The Egyptian Golden Ring with Lapis Lazuli Inscribed Scarab at the Benaki Museum. Was it the Property of a Libyan Pharaoh of Dynasty XXII?

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "The Egyptian Golden Ring with Lapis Lazuli Inscribed Scarab at the Benaki Museum. Was it the Property of a Libyan Pharaoh of Dynasty XXII?"

Transcript

1 AMANDA-ALICE MARAVELIA The Egyptian Golden Ring with Lapis Lazuli Inscribed Scarab at the Benaki Museum. Was it the Property of a Libyan Pharaoh of Dynasty XXII? RINGS ARE normally circular bands used to decorate ears, toes, noses, or, most often, fingers. A finger ring has traditionally been worn for various reasons. It may have a symbolic meaning 1 (as a wedding or a consecration ring); it may identify the wearer or indicate rank or authority (as a signet-ring); it may be thought to have magic powers (as an amuletic ring); or it may be worn merely as an ornament. In ancient Egypt signet-rings 2 bearing carved scarabs 3 (beetles) or engraved hieroglyphs developed from seals carried on cords and were particularly common. 4 The small size of the rings (and of the scarabs) meant that they could travel and be travelled, as is evident from the great number of Egyptian rings uncovered around the Mediterranean, in Crete, Cyprus, Phœnicia, Scythia, Sardinia, Meroe, and elsewhere. 5 The Benaki Museum premises house a small Egyptian collection 6 with several interesting objects, including a faience group 7 which contains finds dating from the Pharaonic period itself. One of the unique non-faience Egyptian objects at the Museum is a charming small ring bearing a scarab (figs 1 a-c), which is examined in detail in this paper: Category: Finger ring with revolving scarab mounted as swivel in funda (Inv. no. B 7335). Typology (Ring / Scarab): Type II [Keel 1995 (n. 4) ] / Type HC.11( ) [?] EP.27( ) SIDE 27( ) [Rowe 1936 (n. 4) cited in Keel 1995 (n. 4) 42, 45, 53 (respectively)]. Date: TIP, Dynasty XXII, belonging to Sheshonq I or II or to somebody of their retinue. Provenance: Egypt (unspecified details). Acquisition: Donation by Lucas Benaki (April 1969). Materials: Gold and lapis lazuli. Weight: 3.4 gr. Colour: Golden metal annulus and lapis blue scarab. Dimensions: H scarab = 1.40 cm, L scarab = 0.90 cm, W scarab = 0.30 cm; D ring, mean = 2.20 cm. Preservation: Quite satisfactory. Hieroglyphic inscription in a rather moderate preservation. Similar Objects: MFA 51.59; BM EA & 57698; Museo Egizio (Firenze), 2790 & 2791; Castellani Collection 335; Newberry (n. 4) 93 & fig. 109; Matouk (n. 4) , & figs 754, , 770, ; Cagliari 21912; Carthage Museum [1190]. Technique: Incision (scarab); hammering (ring). The object studied is a golden (signet-)ring with a lapis lazuli scarab, which is enclosed in a golden funda (in order to protect its edges from possible injuries), and which in turn is mounted as swivel on the (relatively thin) ring by means of perforation threaded with a separate golden wire, the ends of which are tightly wound round the hoop. This type of mounting appears first during Dynasty XIII 8 and continues to be used during the SIP, NK and into the TIP. 9 The hoop of the ring, whose dimensions fit a man s rather than a woman s fingers, is slightly distorted (figs 1 a-c). The funda (or bezel) that holds the scarab in place has the shape of a small cartouche 10 and is made of two oval frames, tightly attached one on the other (fig. 1c). The thinner golden perforation wire passes through the scarab (following the longitudinal 4,

