A Contribution to the Interpretation of the Passage Matt

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "A Contribution to the Interpretation of the Passage Matt. 16.18"

Transcript

1 Main Page Library Homage Seminars Book Reviews ΕΛΛΗΝΙΚΑ ENGLISH FRANÇAIS ESPAÑOL ITALIANO DEUTSCH русский ROMÂNESC БЪЛГАРСКИ LIBRARY ENGLISH TEXT Monday, 19 July 2010 Full Contents Holy Scriptures Previous Page Liturgical texts Spiritual Life Monasticism Patrology Panagiotis Boumis The Rock of Apostle Peter A Contribution to the Interpretation of the Passage Matt Theology Introduction History of Church Canon Law Art Church Services Philosophy Sociology Byzantine Culture Modern Greek History Greek Literature Christian Europe The passages of Holy Scripture with which Roman Catholic theologians usually support the Pope's primacy as heir to the Apostle Peter are the following: Matthew ; Luke and John (1). In this present short study we shall deal mainly with the verse (2) from the section of St Matthew's Gospel (Ι ) mentioned above, following the synoptic process of canonical methodology (3). It is this verse, after all, which is provided as the basis of the Roman Catholic Church's governing system and its claims to Ρapal primacy. The Westerners, using this passage as a basis of primary and major import, found their view concerning the special position and authority of St Peter within the Church. In this verse the Lord addresses Peter saying : «Καγώ δέ σοι λέγω ότι συ ει Πέτρος, και επί ταύτη τη πέτρα oικοδοµήσω µου την εκκλησίαν, και πύλαι άδoυ ου κατισχύσουσιν αυτής». The objector to the views concerning Roman primacy contend, however, that the Lord in this verse does not mean Peter as the rock (πέτρα) of the Church's entire structure, but rather his confession, or otherwise the contents of his confession (that is Jesus Christ) which he had given shortly before οn answering the question which

2 Interchristian Dialogue Bibliographies CONTACT Internet Dept. the Lord posed to his disciples, «Υµείς δε τίνα µε λέγετε είναι;» (Matt ). Peter's confession was the following: «Σύ ει ο Χριστός, ο υιός του Θεoύ του ζώντος» (vs. 16). So the contention between the opposing factions is thus summarised(4). We shall endeavour here to view the above verse in a new light with the help of other passages of Holy Scripture; to examine, that is to say, whether both sides possibly are right, or at least meet each other at a certain point. We think that this will help in the revelation of the actual, or at least the more precise, meaning of the verse in question. SEARCH Ι. The Rock (πέτρα) : Jesus Christ. At first, we cannot exclude the interpretation that the "Πέτρα" named by the Lord is the confession about Christ «ως Υιόν του Θεού του ζώντος», or rather Jesus Christ Himself. This very formulation of the passage allows for just such an interpretation. The passage, as is known. runs, «Καγώ δε σοι λέγω ότι σύ ει Πέτρος, και επί ταύτη τη πέτρα οικοδοµήσω µου την εκκλησίαν» and not «σύ ει Πέτρος και επί σε οικοδοµήσω µου την εκκλησίαν». Thus we can say that this change in the term (Πέτρος - Πέτρα) in the two sentences, in conjunction with the existence of the preceeding confession «Σύ ει ο Χριστός, ο υιός του Θεού του ζώντος», not only provides us with a possibility but also a reason to render this passage in the above mentioned interpretation. Let us, however, look at the matter more analytically. We claim that it is possible to impart this interpretation to the passage in question because apart from the above reason something more supports the fact. This is that the God-inspired writer used the conjunction "και" and not the opposite particle "δέ". That is to say, he says with great care and not without sufficient reason, «Συ ει Πέτρος, και επί ταύτη τη πέτρα, οικοδοµήσω µου την Εκκλησίαν» and not, «Συ ει Πέτρος, επί ταύτη δέ τη Πέτρα, οικοδοµήσω µου τήν Εκκλησίαν». This specific use of "καί"(5) does not oblige us to confine ourselves to one single interpretation, to one single actual "rock" (Πέτρα). Only if it was written «επί ταύτη δε τη Πέτρα, οικοδοµήσω µου την Εκκλησίαν» would we be obliged to comprehend οne explicit Πέτρα and consequently accept one definition. As things stand, however, nobody and nothing compels us to emphasize the «ταύτη τη πέτρα» thus obliging us to bind ourselves to one exclusive definition. Οn the contrary, we are free to restore the suitable position and strength to the "και" of the clause. This means that even if we adopted another view, for example, that "Πέτρα" refers to Peter, the above interpretation would still not be excluded. Indeed, even if the text read «και επί σε (Πέτρε) οικοδοµήσω µου τήν Εκκλησίαν», it still would not detract from the interpretation in question; for it would simply mean that besides frοm the οthers «I will also bild οn you, Peter, as a rοck». The conjunction "και" does not give exclucivity to the rock of Peter (Πέτρα του Πέτρου) and even if Peter were the rock the existance of other rocks would not be excluded. This possibility, or in other words the use of "και"(6) is reinforced by the passage's following clause, «και πύλαι άδου ου κατισχύσουσιν αυτής (= the Church)». Our argument is strengthened here because nobody, we think, would contend this clause discerns that only the gates of Hades would not prevail against the Church. Οn the contrary, it evidently means that apart from all οther likely enemies of the Church, even the gates of Hades (i.e. «the mortal dangers»(7) or «death and organised power of evil»(8)) in addition would not prevail against it. Consequently the possibility exists for us to accept the above interpretation.

