aided by the presence of the law school.
In 1603 Henry IV of France established the Collège Royal de Bourbon in Aix-en-Provence for the study of belles-lettres and philosophy, supplementing the traditional faculties of the university, but not formally a part of it. This college de plain exercice became a significant seat of learning, under the control of the Jesuit order. Throughout the 16th and 17th centuries, the college served as a preparatory, but unaffiliated, school for the university. Only the university was entitled to award degrees in the theology and medicine. Universities accepted candidates who had studied in colleges formally affiliated with them, which in reality required both college and university to be situated in the same city. In 1762 the Jesuits were forced to leave France, in 1763 the Collège Royal de Bourbon was affiliated with the university as a faculty of arts; the addition of the Collège Royal de Bourbon widened the scope of courses provided at the University of Provence. Formal instruction in French was provided at the college, with texts and a structured course of study.
Subsequently, physics became a part of the curriculum at the college as a part of the philosophy course in the 18th century. Equipment for carrying out experiments was obtained and the first course in experimental physics was provided at Aix-e
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every integer greater than 2 can be expressed as the sum of two primes; the conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. Since four is the only number greater than two that requires the prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all integers greater than 4 are Goldbach numbers; the expression of a given number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers: 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7 = 5 + 5 12 = 7 + 5... 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53... The number of ways in which 2n can be written as the sum of two primes is: 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3....
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every integer which can be written as the sum of two primes, can be written as the sum of as many primes as one wishes, until all terms are units. He proposed a second conjecture in the margin of his letter: Every integer greater than 2 can be written as the sum of three primes, he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is: Every integer greater than 5 can be written as the sum of three primes. Euler replied in a letter dated 30 June 1742, reminded Goldbach of an earlier conversation they had, in which Goldbach remarked his original conjecture followed from the following statement Every integer greater than 2 can be written as the sum of two primes,which is, thus a conjecture of Goldbach.
In the letter dated 30 June 1742, Euler stated: "Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." Goldbach's third version is the form in which the conjecture is expressed today. It is known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture; this weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes, appears to have been proved in 2013. The weak conjecture is a corollary of the strong conjecture, as, if n – 3 is a sum of two primes n is a sum of three primes; the converse implication, the strong Goldbach conjecture remain unproven. For small values of n, the strong Goldbach conjecture can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.
With the advent of computers, many more values of n have been checked. One record from this search is that 3,325,581,707,333,960,528 is the smallest number that has no Goldbach partition with a prime below 9781. Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, the more "likely" it becomes that at least one of these representations consists of primes. A crude version of the heuristic probabilistic argument is as follows; the prime number theorem asserts that an integer m selected at random has a 1 / ln m chance of being prime. Thus if n is a large integer and m is a number between 3 and n/2 one might expect the probability of m and n − m being prime to be 1 /. If one pursues this heuristic, one might expect the total number of ways to write a large integer n as the sum of two odd primes to be ∑ m = 3 n / 2 1 ln m 1 ln ≈ n 2 2.
Since this quantity goes to infinity as n increases, we expect that every large integer
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work to solve mathematical problems. Mathematics is concerned with numbers, quantity, space and change. One of the earliest known mathematicians was Thales of Miletus, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number", it was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria, she succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells.
Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages was ongoing throughout the reign of certain caliphs, it turned out that certain scholars became experts in the works they translated and in turn received further support for continuing to develop certain sciences; as these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham; the Renaissance brought an increased emphasis on science to Europe.
During this period of transition from a feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli. As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.” In 1810, Humboldt convinced the King of Prussia to build a university in Berlin based on Friedrich Schleiermacher’s liberal ideas. Thus and laboratories started to evolve. British universities of this period adopted some approaches familiar to the Italian and German universities, but as they enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences that inspired Humboldt.
The Universities of Oxford and Cambridge emphasized the importance of research, arguably more authentically implementing Humboldt’s idea of a university than German universities, which were subject to state authority. Overall, science became the focus of universities in the 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge; the German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of “freedom of scientific research and study.” Mathematicians cover a breadth of topics within mathematics in their undergraduate education, proceed to specialize in topics of their own choice at the graduate level.
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics. Mathematicians involved with solving problems with applications in real life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, localized constructs, applied mathematicians work in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM careers; the discipline of applied mathematics concerns
A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1, not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes, unique up to their order; the property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and n. Faster algorithms include the Miller–Rabin primality test, fast but has a small chance of error, the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical.
Fast methods are available for numbers of special forms, such as Mersenne numbers. As of December 2018 the largest known prime number has 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled; the first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm. Several historical questions regarding prime numbers are still unsolved; these include Goldbach's conjecture, that every integer greater than 2 can be expressed as the sum of two primes, the twin prime conjecture, that there are infinitely many pairs of primes having just one number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers.
Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. A natural number is called a prime number if it is greater than 1 and cannot be written as a product of two natural numbers that are both smaller than it; the numbers greater than 1 that are not prime are called composite numbers. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid, more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, 5 are the prime numbers, as there are no other numbers that divide them evenly. 1 is not prime, as it is excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. The divisors of a natural number n are the numbers.
Every natural number has both itself as a divisor. If it has any other divisor, it cannot be prime; this idea leads to a different but equivalent definition of the primes: they are the numbers with two positive divisors, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, …, n − 1 divides n evenly; the first 25 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. No number n greater than 2 is prime because any such number can be expressed as the product 2 × n / 2. Therefore, every prime number other than 2 is an odd number, is called an odd prime; when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: decimal numbers that end in 0, 2, 4, 6, or 8 are and decimal numbers that end in 0 or 5 are divisible by 5; the set of all primes is sometimes denoted by P or by P.