2 A M A N D A - A L I C E M A R A V E L I A Fig. 1 a-c. (a) The scarab on the ring B 7335, (b) the inscription on the sphragistic surface of the scarab on the same ring, (c) the same ring in profile (photo: K. Manolis). Fig. 2 a-c. (a) Linear drawing of the inscription on the scarab of the ring B 7335, (b) idealized drawing of the inscription on the scarab by the author, (c) linear drawing of the inscription on the scarab of the similar ring MFA (after: E. L. B. Terrace, Ancient Egyptian Jewellery in the Horace L. Mayer Collection, AJA 67/3 [1963] pl. 58 figs 18-19) (drawings: author and Katerina Mavragani). axis) by means of two holes/openings at both ends of the scarab, which are perimetrically covered by two small circular annulets, thus giving it a neater appearance. The scarab carries an incised short hieroglyphic inscription on the sphragistic surface (lower side), while on the upper side the solar creature is depicted in detail, showing clearly all its anatomical details: head, eyes, plates, vertex, clypeus, forelegs, prothorax, elytra and suture (denoted by shallow incisions), the middle and hind legs. 11 It has been argued that the Sn-ring is related to the eternity god, whose notched palm-branch sign (symbolising «years») forms the base; 12 in this aspect the scarab, engraved and «protected» by the oval funda, is related not only to eternity and royal protection, but also to the idea of resurrection and eternal life. Hence, a scarab-ring would be the perfect bearer of this particular symbolism. This gives us a first hint as to the ring s possible owner, whose identity will be founded on the study of its hieroglyphic inscription. In fact this object is not a signet-ring 13 [anc. Eg.: xt m, Dba (w) t; Copt.: TBbe, twb] per se, but rather a bezel-ring of amuletic character, bearing a New Year s inscription for prosperity. This object clearly evokes a double protection for the bearer: the oval cartouche, protectively enclosing a king s name; and the solar regeneration symbolism, relevant also to the beginning of a happy New Year. 14 The inscription on the sphragistic surface of the scarab apparently goes like this: Mwt wp rnpt nfr ^ASA<nq> [= (May) Mñt open a happy New Year 15 (for the Pharaoh) Shesho<nq>! ]. At this point, we have to consider two questions: (i) is this the actual inscription or not, as the partially damaged surface of the back of the scarab renders the reading of the last line somewhat problematic; (ii) if this is the actual inscription, then do we have any clues as to which of the most important pharaohs of Dynasty XXII with this name 16 it refers? Let us examine first of all what is certain about the inscription. It is typical of a New Year s object, beginning with an evocation to Mñt 17, the goddess of Thebes, consort of Amñn and mother of Khonsñ, to offer a happy New Year to the person whose name is in question. Let us call this name N. It seems very probable that the reading of the two identical hieroglyphic signs in the lower row is as shown above: SA-SA. However, supposing that this is not the case, what alternative readings of these signs (if any) do we have? A hypothetical rendering could well be: Mwt wp rnpt nfr SA<a>{SA} [= May (Mut) open a happy New Year s beginning]. If so, the word SAa (= beginning) 18 would present a rather peculiar orthography, which implies that the scribe has made two mistakes simultaneously even without taking the anomalous syntax into account by writing it erroneously and by repeating a similar sign SA after the first. 19 This seems a quite impossible speculation and accordingly should be 24 ΜΟΥΣΕΙΟ ΜΠΕΝΑΚΗ

