The Meeting-Place of the First Ecumenical Council and the Church of the Holy Fathers at Nicaea

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The Meeting-Place of the First Ecumenical Council and the Church of the Holy Fathers at Nicaea"

Transcript

1 The Meeting-Place of the First Ecumenical Council and the Church of the Holy Fathers at Nicaea Cyril MANGO Τόμος ΚΣΤ' (2005) Σελ ΑΘΗΝΑ 2005

2 Cyril Mango THE MEETING-PLACE OF THE FIRST ECUMENICAL COUNCIL AND THE CHURCH OF THE HOLY FATHERS AT NICAEA* X here is no denying the symbolic significance of the First Ecumenical Council. To some of us it may signal the birth of the Byzantine theocracy, but to the Byzantines themselves it marked the condemnation of the mother of all heresies by the 318 'God-bearing' Fathers, who thereby became the guarantors of the established moral order; which is why, I suppose, their curse was so often invoked against wrongdoers, especially those who dared pinch a book from its rightful owner. The figure of 318, recalling the number of Abraham's servants (Gen ) is generally recognised as apocryphal. We do not know the exact number of bishops who participated in the First General Council: estimates vary from 200 to There are other uncertainties that do not concern us here. Why, e.g., was the venue of the Council switched from Ancyra to Nicaea? Was Constantine unaware that the sitting bishop of Nicaea, Theognis, who would normally have acted as host, was an avowed Arian? One thing is clear: Constantine was determined to impose his will on the divided bishops. The sessions were held not in a church, but in the biggest hall of the imperial palace 2. Constantine turned up in full regalia and himself presided over the deliberations. Of course, he got his way: only two of the assembled bishops refused to sign on the dotted line, Theognis being, embarrassingly, one of them. The existence of an imperial palace at Nicaea need not sur- prise us. Roman emperors maintained palaces in many provincial centres where they had occasion to stop on their peregrinations 3. We do not know when the palace of Nicaea was built, but it was still standing, though "collapsed in part", in the reign of Justinian, who ordered its complete restoration 4. Its main hall must have been big enough to afford room for the seated bishops plus the emperor's bodyguard as well as various notables and consultants - in all a gathering of at least 400. As time went by, the Council of Nicaea acquired mythical status as a founding assembly of the Church that had formulated the creed under the guidance of an emperor who was himself a saint. By c. 700 AD at the latest a list of 318 Fathers had been concocted, including some illustrious personages who, it was felt, ought to have been present, even if, in fact, they were not: Sylvester of Rome, Alexander of Constantinople (as bishop), Hypatius of Gangra, Nicholas of Myra, Gregory the Illuminator of Armenia, but not yet Achilleios of Larissa 5. Under these circumstances it would have been natural for the meeting-place of the Council to have been turned into a shrine, and that, indeed, appears to have happened. Its first known visitor was the English pilgrim Willibald (c ) who, after mentioning the synod of 318 bishops convened by Constantine, goes on, "That church is similar to the one on the Mount of Olives, where the Lord ascended to heaven, and in that church [at Nicaea] * The gist of the following argument has been stated by me, very briefly, in 'Notes d'épigraphie et d'archéologie...' TM 12 (1994), trust that George Galavaris, who was keenly interested in the history of the Orthodox Church, would have wished to follow it in greater detail. P.S. Another version of this paper has appeared, without my prior knowledge, in a volume entitled knik throughout History, Istanbul, is Bankasi, 2003, For a good factual account of the Council see, e.g., T.D. Barnes, Constantine and Eusebius, Cambridge MA 1981,214 ff. Eusebius, Vita Constantini, : εν αύτω δη τω μεσαιτάτω οϊκω των βασιλείων, δς δή και ύπερφέρειν έδόκει μεγέθει τους πάντας. 3 See F. Millar, The Emperor in the Roman World, London 1977,51. 4 Procopius, De aedificiis, In commenting on the exceptional character of the Roman fortifications of Nicaea (AD 268), J. Crow has recently remarked that they suggest "that the city was intended as some form of imperial centre", 'Fortifications and Urbanism in Late Antiquity' in L. Lavan (ed.), Recent Research in Late-Antique Urbanism, Portsmouth RI 2001, 91. This may provide a context for the construction of an imperial palace. 5 See E. Honigmann, La liste originale des pères de Nicée, Byz 14 (1939), 58 f. 27

3 CYRIL MANGO were pictures of the bishops who attended the synod" 6. Willibald had himself visited the church of the Ascension at Jerusalem and noted the fact that it was unroofed (ilia aecclesia est desuper patula et sine tectu) and so let in rainwater. In the middle of it stood some kind of bronze altar containing a candle under a glass cover so it could burn in any weather 7. Willibald had, therefore, a good mental picture of the church of the Ascension, which had been more fully described by Arculf ( ) as a rotunda, unroofed in the middle, surrounded by three vaulted galleries 8. Arculf s sketch (Fig. 1) confirms this arrangement. Whether its central space was completely or only partially open to the sky, the church of the Ascension would have borne a generic resemblance to a type of building that had a long tradition in Roman architecture, namely the domed rotunda or octagon with a more or less large oculus at the centre. That formula had featured in temples, palaces (e.g. Nero's Golden House) and mausolea, and was later applied to a few martyria of the Holy Land. In the case of the church of the Ascension a large opening would have been particularly appropriate so as to show visually Christ's elevation to heaven from the spot marked by his footprints. That was not, however, a formula used in Byzantine church architecture 9. It is not unreasonable to suppose that the shrine visited by Willibald had been converted from the palace hall in which the synod was believed to have taken place. In 727, i.e. shortly before or after Willibald's visit, Nicaea was besieged by a strong Arab army. This event - the only recorded siege of Nicaea in the course of the centuries-long Arab-Byzantine conflict 10 - is known to us from two accounts in Syriac and two in Greek. Arabic sources are silent, probably because the attack ended in failure. The Syriac Chronicle of 1234, giving an incorrect date of AD 729, merely reports that Mu'awiya, son of the Caliph Hisham, besieged Nicaea for forty days and withdrew when a Roman army had come into the city on ships, i.e. across lake Asca- nius 11. In the chronicle of Michael the Syrian the story is garbled and the date shifted further to 731: the Arabs under Mu'awiya attack the city and destroy its walls. The Romans flee on boats. The Arabs gain possession of Nicaea and devastate it 12. We shall see that some destruction of the walls did take place. 6 Vita Willibaldi, ed. Ο. Holder-Egger, MGH, Script. XV, 1887, Ibid., De locis Sanctis, 1.23, ed. L. Bieler, CC, ser. Iat. 175, 1965, Versions of Arculf s sketch-plan as reproduced by Adamnan may be found in J. Wilkinson, Jerusalem Pilgrims before the Crusades, Warminster 1979, pi. 2 (whence our Fig. 1); cf They are all fairly similar, showing three concentric galleries, a triple entrance on the south, an altar to the east and eight lamps to the west. The imprint of Christ's feet at the centre appears to have been protected by a brass railing (rota aerea). For the excavations of 1959 see V.C. Corbo, Ricerche archeologiche al Monte degli Ulivi, Jerusalem 1965, With the possible exception of the central octagon of St. Symeon Stylites (Qal'at Siman), the problem of whose roofing is still under debate. Its function, however, was not at all comparable to that of the shrine that concerns us. 10 Agapius of Menbidj, ed. A. Vasiliev, PO VIII, 1912, 501, speaks of Maslama taking many captives at Nicaea in 716, but that is not confirmed by other sources. If true, it may refer to the surrounding area. 11 Chronicon anonymum, trans. J.-B. Chabot, CSCO, Scr. syri, 3rd ser. 14,1937, Trans. J.-B. Chabot, Chronique de Michelle Syrien, II, Paris 1901,