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements proves the infinitude of primes and the fundamental theorem of arithmetic, shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
Virtual International Authority File
The Virtual International Authority File is an international authority file. It is a joint project of several national libraries and operated by the Online Computer Library Center. Discussion about having a common international authority started in the late 1990s. After a series of failed attempts to come up with a unique common authority file, the new idea was to link existing national authorities; this would present all the benefits of a common file without requiring a large investment of time and expense in the process. The project was initiated by the US Library of Congress, the German National Library and the OCLC on August 6, 2003; the Bibliothèque nationale de France joined the project on October 5, 2007. The project transitioned to being a service of the OCLC on April 4, 2012; the aim is to link the national authority files to a single virtual authority file. In this file, identical records from the different data sets are linked together. A VIAF record receives a standard data number, contains the primary "see" and "see also" records from the original records, refers to the original authority records.
The data are available for research and data exchange and sharing. Reciprocal updating uses the Open Archives Initiative Protocol for Metadata Harvesting protocol; the file numbers are being added to Wikipedia biographical articles and are incorporated into Wikidata. VIAF's clustering algorithm is run every month; as more data are added from participating libraries, clusters of authority records may coalesce or split, leading to some fluctuation in the VIAF identifier of certain authority records. Authority control Faceted Application of Subject Terminology Integrated Authority File International Standard Authority Data Number International Standard Name Identifier Wikipedia's authority control template for articles Official website VIAF at OCLC
France the French Republic, is a country whose territory consists of metropolitan France in Western Europe and several overseas regions and territories. The metropolitan area of France extends from the Mediterranean Sea to the English Channel and the North Sea, from the Rhine to the Atlantic Ocean, it is bordered by Belgium and Germany to the northeast and Italy to the east, Andorra and Spain to the south. The overseas territories include French Guiana in South America and several islands in the Atlantic and Indian oceans; the country's 18 integral regions span a combined area of 643,801 square kilometres and a total population of 67.3 million. France, a sovereign state, is a unitary semi-presidential republic with its capital in Paris, the country's largest city and main cultural and commercial centre. Other major urban areas include Lyon, Toulouse, Bordeaux and Nice. During the Iron Age, what is now metropolitan France was inhabited by a Celtic people. Rome annexed the area in 51 BC, holding it until the arrival of Germanic Franks in 476, who formed the Kingdom of Francia.
The Treaty of Verdun of 843 partitioned Francia into Middle Francia and West Francia. West Francia which became the Kingdom of France in 987 emerged as a major European power in the Late Middle Ages following its victory in the Hundred Years' War. During the Renaissance, French culture flourished and a global colonial empire was established, which by the 20th century would become the second largest in the world; the 16th century was dominated by religious civil wars between Protestants. France became Europe's dominant cultural and military power in the 17th century under Louis XIV. In the late 18th century, the French Revolution overthrew the absolute monarchy, established one of modern history's earliest republics, saw the drafting of the Declaration of the Rights of Man and of the Citizen, which expresses the nation's ideals to this day. In the 19th century, Napoleon established the First French Empire, his subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a tumultuous succession of governments culminating with the establishment of the French Third Republic in 1870.
France was a major participant in World War I, from which it emerged victorious, was one of the Allies in World War II, but came under occupation by the Axis powers in 1940. Following liberation in 1944, a Fourth Republic was established and dissolved in the course of the Algerian War; the Fifth Republic, led by Charles de Gaulle, remains today. Algeria and nearly all the other colonies became independent in the 1960s and retained close economic and military connections with France. France has long been a global centre of art and philosophy, it hosts the world's fourth-largest number of UNESCO World Heritage Sites and is the leading tourist destination, receiving around 83 million foreign visitors annually. France is a developed country with the world's sixth-largest economy by nominal GDP, tenth-largest by purchasing power parity. In terms of aggregate household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, human development.
France is considered a great power in global affairs, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a leading member state of the European Union and the Eurozone, a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, La Francophonie. Applied to the whole Frankish Empire, the name "France" comes from the Latin "Francia", or "country of the Franks". Modern France is still named today "Francia" in Italian and Spanish, "Frankreich" in German and "Frankrijk" in Dutch, all of which have more or less the same historical meaning. There are various theories as to the origin of the name Frank. Following the precedents of Edward Gibbon and Jacob Grimm, the name of the Franks has been linked with the word frank in English, it has been suggested that the meaning of "free" was adopted because, after the conquest of Gaul, only Franks were free of taxation.
Another theory is that it is derived from the Proto-Germanic word frankon, which translates as javelin or lance as the throwing axe of the Franks was known as a francisca. However, it has been determined that these weapons were named because of their use by the Franks, not the other way around; the oldest traces of human life in what is now France date from 1.8 million years ago. Over the ensuing millennia, Humans were confronted by a harsh and variable climate, marked by several glacial eras. Early hominids led a nomadic hunter-gatherer life. France has a large number of decorated caves from the upper Palaeolithic era, including one of the most famous and best preserved, Lascaux. At the end of the last glacial period, the climate became milder. After strong demographic and agricultural development between the 4th and 3rd millennia, metallurgy appeared at the end of the 3rd millennium working gold and bronze, iron. France has numerous megalithic sites from the Neolithic period, including the exceptiona