3 The Egyptian Golden Ring with Lapis Lazuli Inscribed Scarab at the Benaki Museum. Was it the Property of a Libyan Pharaoh of Dynasty XXII? rejected. Furthermore, we cannot consider N as being either the word SA (= ordain, predestine) 20 twice repeated, or the word SA(s) (= travel) 21 twice repeated, since the context of the New Year s wish would not justify something like this. Nor can we consider the two signs as being a repetition of the group Hm-kA (= ka-priest), 22 no matter how much they resemble this sign, since the meaning would again not fit the context [namely: Mut, happy New Year s Day; (to the) ka-priest, ka-priest]. Similarly, we must exclude the possibility that the word is a person s (commoner s) name, since no similar entry is found in Ranke s work. 23 Finally, any cryptographic 24 context in this particular inscription must also be excluded. Thus, it seems that the only possible rendering of the inscription is indeed that given in the previous paragraph. It is almost certain that N = ^ASA, and highly probable that N = ^ASA<nq >, referring to the royal name Sheshonq. Now we have to discuss to which of the three pharaohs with the same prenomen 25 it belongs. In certain instances the name of Sheshonq II is written simply as ^ASA, without the final two hieroglyphic signs, using this minimal orthography. 26 However, there are some scarabs of Sheshonq I where the name of the king is also written merely as ^ASA, omitting the final two signs. 27 As for Sheshonq III, the known concomitant scarabs, to the best of our knowledge, show his full name. 28 Thus the ring appears to name either Sheshonq I or Sheshonq II, though which of them cannot be decided with absolute certainty. Last but not least, stylistic reasons 29 imply that it is probably a ring of the Libyan Dynasty. The scarab and its basic anatomical lines are rendered in a particular manner which is reminiscent of the TIP style (cf. also a similar ring: MFA 51.59, already referred to). 30 Additionally, comparison of the scarab with another, also made of lapis lazuli but this time set on a golden bracelet of Sheshonq II which imitates a swivel finger ring, 31 corroborates this evidence. The ring examined here, bearing a royal name, is made of gold and has a finely worked inscribed scarab; however, it is not particularly opulent and it is not made of massive solid gold. It may perhaps have belonged to either Sheshonq I or Sheshonq II, but it seems safer to surmise that it was probably given by the king as a reward to one of his officers or priests. Table 1 shows some interesting parallels to this finger-ring. 32 Dr Amanda-Alice Maravelia Egyptologist/Archaeoastronomer a_maravelia@hotmail.com NOTES * May I thank Prof. Dr Angelos Delivorrias, director of the Benaki Museum, together with the Museum staff, for permitting and encouraging the study and publication of this ring, as well as of the faience collection. Warm thanks for various pieces of information and some bibliographical references are also due to: Dr D. Sweeney (Tel Aviv University), Dr B. Brandl (Israel Antiquities Authority, Jerusalem), Dr N. Guilhou (Université Montpellier III); Mr Ch. Naunton, M. Phil. (The Egypt Exploration Society, London); Prof. Dr E. Cruz Uribe (Department of History, Northern Arizona University). 1. See J. Chevalier A. Gheerbrant, Dictionary of Symbols (UK ) 805ff; G. F. Kunz, Rings for the Finger (New York Philadelphia ). 2. For the symbolism of rings (evoking endless cycles of eternity) in the ancient Egyptian context, see M. Lurker, The Gods and Symbols of Ancient Egypt: An Illustrated Dictionary (London ) 101; R. H. Wilkinson, Reading Egyptian Art: A Hieroglyphic Guide to Ancient Egyptian Painting and Sculpure (London 1992) (cf. also op. cit ). On ancient Egyptian jewellery in general, see C. Andrews, Ancient Egyptian Jewellery (London 1990): for finger rings, cf ; id., in: H. Tait (ed.), Seven Thousand Years of Jewellery (London 1986) 33-36, 42-47; C. Aldred, Jewels of the Pharaohs (London 1971); A. Wilkinson, Ancient Egyptian Jewellery (London 1971); M. Vilímková et al., Egyptian Jewellery (London 1969). 3. For the symbolism of scarabs (scarabeus sacer L.), see LÄ V (1984) s.v. Skarabäus; J. Ward, The Sacred Beetle: A Popular Treatise on Egyptian Scarabs in Art and History (London 1902); C. Andrews, Amulets of Ancient Egypt (London 1994) 50-59; for their funerary use, see M. Malaise, Les Scarabées de cœur dans l Égypte ancienne (Bruxelles 1978); A.-A. Maravelia, Η μαγεία στην αρχαία Αίγυπτο: Μεταφυσική πεμπτουσία της χώρας των θεών (Athens 2003) See, for instance, P. A. Newberry, Egyptian Antiquities: Scarabs. An Introduction to the Study of Egyptian Seals and Signet Rings (London 1906) 62ff. For scarabs, see also W. M. F. Petrie, Scarabs and Cylinders with Names, Illustrated by the Egyptian Collection in University College I-III (London 1917); id., Buttons and Design 4,