4 THE MEETING-PLACE OF THE FIRST ECUMENICAL COUNCIL We now come to the Greek sources. The Patriarch Nicephorus, as usual, is extremely brief: an Arab force, commanded by Mu'awiya and a certain 'Amr, besieges Nicaea for some time, then departs without having accomplished anything 13. A fuller account is given in the Chronicle of Theophanes and may be rendered as follows: "At the summer solstice of the same 10th indiction [June 727]... a multitude of Saracens was drawn up against Nicaea in Bithynia: Amr with 15,000 scouts led the van and surrounded the town, which he found unprepared, while Mu'awiya followed with another 85,000 men 14. After a long siege and a partial desrtruction of the walls, they did not overpower the town thanks to the acceptable prayers addressed to God by the Holy Fathers who are honoured there in a church (wherein their venerable images are set up to this very day and honoured by those who believe as they did) 15. A certain Constantine, however, who was the groom of Artabasdos 16, on seeing an image of the Mother of God that had been set up, picked up a stone and threw it at her. He broke the image and trampled upon it when it had fallen down. He then saw in a vision the Lady standing beside him and saying to him, 'See, what a brave thing you have done to me! Verily, upon your head have you done it'. The next day, when the Saracens attacked the walls and battle was joined, that wretched man rushed to the wall like the brave soldier he was, and was struck by a stone discharged from a siege engine, and it broke his head and face, a just reward for his impiety. After collecting many captives and much booty the Arabs withdrew. In this manner God showed to the impious one [Leo III] that he had overcome his fellow-countrymen 17 not on account of his piety, as he himself boasted, but for some divine cause and inscrutable judgment, whereby so great an Arab force was driven away from the city of the Holy Fathers thanks to their intercession - on account of their most exact likenesses that are honoured therein - and this, too, in reproof and unanswerable condemnation of the tyrant [again Leo III] and in vindication of the true believers From that time on he impudently harassed the blessed Germanus, patriarch of Constantinople, blaming all the emperors, bishops and Christian people who had lived before him for havibg committed idolatry in worshipping the holy and venerable icons, unable as he was to grasp the argument concerning their 'relative' veneration (σχετικής προσκυνήσεως) because of his lack of faith and crass ignoranee.. My translation does not adequately convey the incoherence of the Greek text. Instead of rewriting the whole passage, Theophanes seems to have interpolated his source without regard for proper syntax. Exactly what the source said cannot now be determined, but it probably made no mention of images and may have stated that the deliverance of Nicaea, coming after the defeat of the Helladics, was seen by Leo as a vindication of his religious policy; which is why "from that time on" he increased pressure on the Patriarch Germanus. Unable to explain the success of the 'impious' emperor, Theophanes could only invoke "some divine cause and inscrutable judgment" and, of course, the presence of the Fathers' images. The failed siege of 727 is commemorated by an inscription that is still extant a short distance west of the Istanbul gate on the city side of tower 71 (Fig. 2) 20. Carved in raised letters and pompously, if somewhat ungrammatically worded, it may be translated as follows: "At the place where, with divine help, the insolence of the enemy was put to shame, there our Christ-loving emperors Leo and Constantine restored with zeal the city of Nicaea, having erected in demonstration of their deed a trophy of victory by setting up a kentenarìon tower 21, which Artabasdos, 13 Short History, ed. C. Mango, CFHB 13,1990, ch These figures are certainly greatly inflated. 15 The Greek is almost incomprehensible as it stands: οι [the Arabs] μετά πολιορκίαν πολλήν και καθαίρεσιν των τειχών μερικήν τω των τιμωμένων άγιων πατέρων αυτόθι τεμένει ταύτης μεν ου περιγεγόνασι δια των ευπρόσδεκτων ευχών προς τον θεόν, ένθα και σεβάσμιοι αυτών χαρακτήρες άνεστήλωντο μέχρι νΰν υπό τών όμοφρόνων αυτών τιμώμενοι. Anastasius changes the order: «Et particularem eversionem murorum hanc non optinuerunt, bene acceptis sanctorum precibus patrum ad deum directis et sanctificatione faciente venerabilis templi,» etc. On this basis de Boor proposed to move <έν> τω τών τιμ. άγ. πατ. αυτόθι τεμένει after θεόν. It would be simpler to insert παρά after μερικήν, but that would introduce a localization that may appear arbitrary even if it happens to be true. 16 Son-in-law of Leo III and future emperor. 17 Referring to the defeat of the Helladic revolt described in the previous paragraph. 18 Another confused sentence. 19 Ed. C. de Boor, Leipzig 1883, Cf. trans, by C. Mango and R. Scott, The Chronicle of Theophanes Confessor, Oxford 1997, Photograph reproduced by me, 'Notes d'épigraphie' (as in n.*), fig. 6. In my transcription (ibid., 352) κεντηναρισ' should be corrected to κεντιναρισ". 21 The meaning of this term is unclear to me. 29