3 Νοw we must come to the heart of the matter which impels us to accept the view that the rock (Πέτρα) is the confession of faith in Christ, or rather Christ himself. Apart from the play οn words (Petros-Petra) other passages of Hοly Scripture give us the opportunnity to characterize Christ in this manner. Thus the interpretation given above of the passage Matt harmonizes with these other passages. For instance in Romans 9.33 we have the prophetic passage from Isaiah (28.16 and 8.14) which says of Jesus Christ, «Ιδού τίθηµι εν Σιών λίθον προσκόµµατος και πέτραν σκανδάλου, και ο πιστεύων επ' αυτώ ου καταισχυνθήσεται». Α passage in Corinthians Ι of St Ρaul supports this view to a greater extent, saying, «ου θέλω γαρ υµάς αγνοείν αδελφοί, ότι oι πατέρες ηµών... έπινον εκ πνευµατικής ακολουθούσης πέτρας, η Πέτρα δε ην ο Χρίστος» ( ). This passage explicitly states that Christ is the spiritual rock (η Πέτρα η πνευµατική). Thus, the term Πέτρα is clearly used in conjunction with Christ in Ηοly Scripture. Likewise, St. Peter repeats in his First Epistle General, «Διότι περιέχει εν γραφή. ιδού τίθηµι εν Σιών (9) λίθον εκλεκτόν, ακρογωνιαίον, έντιµον, καί ο πιστεύων επ' αυτώ ου µη καταισχυνθή. Υµίν ουν η τιµή τοις πιστεύουσιν. απιστούσιν δε λίθος ον απεδοκίµασαν oι οικοδοµούντες, ούτος εγενήθη εις κεφαλήν γωνίας και λίθος προσκόµµατος καί πετρα σκανδάλου» (Peter Ι, ). Elsewhere, St Ρaul, addressing the Christians, speaks of Clιrist as the pre-eminent foundation or corner-stone : «Θεού οικοδοµή έστε. Κατά την χάριν του Θεού την δοθείσάν µοι ως σοφός αρχιτέκτων θεµέλιον έθηκα, άλλος δε εποικοδοµεί. Εκαστος δε βλεπέτω πώς εποικοδοµεi. Θεµέλιον γαρ άλλον ουδείς δύναται θείναι παρά τον κείµενον, ος έστιν Ιησούς Χριστός» (Cor.Ι, ).(10) Other passages of Holy Scripture indirectly support the view that Jesus Christ is the foundation stone by characterising the words of the Lord, i.e. the Gospels, as a rock. When we speak of the word of the Lord, we mean the word of God which also means Jesus Christ, the incarnate Word of God, and vice-versa.(11) Ιn this way our argument is reinforced as the Gospels can represent Christ. This principle is confirmed by other passages as well, foremost amongst which is, in our opinion, John 8.25 where the Lord, answering the Jews' question «σύ τίς ει ;», spoke the renown phrase, «Την αρχήν ο,τι και λαλώ υµίν», which is explained: «That which Ι have said unto you from the beginning (or generally, already)(12)». Το wit, «Ι am that which Ι tell you, that which Ι teach yοu». We have used this passage from John here not only because we consider it a foundational principle of the first rank concerning the nature of Christ, but also because the problems and subjects of the passage are almost the same with Matt. 16. l3ff from whence comes the passage under study here. Ιn both cases questions and answers revolve around the same person and problem of Christ, with the difference that in the first case he is asking while in the second he is answering. But the importance of John 8.25, which gives particular weight to the confession therein, is the fact that He who is answering is the person (Christ) about whom the problem is posed «Who are you?» (συ τις ει;). Consequently, we have an authentic self-confession and at the same time confirmation of what we have said above, that by the word of the Lord we mean Jesus Christ. With the above observation we may now indicate the passages of Holy Scripture which liken the Word of God, the words of the Lord (i.e. Christ) with a rock (Πέτρα). Thus we have an excerpt from the Gospel according to St Matthew with the Lord's distinctive words : «Πας ουν όστις ακούει µου τους λόγους τούτους και ποιεί αυτούς, οµοιωθήσεται ανδρί φρονίµω, όστις ωκοδόµησεν αυτού την οικiαν επι την πέτραν. Και κατέβη η βροχή και ήλθον oι ποταµοί και έπνευσαν οι άνεµοι και προσέπεσαν τη οικία εκείνη, και ουκ έπεσεν