4 A M A N D A - A L I C E M A R A V E L I A Scarabs (London 1925); id., Amulets (Warminster ) pls VII-IX; id., Historical Scarabs: A Series of Drawings from the Principal Collections (New York ); C. Blankenberg-van Delden, The Large Commemorative Scarabs of Amenhotep III (Leiden 1969); B. Jaeger, Essai de classification et datation des scarabées Menkheperre (Fribourg 1982); id., Les scarabées à noms royaux du Museo Civico Archeologico de Bologna (Bologna 1993); F. S. Matouk, Corpus du Scarabée égyptien: I. Les scarabées royaux; II. Analyse thématique (Beyrouth ); A. Rowe, A Catalogue of Egyptian Scarabs, Scaraboids, Seals and Amulets in the Palestine Archaeological Museum (Cairo 1936); H. R. Hall, Catalogue of Egyptian Scarabs, & c., in the British Museum: I. Scarabs (London 1913); id., Scarabs (London 1929); E. Hornung E. Stählin, Skarabäen und andere Siegelamulette aus Basler Sammlungen (Mainz 1976); W. A. Ward O. Tufnell, Studies on Scarab Seals I-II (UK ); G. T. Martin, Scarabs, Cylinders and Other Ancient Egyptian Seals (Warminster 1985); for scarabs and rings with scarabs, see also O. Keel, Corpus der Stempelsiegel Amulette aus Palästina/Israel: Von den Anfängen bis zur Perserzeit. Einleitung (Freiburg-Göttingen 1995); id., Corpus der Stempelsiegel-Amulette aus Palästina/Israel: Von den Anfängen bis zur Perserzeit. Katalog Band I: Von Tell Abu Farag bis Atlit (Freiburg-Göttingen 1997). 5. See, for instance, F. H. Marshall, Catalogue of the Finger Rings, Greek, Etruscan, and Roman in the Departments of Antiquities, British Museum (London ; ); E. D. Reeder, Scythian Gold (New York 1999); K.-H. Priese, The Gold of Meroe (Mainz 1993); D. Sweeney, A Lion Hunt Scarab and Other Egyptian Objects from the Late Bronze Fortress at Jaffa, Tel Aviv 30 (2003) 54-65; A. Karetsou (ed.): Krete Aigyptos: Cultural Links through Three Millennia: Catalogue (Herakleion 2000) 304 ff; N. C. Stampolidis (ed.), Sea Routes From Sidon to Huelva: Interconnections in the Mediterranean, 16th-6th c. B.C. (Athens 2003) See, for instance, A.-A. Maravelia, Two Faience Shabtis from the Egyptian Collection at the Benaki Museum, Mouseio Benaki 2 (2002) n See n. 6 and A.-A. Maravelia, Ancient Egyptian Inscribed Faience Objects from the Benaki Museum in Athens, 1, JNES 61/ 2 (2002) See Newberry (n. 4) See for example a similar ring of Sheshonq III (MFA # 51.59) in E. L. B. Terrace, Ancient Egyptian Jewellery in the Horace L. Mayer Collection, AJA 67 (1963) 274 pl. 58: figs On the (solar/cosmic and protective/encircling) symbolism of royal cartouches, see Wilkinson (n. 2) and Lurker (n. 2) 38-39; EG, For a detailed entomological description of the insect, cf. Keel, Corpus 1995 (n. 4) 20ff. 12. See R.H. Wilkinson (n. 2) See, for instance, Wb. III, 350 ff & Wb. V, 566; CD, 199, Among the inscribed faience objects at the Benaki Museum is a New Year s jar, already published by the author [see Maravelia (n. 7) figs 2 a-c], bearing an inscription mentioning Amñn and the New Year s festival. On the importance of New Year s Day to the ancient Egyptian mind, cf. also the reference to the New Year and the heliacal rising of Sirius/SÜthis in the context of ancient Egyptian Love Poems [see e.g.: A.