5 CYRIL MANGO Fig. 2. Nicaea, city side of Tower 71 with Artabasdos in 1996). inscription (as the glorious patrician and curopalates 22, completed by his toil." The inscription marks, therefore, the very spot where the Arabs were "put to shame". Not only tower 71, but the whole stretch of wall between towers 70 and was reconstructed at this time out of re-used marble blocks and column shafts taken from ancient buildings. That was probably the stretch that was partially destroyed, as Theophanes puts it. We are now in a position to analyse an important text that has received insufficient attention. It is a Laudation of the 318 Fathers by a certain Gregory, presbyter of the church of Caesarea in Cappadocia, expressly commissioned by an unnamed bishop of Nicaea 24. It opens with the accession of Constantine and the establishment of Christianity, whereupon peace descends on the Roman Empire. Displeased by this turn of events, the Devil causes Arius to preach his foul heresy. On being informed, Constantine summons a council of bishops to meet at Nicaea. The names and sees of the participants may be found, we are told, in the 'synodal tome' that was set forth by them. They included a number of famous confessors who had suffered in the preceding persecution, like Jacob of Nisibis, Paul of Neocaesarea and Paphnutius, but the Devil was also able to introduce a few wolves in sheep's clothing, among whom our author fails to mention, perhaps from a feeling of tact, Theognis of Nicaea. The venue of the council is described in the following terms: "Having detached from the palace that is there a huge hall - the fairest part (τον όφθαλμόν), as it were, of the imperial apartments, whose decorous beauty has been preserved to our days by the protection of the holy Fathers, the mighty Emperor assigned this hall, like an offering of first-fruits, to the holy synod" 25. Constantine attends the meetings. Arius is condemned and a definition of the creed is formulated. The date of Easter is established and other canons laid down. Constantine demands that the tomos be signed by all the participants, but two of them, Chrysanthus and Musonius, happen to die before having signed. The sealed tomos is brought to their tombs and, when it is opened, their signatures are found inside. At the conclusion of the council Constantine offers a banquet and embraces the confessors. As the Fathers are about to depart and offer prayers for the safety of the city that had welcomed them, "it so happened that a fount of oil gushed out of the so-called mesomphalon [circular plaque in the floor] of the eastern entrance, at the very centre of the apsis [arch or vault] where the choir of the saints had assembled; which fount, still visible today, demonstrates the efficacy of the prayers offered at the time" 26. Another miracle, due to the Fathers' providence, happened 'in our generation'. When the 'Assyrians' were ravaging the Roman Empire, Nicaea was preserved unharmed, "having suffered no loss of men, women or children either by fire or sword" in spite of many enemy attacks conducted both openly and by stealth. Their commander made an attempt against the church of the Fathers and hastened "to celebrate there the detestable rites of his magic", but was stopped by a nocturnal vision and apparitions by daylight. Indeed, he sought to propitiate God by lighting lamps and barred access to the holy church to his 'Babylonians'. Even prisoners taken by 22 The inscription omits the title comes of Opsikion, which appears on some of Artabasdos' lead seals following that of curopalates. Nicaea, of course, was in the Opsikian theme, but I am not sure one can argue from the inscription (which, strictly speaking, is not dated) that Artabasdos was comes at the time it was set up, as does N. Oikonomides, A Collection of Dated Byzantine Lead Seals, Washington, DC 1986, See C. Foss and D. Winfield, Byzantine Fortifications, Pretoria 1986, 90,100. A shorter stretch near the South Lake Gate, including tower 94 and the adjacent wall were rebuilt at the same time. 24 The first ed. by F. Combefis (1648) is reprinted inpg 111, col ; second, but not very satisfactory ed. by J. Compernass, Gregorios Lobrede auf die 318 Väter des Konzils zu Nikaia, diss. Bonn 1908, Ed. Compernass, Ibid.,

6 THE MEETING-PLACE OF THE FIRST ECUMENICAL COUNCIL them escaped injury by claiming they were natives of Nicaea. The bodies of the Fathers have remained uncorrupted. The author can testify to this in the case of Leontius, bishop of Caesarea, his home town. The same applies to Gregory of Armenia, the one who discovered the relics of Ripsime and Gaiane and converted King Tiridates. "Many persons have seen his precious body along with me. It has lost neither its hair nor its fingernails and is suffused with the sweet smell of myrrh". Such in outline is the content of the Laudation which, as its preface makes clear, was spoken at Nicaea itself. At the time of its delivery the palace hall in which the synod had met was still standing in all its beauty. It had allegedly been set apart by Constantine himself and consecrated as a kind of memorial. It had an eastern entrance and, in front of it, a circular plaque in the pavement where a miraculous source of oil gushed out 27. Significantly, our author does not allude to any images. Passing on to the relatively recent enemy attack on Nicaea, we welcome the suggestion by X. Lequeux 28 that the mention of Assyrians and Babylonians echoes the siege of Jerusalem by Sennacherib (IV Reg ). This allusion may have had a further resonance in that it was King Hezekiah, thanks to whose piety Jerusalem was saved, who rid his kingdom of idolatry (συνέτριψεν πάσας τάς στήλας και έξωλόθρευσεν τα άλση και τον οφιν τον χαλκοϋν). The parallel with Leo III may not have been lost on the audience.on the other hand, our author does make it clear that the invaders did actually enter the church of the Fathers and performed their religious rites in it 29, which means, unless the church was extramural, that they broke into Nicaea. Different opinions have been expressed concerning the date of the Laudation. Combefis, its first editor, thought that the Assyrians or Babylonians referred to the Persians, and consequently dated the text to the reign of Heraclius 30. That was not an unreasonable suggestion: the Persians twice advanced as far as Chalcedon, the first time probably in 615, the second time in 626. The reference to 'magical rites' would also fit the Persians. On the other hand, there is no evidence that they ever attacked Nicaea. The next editor, Compernass, dated the Laudation to the tenth century for linguistic and stylistic reasons, i.e. for no objective reasons at all. More recently the late A. Kazhdan argued that the attack on Nicaea was made by Paulician heretics shortly before 869 and that the text was written after that date 31. That, however, can hardly be right, seeing that the Paulicians merely carried out a number of marauding raids, extending as far as Nicomedia, Nicaea and Ephesus 32, whereas the Laudation speaks of a full-scale invasion of the Roman Empire. A date of , which I suggested on a previous occasion, has been endorsed by X. Lequeux 33. Perhaps we can narrow it a little further to for reasons that will soon become apparent. Other possible clues prove elusive. A Gregory presbyter was also the author of a well-known Life of St. Gregory Nazianzen, but it is very unlikely that they were one and the same person. The reference to the uncorrupted body of St. Gregory the Illuminator of Armenia leads nowhere in view of the extremely confused traditions conerning that Saint's relics 34. Nor do we get very far by identifying the textual sources consulted by our Gregory. These were the Ecclesiastical History by Theodoret, perhaps the Life of the Emperor Constantine The existence of this source is confirmed by the legend of Achilleios, bishop of Larissa, which appears to have been concocted in the ninth century. Achilleios, too, is represented as attending the Council and confounds the heretics by causing a stone or rock to exude oil. See ed. of his Life by D.Z. Sophianos, Μεσαιωνικά και νέα ελληνικά 3 (1990), owe this reference to Dr. Olga Karagiorgou. 28 Gregorìi presbyteri Vita S. Gregorii Theologi, CC, ser. gr. 44 = Corpus Naz. 11,2001, Compernass, 30, prints κατατολμήσαι του ναοϋ των άγιων, but the original reading may have been τοϋ άδυτου τοϋ ναοϋ in view of the variants τοϋ άδυτου ναοϋ and αδυνάτου. 30 PG 111, col. 419, n. 1. Attention should be drawn to a curious passage (Compernass, 29-30), which speaks of the ingratitude of the 'Assyrians' for the help they had received from the Roman Empire: των γαρ Ασσυρίων τρόπω χρησαμένων άγνώμονι και δι' ης έσώθησαν των Τωμαίων βασιλείας ταύτην [rather than ταύτη] λυμηναμένων, etc. If the Persians were meant, the reference could be to the help given by the emperor Maurice to place Chosroes II on the throne, but it is more likely that the author is thinking of the removal of the Mardaites from the Lebanon mountains by Justinian II (Theophanes, 363). 31 Constantin imaginaire, Byz 57 (1987), Genesius, ch. 35, ed. A. Lesmueller-Werner and I. Thurn, 1987, 86. For the circumstances see P. Lemerle, L'histoire des Pauliciens dasie Mineure, TM 5 (1973), 97, As inn. 28 above. 34 See M. van Esbroeck, Témoignages littéraires sur la sépulture de S. Grégoire l'illuminateur, AnBoll 89 (1971), Leontius of Caesarea was said to have ordained Gregory the Illuminator bishop. The alleged participation of both of them in the Council of Nicaea is affirmed in a letter of one George, bishop of the Arabs, of the year 714. See ed. and trans, by G. Garitte, Documents pour l'étude du livre d'agathange, ST127 (1946), 411,