4 τεθεµελίωτο γαρ επί την πέτραν» (Matt ). Α similar passage is recognisable in St Luke's Gospel: «Πας ο ερχόµενος προς µε και ακούων µου των λόγων και ποιών αυτοίς όµοιός εστιν ανθρώπω οικοδοµούντι οικίαν, ος έσκαψεν και εβάθυνεν και έθηκεν θεµέλιον επί την Πέτραν» (Luke ). Ιn these passages, then, we have the testimony that Christ, the Son and Word Λόγος of God, is the rock (Πέτρα) uρon which it is possible to build His house safely, so as it stands unshaken. The passages contemplated above not only call to mind but also reinforce the given interpretation of Matt that the rock (Πέτρα) upon which the Church of God would be built so as to withstand Hell, is Christ, the Son and Word of God.(13) II. The Rock: Peter and the other Disciples of Christ. One may come to the acceptance of the view that the passage in question represents the confession in Christ, or rather Christ, the Word of God, Himself, but this does not mean that it is impossible for it to mean St Peter as well.(14) From the beginning the expression of this passage presents the possibility of this interpretation, the reasons for which we shall state here as before: οn one side the alliterative form of Πέτρα-Πέτρος while οn the other the existence, as we noted above, οf the manifold uses and functions of "και" in the Greek language. It is possible, then, for the Lord to build His Church οn other rocks besides His Οwn, and explicitly also οn the rock of St Peter. The above view is significantly strengthened by the fact that we have other passages of Holy Scripture which liken the Apostles (and consequently St Peter) to a foundation stone/ rock (λίθον-πέτραν)(15). Thus our previously mentioned interpretation satisfactorily harmonizes with biblical teaching. Firstly, we have the passage in Ephessians where St Ρaul says to the Christians : «Εστέ συµπολίται των αγίων και οικείοι του Θεού, εποικοδοµηθέντες επι τω θεµελίω των αποστόλων και προφητών, οντος ακρογωνιαίου αυτού Ιησού Χριστού, εν ω πάσα οικοδοµή συναρµολογουµένη αύξει εις ναόν άγιον εν Κυρίω, εν ω και υµείς συνοικοδοµείσθε εις κατοικητήριον του Θεού εν πνεύµατι». Yet again we ascertain here that Jesus Christ is the corner-stone upon which «every structure» (πάσα οικοδοµή) of the Church is assembled, thus producing an «augmentation» (αύξησιν) in the Lord. At the same time, however, we are informed that the Christian community is also built upon «the foundation of the Apostles» (επί τω θεµελίω των αποστόλων), in other words, the Apostles, including, of course, St Peter, constitute the foundation stones of the Church. This is more explicitly formulated and phrased in another section of Holy Scripture, Revelation 21.14, where we read : «Και το τείχος της πόλεως (Holy Jerusalem; i.e. the Church)(16) έχον θεµελίους δώδεκα, και επ' αυτών δώδεκα ονόµατα των δώδεκα αποστόλων του αρνίου».(17) Following from this, we may accept the Roman Catholic view that the rock (πέτρα) is not only the confession in Christ but also the origin οf the confession, St Peter. At the same time, however, we are obliged -as they are- to accept the fact that the Church has as its foundation stone not only St Peter but the other Apostles as well. It is important to note that the disputed passage (Matt ) under examination

5 does not exclude the other Apostles. Indeed, the use of the multi-meaninged "καί" makes it possible for us to understand that the Lord was able to use other "rocks" as well as Peter's earlier or later. Because the Lord said He would build His Church «και επί τη Πέτρα του Πέτρου» it does not follow that it is not possible to build it «και επί ταις πέτραις των άλλων αποστόλων»(18) (οn the stones of the other Apostles as well). Not only the Lord's disciples but also all the believers in Christ and confessors in His word and work constitute the rock of the Church's structure. Peter's words take οn particular importance in this respect when he says: «Eι εγεύσασθε οτι χρηστός ο Κύριος προς ον προσερχόµενοι, λίθον ζώντα, και αυτοί ως λίθοι ζώντες οικοδοµείσθε οίκος πνευµατικός» (Peter, Ι, ). Worthy of note in this respect is that which Origen says: «If we also, like Peter, say "Thou art Christ, the Son of the living God", we become Peter to the extent that the Word says to us "Υοu are Peter" (Συ ει Πέτρος) etc. For every disciple of Christ is a rock after drinking of "that spiritual rock which followed", and οn each such rock is built every ecclesiastical "λόγος" (principle), and in agreement with this the life of the Church.(19) The "Shepherd of Hermas" also speaks revealingly about the Apostles and other true Christians as "square stones", saying, «The square, white stones fitting at their joints ; these are the apostles, bishops, teachers and deacons marching in accordance with the holiness of God, having acted virtuously and decently as bishops, teachers, and ministers οn behalf of the chosen of God».(20) Since the passage Matt has the conjunction "και" («και επί ταύτη τη πέτρα») we are able to apply both the first and the second interpretations. That is to say that both our Lord and.st Peter can be rocks. If the opposite particle "δε" were used («επί ταύτη δε τη πέτρα») then we would be obliged, as was said above, to choose between the two, either the one or the other. Νοw we can freely accept both of them, the one and the other. Upon this conclusion the following arguments are vindicated: 1) Since we accept that the divine guidance accorded the writing of Holy Scripture, it is reasonable to ask ourselves whether, if only one interpretation were right, God would have allowed the passage to read thus so as to appear ambiguous and cause discord? St Ρau1 proclaims: «Πιστός ο Θεός ότι o λόγος ηµών ο προς υµάς ουκ έστιν ναι και ου. Ο του Θεού γάρ υιός Χριστός Ιησούς ο εν υµίν δι' υµών κηρυχθείς ουκ εγένετο ναι και ου» (Corinthians li, ). Should not, then, the writer- Evangelist inspired by the Holy Spirit (21) render himself most explicitly and not place his readers in confusion? 2) The very existence of two views, both with their host of supporters, makes us ask ourselves whether both sides have a right to insist in the validity of their views. 3) We are able to add to the above observation the fact that Origen, one of the men most involved in the interpretation of Holy Scripture intertwines the two interpretations, as is clear in his extended commentary above οn the passage in question.(22) All this promts us to the conclusion that both interpretations have their οwη reasons and supports.