-A. Maravelia, ptri.st mi %pdt xay m-hat rnpt nfrt: Astronomical and Cosmovisional Elements in the Corpus of Ancient Egyptian Love Poems, Lingua Aegyptia 11 (2003) ; pchester Beatty I, v, C1, 1-2]. For the expression rnpt nfr, in opposition to rnpt gab (cf. panastasi IV ), see P. Germond, Les invocations à la bonne année au temple d Edfou, (= Aegyptiaca Helvetica 11, Genève 1986) On some faience rings with New Year s Day wishes, dating from the LP, see B. Latellier, Un souhait de bonne année en faveur d une reine kouchite, RdE 29 (1977) 43-52; pl. 1; L. TÞrÞk, Meroe City: An Ancient African Capital (London 1997) : Inscr. 56a-56d; fig. 122: Inscr. 56a-56d. 16. Of the Libyan Dynasty XXII, we know three important Pharaohs with this nomen: Sheshonq I [ BCE] = (@D xpr Ra, %tp n Ra) (^ASAnq, Mry Imn); Sheshonq II [c. 890?-883 BCE] = (@qa xpr Ra, %tp n Ra) (^ASA<nq>, Mry Imn); Sheshonq III [ BCE] = (Wsr MAat Ra, %tp n Ra) (^ASAnq, Mry Imn). On this, see A.-A. Maravelia, Xρονολογικό μνημόνιο της αρχαίας αιγυπτιακής ιστορίας και των φαραωνικών δυναστειών / Η περίπτωση του φλαμινιανού οβελίσκου, Παρνασσός 45 (2003) , esp For a detailed history of the TIP and the «Sheshonqide» kings, see K. A. Kitchen, The Third Intermediate Period in Egypt ( BC) (Warminster ) , We discard the possibility that the ring under study may date from a slightly later era, e.g. that of Sheshonq V [ BCE] = (Ax xpr Ra) (^AS- Anq). According to Kitchen (op. cit., 88), there is some doubt on the existence of Sheshonq IV [c BCE] = (Wsr MAat Ra, Mry Imn) (^A^Anq); additionally (op. cit., , n. 639), supposed scarabs of Sheshonq V from Palestine are probably not explicitly his. 17. For Mñt, see Lurker (n. 2) 82-83; H. te Velde, Towards a Minimal Definition of the Goddess Mut, JEOL 8/26 ( ) 3-9. In other objects different deities are invoked, e.g. in the Benaki Museum New Year s jar (B18.258), for which see Maravelia (n. 7) 87-88, figs 2a-c, where Amñn is named. Returning to Mñt, we must add that her name is met in some scarabs: see, for instance, Petrie, Buttons (n. 4) 12, 21, 23, 28, pls. IX: # 318, XII: # 700, XIII: # 796, 798, XV: # 1046, 1046 A. 18. See Wb. IV, 406ff; CD, The basic archetype for this very ancient sign (already met in the PT ) is M8 (see EG, 480); in the context of this ring it is rather M8J (less possibly M8H or M8L); cf. N. Grimal J. Hallof D. van der Plas et al. (eds), Hieroglyphica: Sign List (Utrecht- Paris 2000) 1 M Cf. Wb. IV, 402; CD, Cf. Wb. IV, 412; CD, Cf. Wb. III, 90; CD, 169. Further similarly written words exist, but their meaning does not fit the context (cf. e.g.: CD, 261: SASAt = necklace; CT IV, 384a: SAs = escape; CT VII, 397u: SASA = a snake species; & c.). See also Wb. IV, See H. Ranke, Die ägyptischen Personennamen I-III (Hamburg ). There is only a simple form ^A, with sign Hn (M2) as taxogram, dating from the NK (cf. op. cit. I, 12). 26 ΜΟΥΣΕΙΟ ΜΠΕΝΑΚΗ