7 CYRIL MANGO Fig. 3. City plan ofnicaea (after Foss and Winfield, Byzantine Fortifications). edited by Guidi (or something like it) 35 and the catalogue of the Nicaean Fathers, which he calls the 'synodal tome'. The last was not a genuine document of 325, but the interpolated list of 318 names we have already mentioned. It is interesting to note that the list in question contains the names of the two bishops who allegedly died during the synod, Chrysanthus and Musonius, with the comment: "He was buried in the church of Nicaea", and "He, too, was buried in the church" 36. It is possible, therefore, that two such tombs actu- ally existed and were inscribed with the names Chrysanthus and Musonius, thus giving rise to the legend about them 37. In the absence of other evidence it is reasonable to conclude that the foreign attack described in the Laudation was that of 727. Theophanes speaks of a partial destruction of the city walls; Gregory has the enemy entering the church of the Fathers. In both texts a miraculous intervention takes place. Hence the Laudation was composed not long threafter, under iconoclast rule, which is why it says nothing about images. If that is granted and it is true that the Arabs managed to reach the church of the Fathers and offer Muslim prayer in it (which our author calls 'detestable rites'), it would follow that the church in question was close to the breach, i.e. probably a short distance west of the Istanbul gate (Fig. 3) 38. That provides a location for the imperial palace and the venue of the council of 325. The palace's eastern gate would have opened onto the main north-south street. One can only hope that archaeological findings may one day confirm this conclusion. In 740 Nicaea was shaken by a terrible earthquake which "spared only one church" 39. That may be an exaggeration or it may mean that all the city's churches save one sustained some damage. The Laudation does not mention the earthquake, hence may be earlier than 740. Actually, the church of the Fathers did survive, as we shall see, but it may not have been in very good shape, seeing that the Council of 787 met in St. Sophia, although it would have been highly symbolic if it had assembled in the same place as the First Council 40. Indeed, the resemblance of Iconoclasm to Arianism was often stressed on rather shaky doctrinal grounds, and somewhat later the Patriarch Photius was to preach a whole series of sermons on that topic 41. In this context the images of the Fathers came in handy. On the assumption, which no one could dispute at the time, that they had been set up by See P. Heseler, Zum Logos auf das Konzil von Nikaia..., BNJb 5 (1927), It is not clear to me whether the Laudation borrows from the Guidi Life, which also speaks of Chrysanthus and Musonius, or vice versa. The latter text is certainly later, perhaps considerably later than the reign of Heraclius. 36 Honigmann, La liste (as in n. 5), 59. Not knowing our text, he comments: 'Il semble s'agir d'une tradition locale de Nicée, qui n'est pas connue par ailleurs'. The fact that the sees of the two bishops are not specified gives some support to an epigraphic source. 37 The name Chrysanthus does occur in the onomastic of Nicaea. See S. Sahin, Katalog der antiken Inschriften des Museums von Jznik (Nikaia), I (= Inschriften griechischer Städte aus Kleinasien 9), Bonn 1979, no This localization is endorsed with regard to the church of the Fathers by C. Foss,Mcaea. A Byzantine Capital and its Praises, Brookline 1996, Theophanes, Strangely enough, the Life of St. Stephen the Younger, written in 809 by a patriarchal deacon, states that the First Council took place in the church of St. Sophia at Nicaea: ed. M.-F. Auzépy, La Vie d'etienne le Jeune par Etienne le diacre, Aldershot 1997,145, assume that the author, who was anxious to prove that all ecumenical councils had met in churches, confused the First with the Seventh. St. Sophia was the episcopal church (Theophanes, 463) and is usually identified with the standing ruin of a basilica at the centre of the town, although there is no explicit proof of that. See Foss, Nicaea, Iconoclasm likened to Arianism: see, e.g., Nicephorus,^4«ft'/r/ieft'ci/s I, PG 100, col ; id., Apologeticus, ibid., col ; id., Refutatio eteversio definitionis synodalis a. 815, ed. J.M. Featherstone, CC, ser. gr. 33, 1997, ch. 5.5, 8.39, 19.78, etc. Photius: see Homilies 15 and 16 and my comment on them, The Homilies of Photius, Cambridge MA 1958,239 f. 32