6 ΙΙΙ. Α Combination of Both Interpretations. Let us dwell for a little while yet οn these two views, firstly, that the "rock" is Jesus Christ and secondly, that it represents St Peter and the other Apostles. Ιn this way we may get to the core of the passage's meaning and thus complete the picture of the Church's structure.(23) As we saw above, it is possible for both interpreted views to be valid. The νalidity of these views, however, is not the only thing which must be proved; we must also show that it is possible for them to co-exist and harmonize with each other, to complete the full meaning of what is said. In other words, it is possible οn the one hand for the Apostles to constitute the foundation stones (λίθοι) upon which the whole structure of the Church is built, while οn the other the Lord may constitute the rock-ground (Πέτρα-έδαφος) upon which the Church's structure rests. We are able to have the Apostles as foundation-stones, but at the same time we are also able to have the rock, Christ, as the base of the whole structure. The one does not cancel the other. Οn the contrary the above double interpretation enables us to derive a more complete picture of the structure of the Church. As we move towards the consolidation of this picture it is possible to observe the following: It is possible for the foundation stones to cο-exist, or rather exist, as there also exists the rock upon which they are placed, such as those found in a rocky terrain οn a large rock (βράχος). Otherwise they are not able to constitute the foundation -the foundation stones- of the structure. For the foundation stones, to exist and constitute the actual foundation, however, there must exist first and foremost the rock (Πέτρα). This rock, the rocky ground, is the presupposition of the foundation-stones, existence. We consider that this view is expressed or otherwise that this observation is strengthened by the following excerpt from St Luke's Gospel : «Πας ο ερχόµενος προς µε και ακούων µου των λόγων και ποιών αυτούς όµοιός εστιν ανθρώπω οικοδοµούντι οικίαν, ος έσκαψεν και εβάθυνεν και έθηκεν θεµέλιον επί την πέτραν. πληµµύρης δε γενοµένης προσέρρηξεν ο ποταµός τη οικία εκείνη, και ουκ ίσχυσεν σαλεύσαι αυτήν, τεθεµελίωτο γαρ επί την πέτραν. Ο δέ ακούσας και µη ποιήσας όµοιός εστιν ανθρώπω οικοδοµήσαντι oικίαν επί την γην χωρίς θεµελίου, η προσέρρηξεν ο ποταµός, και ευθύς συνέπεσεν, και εγένετο το ρήγµα της οικίας εκείνης µέγα» (Luke, ). It is worthy of note that while Luke in the first case states that the foundation (θεµέλιον) was placed οn the rock (Πέτρα), in other words, he refers to and uses both elements, in the second case, where he does not refer to the rock, he distinctly says that the nοn-believer builds absolutely without («χωρίς») a foundation. He does not state that he at least placed a foundation οn rocky ground but emphasises that he built without foundation, evidently because he did not build οn solid ground, but οn un-sound, sandy ground. We can confirm this idea better if we correlate the above passage from Luke with its counterpart in Matthew which says: «Kαι πας ο ακούων µου τους λόγους τούτους και µη ποιών αυτούς οµοιωθήσεται ανδρί µωρώ, όστις ωκοδόµησεν αυτού την οικίαν επί την άµµον». If one does not place the foundation οn solid ground, οn rocks, but places it οn sandy ground, it is as if he does not place a foundation at all, even if he did place foundation stones. Ιn other words, wherever the rock exists, there also exists the foundation. Where the rock does not exist, there will be no foundation. We confirm this view from a corresponding verse in Matthew (7. 24ff). Ιn this passage we indirectly ascertain the main role of the rock in relation with the foundation stones. We see that the main weight rests οn the rock, on the rocky ground, upon which the