5 The Egyptian Golden Ring with Lapis Lazuli Inscribed Scarab at the Benaki Museum. Was it the Property of a Libyan Pharaoh of Dynasty XXII? 24. For some cryptographic inscriptions, see F. Crevatin, Minor Egyptian Inscriptions (mainly Cryptographic), GM 195 (2003) See n. 16, supra. This name corresponds to the Hellenic names Σέσωγχις/Σεσώγχωσις of ManethÜn. 26. Cf. J. von Beckerath, Handbuch der Ägyptischen Königsnamen (München 1984) 186; S. Quirke, Who were the Pharaohs: A History of their Names with a List of Cartouches (London 1990) 68; Maravelia (n. 16) See also Petrie, Scarabs (n. 4) 1ff. pl. L: See Matouk (n. 4) , 197, # 754, There is another scarab, bearing a New Year s wish for Kara ma (KA-Ra-ma), the wife of Sheshonq II, where the name of the hereditary prince Sheshonq is fully written (see op. cit., 131, 198: # 770; Newberry [n. 4] pl. XL,8]: wp PtH rnpt nfr n (i)r(y) -pa{t} ^ASAnq, maa-xrw mwt KA-Ra-ma [= (May) Ptah open a happy year for (the) hereditary Prince Sheshonq, justified <by his> mother Kara ma ]. See, finally, M.- A. Bonhême, Les noms royaux dans l Égypte de la Troisième Période Intermédiaire (Le Caire 1987) 124; id., Les Chechonquides: Qui, combien?, BSFE 134 (1995) See Terrace (n. 9) 274 pl. 58: figs 19-20; Matouk (n. 4) 131, 198 # Cf. also Petrie, Scarabs (n. 4) passim pl. L: See, however, Bonhême (n. 27) 124 (for a few instances of the minimal orthography, though not on scarabs). The same holds for Sheshonq V (op. cit. 139) and Sheshonq VI, whose existence is dubious (see Kitchen [n. 16] 87, 88). 29. On the dating and stylistic criteria for scarabs and rings, see Ward Tufnell I (n. 4) 20-35; Keel, Corpus 1995 (n. 4) 39-61, See Terrace (n. 9) 274, pl. 58 figs The inscription on that ring goes like this: Mry-Imn, ^ASAnq, za BAstt; Imn-Ra wp rnpt nfr [= (The) beloved of Amñn, Sheshonq, (the) son of Bastet; (may) Amñn RÂ open a happy year (to the king)]. 31. For this exquisite piece from the Egyptian Museum in Cairo (JE 72189), see Andrews, Ancient Egyptian Jewellery (n. 2) 148 fig For more parallels, see Reeder (n. 5) 171 # 60; 217 # 98; Hall, Scarabs (n. 4) 2 and # 2-3, pls II # 2-3, pl. III: # 2936, 4917; Stampolidis (n. 5) 584 and # 1190, for the ring from the National Carthage Museum with no inventory number (together with additional relevant bibliography); the CD-Rom by D. van der Plas (ed.), Egyptian Treasures in Europe. I: Highlights (Utrecht 1999), for the rings 2790 and 2791 of the Archaeological Museum in Florence; Marshall (n. 5) xxxviii (for two bezel-rings with scarabs, namely 1004 & 1007), 60 (for ring 335 of the Castellani Collection). AΜAΝΤΑ-AΛΙΚΗ MΑΡΑΒΕΛΙΑ Ήταν ο χρυσός δακτύλιος με ενεπίγραφο σκαραβαίο από λαζουρίτη του Μουσείου Μπενάκη προσωπικό αντικείμενο Λίβυου Φαραώ της 22ης Δυναστείας; Στο Μουσείο Μπενάκη φυλάσσεται ένας χρυσός δακτύλιος με περιστρεφόμενη σφενδόνη και ενεπίγραφο σκαραβαίο από λαζουρίτη. Ο σκαραβαίος περικλείεται από πλαίσιο (funda) σε ελλειψοειδές σχήμα φαραωνικής δέλτου και είναι προσδεδεμένος στον δακτύλιο με λεπτό μεταλλικό σύρμα που διαπερνά τον κάθετο άξονά του. Κατά πάσα πιθανότητα το αντικείμενο αυτό, το οποίο φέρει χαρακτηριστική ευχή για το Νέο Έτος (wp rnpt n fr), χρονολογείται από την 22η Δυναστεία, ενώ μπορεί να ανήκε, είτε στον Φαραώ Σέσωγχι Ι ( ΠΚΕ), είτε στον Φαραώ Σέσωγχι ΙΙ (περ. 890;-883 ΠΚΕ). Υποθέτουμε ότι το πιθανότερο είναι πως ήταν δώρο ενός από τους δύο αυτούς φαραώ προς κάποιον υπήκοό τους (αξιωματούχο ή ιερέα). Η ευχή αναφέρεται στη θεά Μουτ (σύζυγο του Άμμωνα), την οποία επικαλείται ούτως ώστε να χαρίσει στον Σέσωγχι ευτυχισμένη (πρωτο)χρονιά. Στην εργασία αυτή μελετάται ακροθιγώς η επιγραφή και αποκλείονται συγκεκριμένες πιθανές αποδόσεις της (εξαιτίας των δυσανάγνωστων ιερογλυφικών στην τελευταία γραμμή της), ενώ ταυτόχρονα δίνεται πλήρης περιγραφή του σφυρήλατου δακτυλίου. Η χρήση του δακτυλίου εν είδει περιάπτου ευημερίας και προστασίας θα χάριζε στον κάτοχο (πρεσβείαις της Μουτ) ευτυχία και ευμάρεια για τη νέα χρονιά. Ο σκαραβαίος, ηλιακό σύμβολο αναγέννησης και ανάστασης, κατείχε στη σκέψη των Αιγυπτίων εξέχουσα θέση ως αρχέτυπο. Δακτύλιοι όπως αυτός, αλλά και απειράριθμοι σκαραβαίοι, έχουν εντοπισθεί σε πολλά σημεία ανά τη λεκάνη της Μεσογείου, γεγονός που καταδεικνύει τη φήμη των αιγυπτιακών περιάπτων κατά την αρχαιότητα. Το συγκεκριμένο αντικείμενο φέρει εγχάρακτη επιγραφή στην οποία το όνομα Σέσωγχις δεν είναι γραμμένο εξολοκλήρου (απουσιάζουν τα δύο τελευταία ιερογλυφικά), γεγονός σύνηθες, τόσο για τον Φαραώ Σέσωγχι Ι, όσο και για τον Σέσωγχι ΙΙ. 4,