8 THE MEETING-PLACE OF THE FIRST ECUMENICAL COUNCIL Constantine himself, one could assert that the first Christian emperor had given his sanction to the practice of sacred painting. The images of the Fathers that were seen by Willibald in the 720's were certainly a group representation, hence a historical rather than a devotional picture. There is a possible reference to it in the Admonition of an Old Man Concerning the Holy Icons of c. 750, but the text is incomplete and unclear 42. Theophanes, as we have seen, refers to it as still extant in c A few years later ( ) the Patriarch Nicephorus, who had certainly visited Nicaea, speaks twice of the same composition. The church built in honour of the Fathers, he says, preserves to this day, in addition to other holy representations, the images of the Fathers and of Constantine in brilliant mosaic. The iconoclasts, he claims, tried to remove them, but did not succeed in doing so 43. It is difficult to imagine that the iconoclasts, who saw themselves as standing in the tradition of Nicaea I 44, would have wanted to do such a thing. Representations of church councils are recorded in the Byzantine world from c. 700 onwards, but had probably existed earlier. The best known example was in the vault of the Milion at Constantinople. Here were represented the first five Councils, i.e. down to 553. The Sixth (680/81), also represented in the guards' quarter (Scholae) of the Imperial Palace, was added to the Milion series some time after 713. Pictures of all six councils were displayed in St. Peter's, Rome, in Such images forming a series were propagandistic rather than commemorative: they served to underline the adherence on the part of the emperor or the Pope to Christian dogma as defined by the councils and the rejection of such heresies as had been condemned by them. The mosaic at Nicaea appears on the other hand to have been commemora- tive: it represented an event that had taken place at that very spot. We do not know when it was set up nor what it looked like, i.e. whether it followed what was to become traditional Byzantine iconography, of which the earliest preserved representative, depicting the Second Council, is in the Paris, gr. 510 of c A slightly earlier example (second quarter of the ninth century) is, however, preserved in a Carolingian manuscript and represents precisely Nicaea I 46. Its iconography is a little different: Constantine, surrounded by his bodyguard, is on the left, a group of bishops on the right, while in the foreground a heap of books is being burnt. The copyist of the manuscript understood this act of incineration to depict the destruction of heretical writings (witness the label HereticiArriani damnati), but it may show instead a famous incident of the Council: presented with denunciations of certain bishops, Constantine ordered them to be burnt unopened. We shall not be concerned here with the subsequent history of the church of the Fathers, which was shaken down, along with St. Sophia, by the earthquake of It was probably repaired and re-appears later as a monastery 48. Two intriguing references are, however, worth mentioning. An ecclesiastical council met at Nicaea in 1232 "in the domed oaton" of what was then the Greek Patriarchate in exile 49. The name oaton (ovatum, egg-shaped) was also applied to a hall in the Imperial Palace at Constantinople. It was there that both the Sixth Ecumenical Council (680/81) and the Quinisext (692) assembled. Could the oaton of Nicaea have been the patched up rotunda of Constantine's palace? Two years later (1234) a Latin delegation came to Nicaea and were shown the church in which the First Council had purportedly met. In it they saw a painting representing that assembly 50. Was it the mosaic we have been discussing or a later substitute for it? Ed. B.M. Melioranskij, Georgij Κίρήαηίη I Ioann Ierusalimljanin, St. Petersburg 1901, xxv. The iconoclasts, argues the Old Man, are opposing the great Constantine by calling Christ's image an idol, ην αυτός έν Νικαία τη πόλει μεγάλως Ιστόρησεν δεϋρο και ϊδε την άγίαν... πρώτην σύνοδον, ην αυτός τη του θεού κελεύσει συνήθροισεν... και αυτός ό ευλαβέστατος βασιλεύς δεχόμενος άνωθεν στέφος υπό αγίου αγγέλου, τη πνευματική χρωματουργία [lacuna] και παρέδωκεν αυτά τη καθολική... εκκλησία. The author appears to be saying that Constantine was depicted in the act of receiving a crown from an angel (a flying victory?), but an image of Christ would not normally have been part of the same composition. On p. xxxv of the same text is a reference to a mosaic representing the Second Council. 43 Refutatio et eversio, ch ff., 74.1 ff. 44 See Hows of 754, Mansi XIII, 233B ff. On this subject see esp. Chr. Walter, L'iconographie des conciles dans la tradition byzantine, Paris On the political message conveyed by images of councils cf. A. Grabar, L'iconoclasme byzantin 2, Paris 1984,65 ff. 46 Chr. Walter, Les dessins carolingiens d'un manuscrit de Verceil, CahArch 18 (1968), Michael Attaliates, Bonn ed., As shown by the inscription discussed by me in 'Notes d'épigraphie' (as inn.*), 354 ff. 4 V. Laurent, Les régestes des actes du Patriarcat de Constantinople, 1/4, Paris 1971, no Cf. Foss,Nicaea, Foss, ibid. J.P.A. Van den Vin, Travellers to Greece and Constantinople, Leiden 1980,1,297-8, is mistaken in connecting this painting with the church of the Dormition. 33