7 structure is built, whereas the foundation stones play the secondery role. Thus the word foundation is not mentioned explicitly. These verses run as follows: «Πας ουν όστις ακούει µου τους λόγους τούτους και ποιεί αυτούς, οµοιωθήσεται ανδρί φρονίµω, όστις ωκοδόµησεν αυτού την oικίαv επί την πέτραν. Kαι κατέβη η βροχή και ήλθον oι ποταµοί και έπνευσαν oι άνεµοι και προσέπεσαν τη οικία εκείνη, και ουκ έπεσεν. τεθεµελίωτο γαρ επί την πέτραν». We said that a foundation (θεµέλιον) is not explicitly written here as in Luke, not, however, that it is not at all mentioned therefore cannot exist. Οn the contrary, we discern its existence in the word "τεθεµελίωτο".(24) Consequently, the foundation exists, but its existence is obscured by the existence of the rock (Πέτρα) and thus we verify the main and decisive role of the rock, that of a rocky ground. The rock, then, is the indispensable presupposition of the foundation stones' existence. Ιn this way the Apostles (including St Peter, of course) are not a foundation without the rock, without Christ; it would not be possible to use them as foundation stones upon which would be built the structure of the Church. Even if the role of foundation stone is secondary, however, it does not cease to be essential. It is easily understood that even should we wish to build a house οn rocks, upon βράχοι, but without foundations, without foundation stones, then the whole structure would still have flaws. With the first flood and gales it would be in danger of falling. The role of the Apostles is, then, a necessary, decisive foundation in the structure of the Church of Christ.(25) Αll this concerns the combination of the two interpretations. Concluding, then, at the end of this section and after the presentation of all the above verses, we venture to say that it is not only possible to accept both of the above mentioned interpretations, but it is also obligatory to do so. CONCLUSION Ιn retrospect, one may say that the clearest interpretation of «και επί ταύτη τη πέτρα οικοδοµήσω µου την Εκκλησίαν» is the following: «Ι will build my Church also οn this rock, in which you confessed, that Jesus Christ, as upon solid ground and οn you, Peter, as confessing in Him and constituting thus a foundation stone οn this ground, without excluding but rather understanding that Ι will also build οn other foundation stones, namely the rest of the Apostles, who will proclaim the same truth in Christ upon whom will rest the teachings of the Church concerning me».(26) lt is therefore possible for all the above mentioned interpretations to cο-exist and, indeed, to harmonize and complete the meaning of the verse. Since we place these things οn solid ground, οn rock, as it were, we can thrust aside the incidental dissentions which are rarely absent from personal or dogmatic suppositions and presuppositions.(27) FOOTNOTES

8 1. See, Ed. Eichmann-Κ1. Morsdort, Lehrbuch des Kirchenrechts (auf Grund des Codex Juris Canonici), 1. Band, 11. Aufl., Munchen- Paderborn-Wien 1964, p. 343 ; Μ. Schmaus, Katholische Dogmatik, Band ΙΙΙ, 1: Die Lehre von der Kirche, 5 Aufl., Munchen 1958, p. 156 ff.; Μ. Kaiser, Die Einheit der Kirchengewalt nach dem Zeugnis des Neuen Testaments und der Apostolischen Vater, Munchen 1956, p. 45 ff.; Ιω. Καρµίρη, Ορθόδοξος Εκκλησιολογία, Δογµατικής Τµήµα Ε', Athens 1973, p. 555 ff and Ο. Gullmann, Ρetrus, Juηg e r- Αpοstel-Μartyrer, 2. Aufl., Zurich- Stuttgart Furthermore, we think it neither advisable nor advantageous to deal with this passage by scrutinizing the interpretation given from time tο time by the supporters of Ρapal Primacy or their opponents, and to allign ourselves either with one or the other factions and interpretations. Οn the related "options" and refutations it is possible to find a rich bibliography in the following studies: Χρυσοστόµου Παπαδοπούλου, Το Πρωτείον του Επισκόπου Ρώµης, Athens 1930; second ed. Athens 1964; Μ. Schmaus, οp. cit., p. 157ff. and 852ff ; Λεων. Φιλιππίδου, Πέτρα-Πέτρος (= λίθος) καί η Πέτρα ( = ο βράχος), Athens, Archim. Σπ. Μπιλάλη, Ορθοδοξία και Παπισµός, 2 vols., Athens 1969 and Ιω. Καρµίρη, οp.cit., p. 548 ff., 555ff. With the help of the above works a host of other works can be found. 3. We dealt with the interpretation of John in an earlier work : Συνέπειαι της άρσεως των αναθεµάτων Ρώµης-Κωνσταντινουπόλεως, Athens 1976, p.176 ff. Ιn this present work we shall deal, God willing, with other relevant passages ot Holy Scripture and the Holy Law of our Church which are relevant to the topic in question. 4. Λεων. Φιλιππίδης (οp cit., p. 57) speaks οf the "many intricate and much seasoned antibiblical, wrongly-interpreted babble". 5. As is well known, the use and function of "και" in the Greek language is extremely broad. See J. Humbert-Γ. Κουρµούλη, Συντακτικόν της αρχαίας Ελληνικής Γλώσσης, Athens, p. 354 ff. and 394 ff. 6. See footnote 5 above. 7. See John Chrysostom, PG and Λεων. Φιλιππίδου, οp. cit., p See Παν.Τρεµπέλα, Υπόµνηµα εις το κατά, Ματθαίον Ευαγγέλιον, Athens 1951, p From this passage, in combination, of course, with the above confession of St Peter concerning Jesus, it is possible for one to say that the episcopate οf Jerusalem, where οn one hand «this stone was dissalowed by all men» (απεδοκιµάσθη υπό των ανθρώπων) (1. Peter, 2.4), while οn the other hand it «was made the heαd of the corner (εγενή0η εις κεφαλήν γωνίας) (1. Peter, 2.7), can claim also the primacy of the Church. 10. See John 1. 1 : «Εν αρχη ην ο Λόγος και ο Λόγος ην προς τον Θεόν και Θεός ην ο λόγος» and John 1.14 : «Kαι ο λόγος σαρξ εγένετο». 11. As concerns the above excerpt (Corinthians Ι, ). Prof. Trebelas remarks are worthy of note : «Christ is the stone, the corner-stone (the one that is rejected as being impossiblc to build into the structure οf Israel) upon which we are called to be built. When we speak ot Jesus Christ as the foundation we mean all the dogmatical teachings of the Gospels» (Παν. Τρεµπέλα, Υπόµνηµα εις τας επιστολάς της