6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

«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

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

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

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

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

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

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

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

Advanced Subsidiary Unit 1: Understanding and Written Response

Advanced Subsidiary Unit 1: Understanding and Written Response Write your name here Surname Other names Edexcel GE entre Number andidate Number Greek dvanced Subsidiary Unit 1: Understanding and Written Response Thursday 16 May 2013 Morning Time: 2 hours 45 minutes

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

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

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

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

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

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

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

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

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

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ Μελέτη των υλικών των προετοιμασιών σε υφασμάτινο υπόστρωμα, φορητών έργων τέχνης (17ος-20ος αιώνας). Διερεύνηση της χρήσης της τεχνικής της Ηλεκτρονικής Μικροσκοπίας

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

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

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

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

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013 Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering

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

Paper Reference. Paper Reference(s) 1776/04 Edexcel GCSE Modern Greek Paper 4 Writing. Thursday 21 May 2009 Afternoon Time: 1 hour 15 minutes

Paper Reference. Paper Reference(s) 1776/04 Edexcel GCSE Modern Greek Paper 4 Writing. Thursday 21 May 2009 Afternoon Time: 1 hour 15 minutes Centre No. Candidate No. Paper Reference(s) 1776/04 Edexcel GCSE Modern Greek Paper 4 Writing Thursday 21 May 2009 Afternoon Time: 1 hour 15 minutes Materials required for examination Nil Paper Reference

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

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds! MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.

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

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

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

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

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

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

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

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

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

Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ

Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ ΠΗΓΩΝ Θεόδωρος Μαρκόπουλος University of Uppsala thodorismark@yahoo.gr Abstract This paper discusses methodological

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

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

Lecture 15 - Root System Axiomatics

Lecture 15 - Root System Axiomatics Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the

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

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

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

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

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

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

ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΛΛΗΝΙΔΕΣ

ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΛΛΗΝΙΔΕΣ ΕΛΛΗΝΙΔΕΣ ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΚΔΟΣΗ ΤΗΣ ΣΟΡΟΠΤΙΜΙΣΤΙΚΗΣ ΕΝΩΣΗΣ ΕΛΛΑΔΟΣ - ΤΕΥΧΟΣ Νο 110 - Δ ΤΡΙΜΗΝΟ 2014 Το πρώτο βραβείο κέρδισε η Ελλάδα για την φωτογραφία Blue + Yellow = Green στον διαγωνισμό 2014 του

Διαβάστε περισσότερα

ΜΕΤΑΠΤΥΧΙΑΚΗ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ «ΘΕΜΑ»

ΜΕΤΑΠΤΥΧΙΑΚΗ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ «ΘΕΜΑ» ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ ΣΧΟΛΗ ΑΝΘΡΩΠΙΣΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΤΗΣ ΠΡΟΣΧΟΛΙΚΗΣ ΑΓΩΓΗΣ ΚΑΙ ΤΟΥ ΕΚΠΑΙΔΕΥΤΙΚΟΥ ΣΧΕΔΙΑΣΜΟΥ Π.Μ.Σ. «ΠΕΡΙΒΑΛΛΟΝΤΙΚΗ ΕΚΠΑΙΔΕΥΣΗ» ΜΕΤΑΠΤΥΧΙΑΚΗ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ «ΘΕΜΑ» «Εφαρμογή

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

FACE VALUE: THE POWER OF IMAGES AT APHRODISIAS

FACE VALUE: THE POWER OF IMAGES AT APHRODISIAS FACE VALUE: THE POWER OF IMAGES AT APHRODISIAS H O W D I G I T A L R E S O U R C E S C A N T R A N S F O R M O U R U N D E R S T A N D I N G O F P A L A E O G R A P H Y KEY QUESTIONS: What elements of

Διαβάστε περισσότερα

þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼

þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼ Neapolis University HEPHAESTUS Repository School of Health Sciences http://hephaestus.nup.ac.cy Master Degree Thesis 2015 þÿ»±íº »¹ Áà  : É º±¹ Ä þÿ Á³ Ä Å : ¼¹± ºÁ¹Ä¹º ±À Ä ¼ þÿ Ä Æ Á Â, Á ÃÄ Â þÿ Á̳Á±¼¼±