9 CYRIL MANGO In conclusion I should like to stress once again the importance of the siege of 727 in leading up to the outbreak of Iconoclasmi. The event that is usually adduced as swaying Leo's mind, namely the eruption of Thera in 726, was, to use modern jargon, non-specific. The same may be said of the defeat of the Helladics. But the deliverance of Nicaea, especially if it was seen as miraculous, carried the express blessing of the 318 Fathers, i.e. the highest authority in the formulation of Orthodoxy. Leo could now be certain that he was on the right course. Cyril Mango Ο ΤΟΠΟΣ ΣΥΓΚΛΗΣΗΣ ΤΗΣ Α' ΟΙΚΟΥΜΕΝΙΚΗΣ ΣΥΝΟΔΟΥ ΚΑΙ Ο ΝΑΟΣ ΤΩΝ ΑΓΙΩΝ ΠΑΤΕΡΩΝ ΣΤΗ ΝΙΚΑΙΑ -ΓΤ Α' Οικουμενική Συνοδός (325) πραγματοποιήθηκε στην κύρια αίθουσα του αυτοκρατορικού ανακτόρου της Νίκαιας. Στο παρόν άρθρο υποστηρίζεται ότι πριν από τον 8ο αιώνα η αίθουσα αυτή μετατράπηκε σε ναό αφιερωμένο στους άγιους Πατέρες που έλαβαν μέρος στη Σύνοδο και κοσμήθηκε με ψηφιδωτό στο οποίο εικονιζόταν η Σύνοδος. Ο πρώτος γνωστός επισκέπτης, ο άγγλος προσκυνητής Willibald, ο οποίος επισκέφθηκε τη Νίκαια περί το , περιγράφει την αίθουσα ως ροτόντα με ομφάλιο, έναν αρχιτεκτονικό τύπο που ταιριάζει σε αίθουσα ανακτόρου. Το 727 η Νίκαια πολιορκήθηκε από τους Άραβες, που κατόρθωσαν να καταστρέψουν τμήμα των τειχών, αλλά απέτυχαν να καταλάβουν την πόλη. Το ακριβές σημείο καταστροφής αναφέρεται σε επιγραφή του Αρταβάσδου (Εικ. 2), εντειχισμένη σε μικρή απόσταση δυτικά της πύλης της Istanbul στη Νίκαια (Εικ. 3) στην επιγραφή αναφέρεται ότι η σωτηρία της πόλης οφείλεται σε θεία παρέμβαση. Στην πληρέστερη αφήγηση αυτής της πολιορκίας από τον Θεοφάνη, τα γεγονότα παρουσιάζονται σαφώς παραποιημένα, ώστε η σωτηρία της πόλης να αποδίδεται στις απεικονιζόμενες μορφές των αγίων Πατέρων. Σε ένα εγκώμιο των 318 Πατέρων από τον άγνωστο από αλλού πρεσβύτερο Γρηγόριο, που χρονολογείται μεταξύ των ετών 727 και 740, γίνεται επίσης αναφορά σε μία «πρόσφατη» πολιορκία, στη διάρκεια της οποίας οι εχθροί εισέβαλαν στο ναό των Αγίων Πατέρων και τέλεσαν εκεί τελετουργίες της θρησκείας τους, αλλά εκδιώχθηκαν με θαυμαστό τρόπο. Την εποχή εκείνη το οικοδόμημα διετηρείτο ακόμη σε όλη του την αίγλη και είχε μία κρήνη ελαίου κοντά στην ανατολική είσοδο του. Ο Γρηγόριος δεν αναφέρεται στην απεικόνιση της Συνόδου. Ο ναός των Αγίων Πατέρων πιθανότατα υπέστη καταστροφές από το σεισμό του 740 (για το λόγο αυτό η Σύνοδος του 787 έγινε στον καθεδρικό ναό της Αγίας Σοφίας επίσης στη Νίκαια) αλλά συνέχισε να υπάρχει με τη μία ή την άλλη μορφή του έως την τουρκική κατάκτηση. Συμπερασματικά επισημαίνονται τα εξής: 1. Το αυτοκρατορικό ανάκτορο με τη ροτόντα βρισκόταν πιθανότατα στο βορειοδυτικό τμήμα της πόλης, κοντά στην πύλη της Istanbul. 2. Η με θαυμαστό τρόπο διάσωση της Νίκαιας το 727 θεωρήθηκε από τον Λέοντα Γ ένδειξη θείας επιδοκιμασίας της θρησκευτικής πολιτικής του, γεγονός πιο ουσιαστικό και πιο σημαντικό από τη συχνά αναφερόμενη έκρηξη του ηφαιστείου της Θήρας (726), καθώς έφερε τη σφραγίδα των 318 Πατέρων που είχαν διατυπώσει το ορθόδοξο δόγμα. Οι εικονόφιλοι, από την πλευρά τους, κατέβαλαν μια κάπως αδέξια προσπάθεια (όπως μαρτυρεί ο Θεοφάνης) να παρουσιάσουν το θαύμα με τρόπο που να στηρίζει τις δικές τους θέσεις. 34

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

Right Rear Door. Let's now finish the door hinge saga with the right rear door

Right Rear Door. Let's now finish the door hinge saga with the right rear door Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

Οι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού)

Οι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού) Οι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού) Προσδοκώμενα αποτελέσματα Περιεχόμενο Ενδεικτικές δραστηριότητες

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple

7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple A/ Ονόματα και ένα παράδειγμα 1 Present Simple 7 Present PERFECT Simple 2 Present Continuous 8 Present PERFECT Continuous 3 Past Simple (+ used to) 9 Past PERFECT Simple she eats she is eating she ate

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

1999 MODERN GREEK 2 UNIT Z

1999 MODERN GREEK 2 UNIT Z STUDENT NUMBER CENTRE NUMBER HIGHER SCHOOL CERTIFICATE EXAMINATION 1999 MODERN GREEK 2 UNIT Z (55 Marks) Time allowed Two hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Write your Student

Διαβάστε περισσότερα

Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

the total number of electrons passing through the lamp.

the total number of electrons passing through the lamp. 1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy

Διαβάστε περισσότερα

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ EΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΕΚΠΑΙΔΕΥΤΙΚΟ ΤΕΧΝΟΛΟΓΙΚΟ ΙΔΡΥΜΑ ΤΕΙ ΙΟΝΙΩΝ ΝΗΣΩΝ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : Λεωφ. Αντ.Τρίτση, Αργοστόλι Κεφαλληνίας Τ.Κ. 28 100 τηλ. : 26710-27311 fax : 26710-27312

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

Assalamu `alaikum wr. wb.

Assalamu `alaikum wr. wb. LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump

Διαβάστε περισσότερα

Συστήματα Διαχείρισης Βάσεων Δεδομένων

Συστήματα Διαχείρισης Βάσεων Δεδομένων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

Finite Field Problems: Solutions

Finite Field Problems: Solutions Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

ICTR 2017 Congress evaluation A. General assessment

ICTR 2017 Congress evaluation A. General assessment ICTR 2017 Congress evaluation A. General assessment -1- B. Content - 2 - - 3 - - 4 - - 5 - C. Speakers/ Presentations/ Sessions - 6 - - 7 - D. Posters/ Poster sessions E. Organisation and coordination

Διαβάστε περισσότερα

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11 Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and

Διαβάστε περισσότερα

LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV. 4 February 2014

LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV. 4 February 2014 LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV 4 February 2014 Somewhere κάπου (kapoo) Nowhere πουθενά (poothena) Elsewhere αλλού (aloo) Drawer το συρτάρι (sirtari) Page η σελίδα (selida) News τα νέα (nea)

Διαβάστε περισσότερα

Writing for A class. Describe yourself Topic 1: Write your name, your nationality, your hobby, your pet. Write where you live.

Writing for A class. Describe yourself Topic 1: Write your name, your nationality, your hobby, your pet. Write where you live. Topic 1: Describe yourself Write your name, your nationality, your hobby, your pet. Write where you live. Χρησιμοποίησε το and. WRITE your paragraph in 40-60 words... 1 Topic 2: Describe your room Χρησιμοποίησε

Διαβάστε περισσότερα

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 WEEK 21 The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 I. The Feast of Firstfruits signifies the resurrected Christ as the firstfruits for our enjoyment as

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: Discrete-time systems and computer control by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

department listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι

department listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι She selects the option. Jenny starts with the al listing. This has employees listed within She drills down through the employee. The inferred ER sttricture relates this to the redcords in the databasee

Διαβάστε περισσότερα

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ Ενότητα 1β: Principles of PS Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης

Διαβάστε περισσότερα

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ ΝΟΜΙΚΟ ΚΑΙ ΘΕΣΜΙΚΟ ΦΟΡΟΛΟΓΙΚΟ ΠΛΑΙΣΙΟ ΚΤΗΣΗΣ ΚΑΙ ΕΚΜΕΤΑΛΛΕΥΣΗΣ ΠΛΟΙΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που υποβλήθηκε στο