9 Καινής Διαθήκης, νοl. 1, 2nd ed., Athens 1954, p. 261). 12. The varied views οn the meaning of the term «Tηv αρχήν» are well known, See Παν. Τρεµπέλα, Υπόµνηµα εις το κατά Ιωάννην Ευαγγέλιον, Athens 1954, p More generally concerning the interpretation of the passage in question, see Στ.Σάκκου, lωάννου Συµβολή εις την γλωσσικήν εξέτασιν της Καινής Διαθήκης, Thessalonica Certainly this implication is given especially by the above passage οf Matthew since it is of the same author (the same who wrote the contested passage, Matt ). Thus one has right, after analysis, to say that the author speaks about this rock. 14. Ιn an earlier work of ours (Τα εκκλησιαστικά οφφίκια ως τα κατά κόσµον αξιώµατα - Συµβολή εις την ερµηνείαν τού ζ' καν. της Πενθέκτης Oικ. Συν. Athens 1970, p.20) stimulated by other motives, we wrote «In this passage it is possible to accept two or more interpretations, since the formulation permits it». We followed this principle in another explanatory work οf ours : «Περί την επικοινωνίαν ηµών µετά των ετεροδόξων» - Συµβολή εις την ερµηνείαν του χωρίου Β' Ιω.10, Athens 1972, p. 22, where we wrote, «From all that we have studied we come to the conclusion... that we are not obliged to confine ourselves to one interpretation, so much so that the Holy Scripture helps us and encourages us to do sο». As is logical, the two interpretations should not collide, but neither sbould they come into conflict with other passages of Holy Scripture. 15. Concerning the alteration of the tcrms "λίθος-πέτρα" see the passages in Romans 9.33 and Ι. Peter See Παν. Μπρατσιώτου, Η Αποκάλυψις του Ιωάννου, Athens 1949, p. 300 : «The new and eternal city, holy Jerusalem, the symbol οf the kingdom of God, His true Church». Also p. 308 : «The introductory verse nine shows the topic of the "holy city" Jerusalem, i.e. the Church». 17. See Hebrews : «Εξεδέχετο γάρ (Αβραάµ) την τούς θεµελίους έχουσαν πόλιν, ης τεχνίτης και δηµιουργός ο Θεός». 18. The Apostle Peter speaks (confesses) at that time and for that reason Christ addresses Himself to him. If another Apostle had been speaking (confessing) then would not the Lord have addressed him? 19. Origen, The Explanation of St Matthew's Gospel, PG Ι Vision 3 (V,1) Publish. Ministry of the Apostolic Library of the Hellenic fathers and Church authors, 3rd Vol., p. 45. col. 5ff. See Παν. Μπούµη, Η oικοδoµή της αιωνίας πόλεως, ή ο τετραγωνισµός του κύκλου (offprint from «Κοινωνία»), 2nd Ed., Athens 1976, p See Peter ΙΙ, Likewise it is also worth adding this: Prof. Λεων. Φιλιππίδης (οp. cit. ρ. 5lff) having previously supplied various extracts from Origen's interpretations and scholia dealing with the above passage, says «The interpretive fluctuations leading even to contradictions οn this subject are obvious in the above passages». Continuing, οn the other hand, after having added other passages from different interpretations, he says «that these as a sampling of the oldest most distinguished clarifications, throw light upon the fluctuatίons concerning the subject of

10 Hermeneutics» (p. 56). See the observations of Arch. Σπυρ. Μπιλάλη (οp. cit., Vol. Ι, p. 100) concerning the holy Augustin. 23. That the term πέτρα can take on the meaning of rocky ground or of a large rock (βράχος) which can hold other stones (λίθοι), can be seen from the following Scriptural passages: A) «Εξήλθεν ο σπείρων του σπείραι τον σπόρον αυτού, και εν τω σπείρειν έτερον κατέπεσεν επί την πέτραν και φυέν εξηράνθη δια το µη έχειν ικµάδα» (Luke 8: 5-6). See Matt. 13:15 and 13:20 and Mark 4-5 and 4.16 : «επί τα πετρώδη». Β) «Και λαβών το σώµα ο Ιωσήφ έθηκεν αυτό εν τω καινώ αυτού µνηµείω ο ελατόµησεν εν τη πέτρα» (Matt. 27:59-60) and See Mark and Revelation 6: If this did not exist, at all one might perhaps yield to the temptation to say that there exists a contradiction between Matthew and Luke and thus a refutation of the above conclusion. Everything in Holy Scripture, however, is wτitten in wisdom. 25. Suffice it here to note that the role of the Αpostolic tradition and succession («αποστολικών διαδοχών» according to Prof. Γερ. Κονιδάρη, Περί το πρόβληµα της ενώσεως των Εκκλησιών, Athens P. 9, 15ff. is basic and most important. 26. For that reason this is named «αποστολική παράδοσις» (apostolic tradition) and not «Πέτρειος παράδοσις» (Peter's tradition). 27. Οn the other hand, however, a disagreement may lead us to search better and interpret things in a clearer light, that is to re-examine the relevant passage for better comprehension. We think that necessary when the proposed interpretation agrees with Holy Scripture and the whole spirit harmonises with the holy canons and best responds to the demands of modern times. This improvement is (or, at least, should be) a side οf a certain part in the Church congregation's search for and pursual of renewal. See the contents of another work of ours: «Περί την επικοινωνίαν ηµών µετά των ετεροδόξων», p Previous Page

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

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

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

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

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

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

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

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

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

Chapter 2 * * * * * * * Introduction to Verbs * * * * * * *

Chapter 2 * * * * * * * Introduction to Verbs * * * * * * * Chapter 2 * * * * * * * Introduction to Verbs * * * * * * * In the first chapter, we practiced the skill of reading Greek words. Now we want to try to understand some parts of what we read. There are a

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

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 1: Elements of Syntactic Structure Το περιεχόμενο του μαθήματος διατίθεται με άδεια

Διαβάστε περισσότερα

Lecture 2. Soundness and completeness of propositional logic

Lecture 2. Soundness and completeness of propositional logic Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

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

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

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

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

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

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

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

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

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

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

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

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

1 John 5. LGNT - September 6, 2019 Page 1. Book Chapter Verse. Word # 1 John Greek. # of words. Numeric value. Interlinear English.

1 John 5. LGNT - September 6, 2019 Page 1. Book Chapter Verse. Word # 1 John Greek. # of words. Numeric value. Interlinear English. Book Chapter Verse Interlinear English Word # 1 John Greek # of letters # of words Numeric value 230501 everyone 01709 ΠΑΣ 3 281 230501 01710 Ο 1 70 230501 believing 01711 ΠΙΣΤΕΥΩΝ 8 1845 230501 that 01712

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

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

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

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

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

Section 7.6 Double and Half Angle Formulas

Section 7.6 Double and Half Angle Formulas 09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

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

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

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

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

Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ

Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Τ.Ε.Ι. ΔΥΤΙΚΗΣ ΜΑΚΕΔΟΝΙΑΣ ΠΑΡΑΡΤΗΜΑ ΚΑΣΤΟΡΙΑΣ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Η προβολή επιστημονικών θεμάτων από τα ελληνικά ΜΜΕ : Η κάλυψή τους στον ελληνικό ημερήσιο τύπο Σαραλιώτου

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

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

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

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

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

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

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

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

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

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

Instruction Execution Times

Instruction Execution Times 1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables

Διαβάστε περισσότερα

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

I haven t fully accepted the idea of growing older

I haven t fully accepted the idea of growing older I haven t fully accepted the idea of growing older 1 Peter Authorship Questions Πέτρος ἀπόστολος Ἰησοῦ Χριστοῦ ἐκλεκτοῖς παρεπιδήμοις διασπορᾶς Πόντου, Γαλατίας, Καππαδοκίας, Ἀσίας 1 Peter Authorship

Διαβάστε περισσότερα

Η ΠΡΟΣΩΠΙΚΗ ΟΡΙΟΘΕΤΗΣΗ ΤΟΥ ΧΩΡΟΥ Η ΠΕΡΙΠΤΩΣΗ ΤΩΝ CHAT ROOMS

Η ΠΡΟΣΩΠΙΚΗ ΟΡΙΟΘΕΤΗΣΗ ΤΟΥ ΧΩΡΟΥ Η ΠΕΡΙΠΤΩΣΗ ΤΩΝ CHAT ROOMS ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ Ι Ο Ν Ι Ω Ν Ν Η Σ Ω Ν ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : ΑΤΕΙ Ιονίων Νήσων- Λεωφόρος Αντώνη Τρίτση Αργοστόλι Κεφαλληνίας, Ελλάδα 28100,+30

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

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

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

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 9: Inversion Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons εκτός

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

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011

ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011 Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι

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

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΝΟΜΙΚΗ ΣΧΟΛΗ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΙΣΤΟΡΙΑΣ ΦΙΛΟΣΟΦΙΑΣ ΚΑΙ ΚΟΙΝΩΝΙΟΛΟΓΙΑΣ ΤΟΥ ΔΙΚΑΙΟΥ Διπλωματική εργασία στο μάθημα «ΚΟΙΝΩΝΙΟΛΟΓΙΑ ΤΟΥ ΔΙΚΑΙΟΥ»

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

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0. DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec

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

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

ΑΓΓΛΙΚΑ IV. Ενότητα 8: Analysis of Consumerism and Consumers Rights. Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών

ΑΓΓΛΙΚΑ IV. Ενότητα 8: Analysis of Consumerism and Consumers Rights. Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Ενότητα 8: Analysis of Consumerism and Consumers Rights Ιφιγένεια Μαχίλη Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπως εικόνες, που

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

Solutions to Exercise Sheet 5 Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X

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

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

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

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

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

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

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

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

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

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

Chapter 3: Ordinal Numbers

Chapter 3: Ordinal Numbers Chapter 3: Ordinal Numbers There are two kinds of number.. Ordinal numbers (0th), st, 2nd, 3rd, 4th, 5th,..., ω, ω +,... ω2, ω2+,... ω 2... answers to the question What position is... in a sequence? What

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

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΜΗΧΑΝΙΚΩΝ Η/Υ & ΠΛΗΡΟΦΟΡΙΚΗΣ. του Γεράσιμου Τουλιάτου ΑΜ: 697

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΜΗΧΑΝΙΚΩΝ Η/Υ & ΠΛΗΡΟΦΟΡΙΚΗΣ. του Γεράσιμου Τουλιάτου ΑΜ: 697 ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΜΗΧΑΝΙΚΩΝ Η/Υ & ΠΛΗΡΟΦΟΡΙΚΗΣ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΣΤΑ ΠΛΑΙΣΙΑ ΤΟΥ ΜΕΤΑΠΤΥΧΙΑΚΟΥ ΔΙΠΛΩΜΑΤΟΣ ΕΙΔΙΚΕΥΣΗΣ ΕΠΙΣΤΗΜΗ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑ ΤΩΝ ΥΠΟΛΟΓΙΣΤΩΝ του Γεράσιμου Τουλιάτου

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

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

Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα

Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα [ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη

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

2nd Training Workshop of scientists- practitioners in the juvenile judicial system Volos, EVALUATION REPORT

2nd Training Workshop of scientists- practitioners in the juvenile judicial system Volos, EVALUATION REPORT 2nd Training Workshop of scientists- practitioners in the juvenile judicial system Volos, 26-6-2016 Can anyone hear me? The participation of juveniles in juvenile justice. EVALUATION REPORT 80 professionals

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

ΦωτίστεΤονΚόσμο. ΠΡΩΤΟΒΟΥΛΙΑ ΧΡΙΣΤΟΥΓΕΝΝΩΝ Νοεμβρίου Δεκεμβρίου 2017

ΦωτίστεΤονΚόσμο. ΠΡΩΤΟΒΟΥΛΙΑ ΧΡΙΣΤΟΥΓΕΝΝΩΝ Νοεμβρίου Δεκεμβρίου 2017 ΦωτίστεΤονΚόσμο ΠΡΩΤΟΒΟΥΛΙΑ ΧΡΙΣΤΟΥΓΕΝΝΩΝ 2017 24 Νοεμβρίου -- 25 Δεκεμβρίου 2017 ΘΕΜΑ «Καθώς ακολουθούμε το παράδειγμα του Σωτήρος και ζούμε όπως ζούσε Εκείνος και όπως δίδασκε, αυτό το φως θα καίει μέσα

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

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ 0 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Επιβλέπων Καθηγητής: Δρ. Βασίλειος Ραφτόπουλος ΔΙΕΡΕΥΝΗΣΗ ΤΗΣ ΣΤΑΣΗΣ ΤΩΝ ΝΟΣΗΛΕΥΤΩΝ ΓΙΑ ΤΗΝ ΕΥΘΑΝΑΣΙΑ Ονοματεπώνυμο:

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

ΑΓΓΛΙΚΑ ΙΙΙ. Ενότητα 12b: The Little Prince. Ζωή Κανταρίδου Τμήμα Εφαρμοσμένης Πληροφορικής

ΑΓΓΛΙΚΑ ΙΙΙ. Ενότητα 12b: The Little Prince. Ζωή Κανταρίδου Τμήμα Εφαρμοσμένης Πληροφορικής Ενότητα 12b: The Little Prince Τμήμα Εφαρμοσμένης Πληροφορικής Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπως εικόνες, που υπόκειται

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

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support

105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support 105 Verses in the Critical Greek Text of the New Testament Without Any Manuscript Support These notes were sent to me by Dr. Maurice Robinson and detail 105 places in the Critical Greek Text of the New

Διαβάστε περισσότερα

Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author.

Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author. Κάθε γνήσιο αντίγραφο φέρει υπογραφή του συγγραφέα. / Each genuine copy is signed by the author. 2012, Γεράσιμος Χρ. Σιάσος / Gerasimos Siasos, All rights reserved. Στοιχεία επικοινωνίας συγγραφέα / Author

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

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

( ) 2 and compare to M.

( ) 2 and compare to M. Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

F-TF Sum and Difference angle

F-TF Sum and Difference angle F-TF Sum and Difference angle formulas Alignments to Content Standards: F-TF.C.9 Task In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when

Διαβάστε περισσότερα