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

6.3 Forecasting ARMA processes

6.3 Forecasting ARMA processes 122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

LESSON 16 (ΜΑΘΗΜΑ ΔΕΚΑΕΞΙ) REF : 102/018/16-BEG. 4 March 2014

LESSON 16 (ΜΑΘΗΜΑ ΔΕΚΑΕΞΙ) REF : 102/018/16-BEG. 4 March 2014 LESSON 16 (ΜΑΘΗΜΑ ΔΕΚΑΕΞΙ) REF : 102/018/16-BEG 4 March 2014 Family η οικογένεια a/one(fem.) μία a/one(masc.) ένας father ο πατέρας mother η μητέρα man/male/husband ο άντρας letter το γράμμα brother ο

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

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ Πτυχιακή εργασία ΓΝΩΣΕΙΣ ΚΑΙ ΣΤΑΣΕΙΣ ΝΟΣΗΛΕΥΤΩΝ ΠΡΟΣ ΤΟΥΣ ΦΟΡΕΙΣ ΜΕ ΣΥΝΔΡΟΜΟ ΕΠΙΚΤΗΤΗΣ ΑΝΟΣΟΑΝΕΠΑΡΚΕΙΑΣ (AIDS) Αλέξης Δημήτρη Α.Φ.Τ: 20085675385 Λεμεσός

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

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

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

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

Chapter 29. Adjectival Participle

Chapter 29. Adjectival Participle Chapter 29 Adjectival Participle Overview (29.3-5) Definition: Verbal adjective Function: they may function adverbially or adjectivally Forms: No new forms because adverbial and adjectival participles

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

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

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

ΣΧΕΔΙΑΣΜΟΣ ΔΙΚΤΥΩΝ ΔΙΑΝΟΜΗΣ. Η εργασία υποβάλλεται για τη μερική κάλυψη των απαιτήσεων με στόχο. την απόκτηση του διπλώματος

ΣΧΕΔΙΑΣΜΟΣ ΔΙΚΤΥΩΝ ΔΙΑΝΟΜΗΣ. Η εργασία υποβάλλεται για τη μερική κάλυψη των απαιτήσεων με στόχο. την απόκτηση του διπλώματος ΣΧΕΔΙΑΣΜΟΣ ΔΙΚΤΥΩΝ ΔΙΑΝΟΜΗΣ Η εργασία υποβάλλεται για τη μερική κάλυψη των απαιτήσεων με στόχο την απόκτηση του διπλώματος «Οργάνωση και Διοίκηση Βιομηχανικών Συστημάτων με εξειδίκευση στα Συστήματα Εφοδιασμού

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

Modbus basic setup notes for IO-Link AL1xxx Master Block

Modbus basic setup notes for IO-Link AL1xxx Master Block n Modbus has four tables/registers where data is stored along with their associated addresses. We will be using the holding registers from address 40001 to 49999 that are R/W 16 bit/word. Two tables that

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

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

Commutative Monoids in Intuitionistic Fuzzy Sets

Commutative Monoids in Intuitionistic Fuzzy Sets Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,

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

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

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

Notes on the Open Economy

Notes on the Open Economy Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.

Διαβάστε περισσότερα

1) Abstract (To be organized as: background, aim, workpackages, expected results) (300 words max) Το όριο λέξεων θα είναι ελαστικό.

1) Abstract (To be organized as: background, aim, workpackages, expected results) (300 words max) Το όριο λέξεων θα είναι ελαστικό. UΓενικές Επισημάνσεις 1. Παρακάτω θα βρείτε απαντήσεις του Υπουργείου, σχετικά με τη συμπλήρωση της ηλεκτρονικής φόρμας. Διευκρινίζεται ότι στα περισσότερα θέματα οι απαντήσεις ήταν προφορικές (τηλεφωνικά),

Διαβάστε περισσότερα

ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα

ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα ΖητΗματα υποδοχης και προσληψης του Franz Kafka στην ΕλλΑδα Ζητήματα υποδοχής και πρόσληψης του Franz Kafka στην Ελλάδα Μιχ. Γ. Μπακογιάννης Aspects of Kafka s reception in Modern Greek Literature: The

Διαβάστε περισσότερα