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή

Διαβάστε περισσότερα

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014 LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV 18 February 2014 Slowly/quietly Clear/clearly Clean Quickly/quick/fast Hurry (in a hurry) Driver Attention/caution/notice/care Dance Σιγά Καθαρά Καθαρός/η/ο

Διαβάστε περισσότερα

FINAL TEST B TERM-JUNIOR B STARTING STEPS IN GRAMMAR UNITS 8-17

FINAL TEST B TERM-JUNIOR B STARTING STEPS IN GRAMMAR UNITS 8-17 FINAL TEST B TERM-JUNIOR B STARTING STEPS IN GRAMMAR UNITS 8-17 Name: Surname: Date: Class: 1. Write these words in the correct order. /Γράψε αυτέσ τισ λέξεισ ςτη ςωςτή ςειρά. 1) playing / his / not /

Διαβάστε περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8  questions or comments to Dan Fetter 1 Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

Ακαδημαϊκός Λόγος Εισαγωγή

Ακαδημαϊκός Λόγος Εισαγωγή - In this essay/paper/thesis I shall examine/investigate/evaluate/analyze Γενική εισαγωγή για μια εργασία/διατριβή Σε αυτήν την εργασία/διατριβή θα αναλύσω/εξετάσω/διερευνήσω/αξιολογήσω... To answer this

Διαβάστε περισσότερα

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr 9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3) Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)

Διαβάστε περισσότερα

Section 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016

Section 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Section 1: Listening and Responding/ Aκουστική εξέταση Στο πρώτο μέρος της

Διαβάστε περισσότερα

ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α. Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:.

ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α.  Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:. ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α 2 ειδήσεις από ελληνικές εφημερίδες: 1. Τα Νέα, 13-4-2010, Σε ανθρώπινο λάθος αποδίδουν τη συντριβή του αεροσκάφους, http://www.tanea.gr/default.asp?pid=2&artid=4569526&ct=2 2. Τα Νέα,

Διαβάστε περισσότερα

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΨΥΧΟΛΟΓΙΚΕΣ ΕΠΙΠΤΩΣΕΙΣ ΣΕ ΓΥΝΑΙΚΕΣ ΜΕΤΑ ΑΠΟ ΜΑΣΤΕΚΤΟΜΗ ΓΕΩΡΓΙΑ ΤΡΙΣΟΚΚΑ Λευκωσία 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ

Διαβάστε περισσότερα

Ακαδημαϊκός Λόγος Εισαγωγή

Ακαδημαϊκός Λόγος Εισαγωγή - Σε αυτήν την εργασία/διατριβή θα αναλύσω/εξετάσω/διερευνήσω/αξιολογήσω... Γενική εισαγωγή για μια εργασία/διατριβή In this essay/paper/thesis I shall examine/investigate/evaluate/analyze Για να απαντήσουμε

Διαβάστε περισσότερα

Δελτίον της Χριστιανικής Αρχαιολογικής Εταιρείας

Δελτίον της Χριστιανικής Αρχαιολογικής Εταιρείας Δελτίον της Χριστιανικής Αρχαιολογικής Εταιρείας Τομ. 26, 2005 The Meeting-Place of the First Ecumenical Council and the Church of the Holy Fathers at Nicaea MANGO Cyril http://dx.doi.org/10.12681/dchae.425

Διαβάστε περισσότερα

The challenges of non-stable predicates

The challenges of non-stable predicates The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates

Διαβάστε περισσότερα

Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

Modern Greek Extension

Modern Greek Extension Centre Number 2017 HIGHER SCHOOL CERTIFICATE EXAMINATION Student Number Modern Greek Extension Written Examination General Instructions Reading time 10 minutes Working time 1 hour and 50 minutes Write

Διαβάστε περισσότερα

ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ

ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ ΠΑΝΔΠΗΣΖΜΗΟ ΠΑΣΡΩΝ ΣΜΖΜΑ ΖΛΔΚΣΡΟΛΟΓΩΝ ΜΖΥΑΝΗΚΩΝ ΚΑΗ ΣΔΥΝΟΛΟΓΗΑ ΤΠΟΛΟΓΗΣΩΝ ΣΟΜΔΑ ΤΣΖΜΑΣΩΝ ΖΛΔΚΣΡΗΚΖ ΔΝΔΡΓΔΗΑ Γηπισκαηηθή Δξγαζία ηνπ Φνηηεηή ηνπ ηκήκαηνο Ζιεθηξνιόγσλ Μεραληθώλ θαη Σερλνινγίαο Ζιεθηξνληθώλ

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

Πτυχιακή Εργασία. Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους

Πτυχιακή Εργασία. Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους ΑΛΕΞΑΝΔΡΕΙΟ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΣΧΟΛΗ ΤΕΧΝΟΛΟΓΙΑΣ ΤΡΟΦΙΜΩΝ ΚΑΙ ΔΙΑΤΡΟΦΗΣ ΤΜΗΜΑ ΔΙΑΤΡΟΦΗΣ ΚΑΙ ΔΙΑΙΤΟΛΟΓΙΑΣ Πτυχιακή Εργασία Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους Εκπόνηση:

Διαβάστε περισσότερα

Μηχανική Μάθηση Hypothesis Testing

Μηχανική Μάθηση Hypothesis Testing ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider

Διαβάστε περισσότερα

ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ

ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ Ἀναπάντεχες λεπτομέρειες γιὰ τὸν βίο καὶ τὴν πολιτεία τοῦ Ἰωάννη Λάσκαρη μᾶς εἶναι γνωστὲς ἀπὸ μιὰ σειρὰ ἐγγράφων (ποὺ ἀνακάλυψε στὰ κρατικὰ ἀρχεῖα

Διαβάστε περισσότερα

Strain gauge and rosettes

Strain gauge and rosettes Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

Policy Coherence. JEL Classification : J12, J13, J21 Key words :

Policy Coherence. JEL Classification : J12, J13, J21 Key words : ** 80%1.89 2005 7 35 Policy Coherence JEL Classification : J12, J13, J21 Key words : ** Family Life and Family Policy in France and Germany: Implications for Japan By Tomoko Hayashi and Rieko Tamefuji

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education www.xtremepapers.com UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *6301456813* GREEK 0543/03 Paper 3 Speaking Role Play Card One 1 March 30

Διαβάστε περισσότερα

Calculating the propagation delay of coaxial cable

Calculating the propagation delay of coaxial cable Your source for quality GNSS Networking Solutions and Design Services! Page 1 of 5 Calculating the propagation delay of coaxial cable The delay of a cable or velocity factor is determined by the dielectric

Διαβάστε περισσότερα

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ»

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ» I ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΝΟΜΙΚΩΝ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΠΟΛΙΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΟΙΚΟΝΟΜΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ «ΔΙΟΙΚΗΣΗ ΚΑΙ ΟΙΚΟΝΟΜΙΑ» ΚΑΤΕΥΘΥΝΣΗ: ΟΙΚΟΝΟΜΙΚΗ

Διαβάστε περισσότερα

ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS

ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS ΟΜΗΡΟΥ ΙΛΙΑΔΑ ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS www.scooltime.gr [- 2 -] The Project Gutenberg EBook of Iliad, by Homer This ebook is for the use of anyone anywhere at no cost and with almost no restrictions

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

Συντακτικές λειτουργίες

Συντακτικές λειτουργίες 2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.

Διαβάστε περισσότερα

Problem Set 3: Solutions

Problem Set 3: Solutions CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)

Διαβάστε περισσότερα

5.4 The Poisson Distribution.

5.4 The Poisson Distribution. The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Αναερόβια Φυσική Κατάσταση

Αναερόβια Φυσική Κατάσταση Αναερόβια Φυσική Κατάσταση Γιάννης Κουτεντάκης, BSc, MA. PhD Αναπληρωτής Καθηγητής ΤΕΦΑΑ, Πανεπιστήµιο Θεσσαλίας Περιεχόµενο Μαθήµατος Ορισµός της αναερόβιας φυσικής κατάστασης Σχέσης µε µηχανισµούς παραγωγής

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ ΣΥΓΚΡΙΤΙΚΗ ΜΕΛΕΤΗ ΤΗΣ ΣΥΓΚΡΑΤΗΤΙΚΗΣ ΙΚΑΝΟΤΗΤΑΣ ΟΡΙΣΜΕΝΩΝ ΠΡΟΚΑΤΑΣΚΕΥΑΣΜΕΝΩΝ ΣΥΝΔΕΣΜΩΝ ΑΚΡΙΒΕΙΑΣ

Διαβάστε περισσότερα

LESSON 6 (ΜΑΘΗΜΑ ΕΞΙ) REF : 201/045/26-ADV. 10 December 2013

LESSON 6 (ΜΑΘΗΜΑ ΕΞΙ) REF : 201/045/26-ADV. 10 December 2013 LESSON 6 (ΜΑΘΗΜΑ ΕΞΙ) REF : 201/045/26-ADV 10 December 2013 I get up/i stand up I wash myself I shave myself I comb myself I dress myself Once (one time) Twice (two times) Three times Salary/wage/pay Alone/only

Διαβάστε περισσότερα

Section 9.2 Polar Equations and Graphs

Section 9.2 Polar Equations and Graphs 180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

TMA4115 Matematikk 3

TMA4115 Matematikk 3 TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet

Διαβάστε περισσότερα

Adjectives. Describing the Qualities of Things. A lesson for the Paideia web-app Ian W. Scott, 2015

Adjectives. Describing the Qualities of Things. A lesson for the Paideia web-app Ian W. Scott, 2015 Adjectives Describing the Qualities of Things A lesson for the Paideia web-app Ian W. Scott, 2015 Getting Started with Adjectives It's hard to say much using only nouns and pronouns Simon is a father.

Διαβάστε περισσότερα

«ΨΥΧΙΚΗ ΥΓΕΙΑ ΚΑΙ ΣΕΞΟΥΑΛΙΚΗ» ΠΑΝΕΥΡΩΠΑΪΚΗ ΕΡΕΥΝΑ ΤΗΣ GAMIAN- EUROPE

«ΨΥΧΙΚΗ ΥΓΕΙΑ ΚΑΙ ΣΕΞΟΥΑΛΙΚΗ» ΠΑΝΕΥΡΩΠΑΪΚΗ ΕΡΕΥΝΑ ΤΗΣ GAMIAN- EUROPE «ΨΥΧΙΚΗ ΥΓΕΙΑ ΚΑΙ ΣΕΞΟΥΑΛΙΚΗ» ΠΑΝΕΥΡΩΠΑΪΚΗ ΕΡΕΥΝΑ ΤΗΣ GAMIAN- EUROPE We would like to invite you to participate in GAMIAN- Europe research project. You should only participate if you want to and choosing

Διαβάστε περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 11: The Unreal Past Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons

Διαβάστε περισσότερα

Galatia SIL Keyboard Information

Galatia SIL Keyboard Information Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

Διαβάστε περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 4: English a Language of Economy Το περιεχόμενο του μαθήματος διατίθεται με άδεια

Διαβάστε περισσότερα

ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας»

ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας» ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ ΣΧΟΛΗ ΑΝΘΡΩΠΙΣΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΤΗΣ ΠΡΟΣΧΟΛΙΚΗΣ ΑΓΩΓΗΣ ΚΑΙ ΤΟΥ ΕΚΠΑΙ ΕΥΤΙΚΟΥ ΣΧΕ ΙΑΣΜΟΥ «ΠΑΙ ΙΚΟ ΒΙΒΛΙΟ ΚΑΙ ΠΑΙ ΑΓΩΓΙΚΟ ΥΛΙΚΟ» ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που εκπονήθηκε για τη

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

LECTURE 2 CONTEXT FREE GRAMMARS CONTENTS

LECTURE 2 CONTEXT FREE GRAMMARS CONTENTS LECTURE 2 CONTEXT FREE GRAMMARS CONTENTS 1. Developing a grammar fragment...1 2. A formalism that is too strong and too weak at the same time...3 3. References...4 1. Developing a grammar fragment The

Διαβάστε περισσότερα

A Note on Intuitionistic Fuzzy. Equivalence Relation

A Note on Intuitionistic Fuzzy. Equivalence Relation International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

Διαβάστε περισσότερα

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω 0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής

Διαβάστε περισσότερα

Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your

Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your GP practice in Islington Σε όλα τα Ιατρεία Οικογενειακού

Διαβάστε περισσότερα

Code Breaker. TEACHER s NOTES

Code Breaker. TEACHER s NOTES TEACHER s NOTES Time: 50 minutes Learning Outcomes: To relate the genetic code to the assembly of proteins To summarize factors that lead to different types of mutations To distinguish among positive,

Διαβάστε περισσότερα

14 Lesson 2: The Omega Verb - Present Tense

14 Lesson 2: The Omega Verb - Present Tense Lesson 2: The Omega Verb - Present Tense Day one I. Word Study and Grammar 1. Most Greek verbs end in in the first person singular. 2. The present tense is formed by adding endings to the present stem.

Διαβάστε περισσότερα

[1] P Q. Fig. 3.1

[1] P Q. Fig. 3.1 1 (a) Define resistance....... [1] (b) The smallest conductor within a computer processing chip can be represented as a rectangular block that is one atom high, four atoms wide and twenty atoms long. One

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα