Daniel Kölligan Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "Daniel Kölligan Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται"

Transcript

1 Kadmos 2015; 54(1/2): Daniel Kölligan Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται DOI /kadmos Abstract: The Myc. PN a-o-ri-me-ne may be interpreted as a possessive compound ʻwho has the μένος of the sword, or as ʻwho has the μένος in the swordʼ, if the compounding vowel -i- was still functional as a locative marker, as Ruijgh proposed, or simply as the combination of the two elements [rage] and [sword]. The interpretation is supported by collocations of words for ʻspearʼ and ʻrageʼ in the epic language. At least in the Homeric world the μένος that temporarily resides in a weapon is probably due to the divine influence of Ares whose fury may enter both the weapon and its possessor. Keywords: Personal name, onomastics, phraseology, Ares. 1. In a series of tablets from Pylos recording various offerings, Qa 1296 gives us the name of a priest a-o-ri-me-ne: (1) PY Qa 1296 a-o-ri-me-ne, i-je-re-u *189 [ The priest A-o-ri-menēs... While there is a general agreement that the second member of the compound may be interpreted as /-menēs/ from μένος,1 there has been some hesitation as to the first part; both ὥρα and ἄορ have been suggested, and although most scholars now seem to assume that the latter lies behind the spelling a-o-ri, the semantic relation between the name s compositional members has been interpreted in various ways. After a short review of the doxography (2),2 phraseological evidence will be presented that may help us to understand this relation more precisely (3 4). 1 Cf. Alph.-Gk. εὐμενής, etc., and the numerous names in -μένης in Bechtel Cf. also Aura Jorro Article note: English translations of Greek authors are those of the Loeb series (HUP). *Corresponding author: Daniel Kölligan, Institut für Linguistik, Historisch-Vergleichende Sprachwissenschaft, Universität zu Köln, Köln, d.koelligan@uni-koeln.de.

2 32 Daniel Kölligan 2. While Chadwick and Baumbach (1963) and Baumbach (1971) were reluctant to make a decision between the two options noted above,3 Gallavotti 1961: 163 favoured the interpretation with ἄορ, pointing out that μένος describes a disposizione d animo attiva [...] e specialmente lo spirito battagliero. He compared the compounds μενεπτόλεμος, μενέχαρμος, and μεναίχμης, and the PN Μέναιχμος and Δορυμένης (Polyb. 5.61) and assumed a mutual interference between compounds based on μένος and μένω to stayʼ. Docs. 2 4 gives the name with a question mark as A(h)orimenēs? [ἄορ, μένος]. Lejeune 1972: 251f. also rejected the interpretation with ὥρα one would have to assume that a-o-ri- consists of the negative prefix ἀ- and ὥρα, i.e. ἀωρί ʻuntimelyʼ, which is attested rather late (E.+) but he doubted Gallavotti s explanation, preferring to derive ἄορ from ἀείρω ʻto hangʼ5 instead of the more traditional interpretation as *n sor or *n sr related to Lat. ēnsis ʻswordʼ and Skt. así- ʻid.ʼ6 The latter was taken up again by Ruijgh 1985: 149ff. who assumed ἄορ to show the reflex /or/ of earlier /r / typical for Mycenaean and Aeolic, hence Homeric ἄορ, and proposed to understand a-o-ri-me-ne as ʻqui a de la force (de l élan) dans son armeʼ similar to a-re-i-me-ne/a-re-me-ne ʻqui a de l élan à la guerreʼ.7 Further, García Ramón 2008: has argued that the latter name contains the elements Ἄρης8 and μένος and has drawn attention to the combination of these two nouns in the epic language, cf. (2) ʻthe fury of war/aresʼ: Il μένος Ἄρηος, Od μένος (κρίνηται) Ἄρηος (3) ʻAres kindled his furyʼ: Il τοῦ δ ὄτρυνεν μένος Ἄρης. The idea of ʻraging Aresʼ may also be expressed with the cognate verb μαίνομαι: 3 Cf. Chadwick and Baumbach 1963: 220 ( Aōrimenēs ), 259 ( alternatively from ἄορ + μένος ). Baumbach 1971: 155: a-o-ri-me-ne PY QA 1296 is as likely to be from ἄορ as from ὥρα. 4 Chadwick 1973: For this one might also consider the divine epithet χρυσάωρ which could mean ʻwith a golden swordʼ or ʻwith a golden pendantʼ. Cf. also the name of the tribe Ἀϝοροί on Corcyra. Cf. Beekes 2010: He suspected *n si- to be attested in Myc. PN a-i-qe-u/a h ik wh eu s/ (PY En a-i -qe-wo, Ep a-i-qe-u, etc.), which according to him could be a short form of a compound noun in /-k wh ontās/, i.e. /a h ik wh ontās/. 7 Cf. Ruijgh 1985: On Ares in Mycenaean and classical Greece cf. also Gulizio For a recent etymological proposal (PIE *h 2 reu - to ripʼ, lat. ruere, Ved. subj. ravat will harmʼ, etc.) cf. Willi 2014.

3 Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται 33 (4) Od μαίνεται Ἄρης.9 It was Lejeune who brought into the discussion the interpretation of a-o-ri- as being related to Alph.-Gk. ἦρι ʻin the morningʼ from *ā i eri, which is found also in ἄριστον [ā-] ʻbreakfastʼ < *ai eri-h 1 d-to-. According to him, a-o-ri- would be an ablaut variant of a-e-ri found in the PN a-e-ri-qo-ta (PY An 192.7, 209.6, , Aq 218.5).10 But, as argued by Peters 1980: 32ff., ἦρι is likely to be the contracted form of *ἠερι still found in the adj. ἠέριος ʻearlyʼ and not of *αερ- (> **ᾱρ-). The traditional connection with ἄριστον could only be upheld if one assumed either two pre-forms *āi eri- and *ai eri-, which would be morphologically unsatisfactory and clearly ad hoc, or a metrical lengthening of *ai eri, but the contraction to ἦρι indicates that *ἠερι is not an artificial creation of the epic language. The alternative explanation of *ἠερι as deriving from a locative *h 2 (e)us-er-i > Gk. *awseri (cf. RV us ar-budh- ʻawaking earlyʼ) in turn rules out a connection of a-e-ri- with this root, as one would expect *u to be written in Mycenaean, i.e. <a-we-ri > or <a 4 -e-ri->.11 Hajnal 1992 therefore, discussing the fate of intervocalic PIE *i in Mycenaean and finding no conclusive evidence for a development of *i > h in this position cf. e.g. to-ro-qe-jo-me-no /trok w ei omeno-/ (PY Eq 213.1)12, concluded that the PN a-e-ri-qo-ta is more likely to contain a first member /aheri-/, i.e. a locative or rather compound form with /i/ used as compositional vowel of what in Alph.-Gk. is ἄορ, meaning ʻkilling with the swordʼ. The related name a-ori-me-ne would then show a secondary o-ablaut grade in the suffix taken from the nominative/accusative. As for the meaning of the name, Hajnal prefers an interpretation of derjenige, welcher seinen Sinn auf das Schwert = den Kampf richtet ( the one who directs his mind to the sword = battle ), understanding μένος as ʻstriving, eagerness for actionʼ and comparing it with the meaning of the pf. μέμονα ʻI am eager to do s.th.ʼ. Hence, while it seems that there is a consensus 9 Cf. also Il f. μαίνετο δ ὡς ὅτ Ἄρης ἐγχέσπαλος ἢ ὀλοὸν πῦρ οὔρεσι μαίνηται And he was raging like Ares, wielder of the spear, or as when destructive fire rages among the mountains in the thickets of a deep wood. 10 If one takes, as Lejeune does, the second member of the compound as / k wh ontās/ ʻstriking, killingʼ, a-e-ri-qo-ta could be compared with the Vedic epithet vasar-hā ʻkilling (the demons) in the morningʼ said of the wind (vā ta-) in RV However, vasar-hā may also be derived from hā ʻto moveʼ, i.e. ʻrising earlyʼ which seems to make better sense as an epithet of the wind, cf. Jamison and Brereton 2014: I.284 and Jamison s online commentary, alc.ucla.edu/ (accessed 1/5/2016). 11 This does not leave ἄριστον without explanation, however, as it may be connected with Av. aiiarǝ ʻdayʼ and Goth. air ʻearly, beforeʼ. 12 Cf. Hajnal 1992: 294 for further examples.

4 34 Daniel Kölligan in recent research on the nouns underlying the PN a-o-ri-me-ne, their semantic relationship is still open for discussion. It may therefore be helpful to look for combinations of these elements in later Greek, especially the epic language. 3. As in the case of a-re-(i-)-me-ne, which, as argued by García Ramón 2008 (cf. 2), corresponds to the epic collocations μένος Ἄρηος and Ἄρης μαίνεται and hence combines the notions of [rage] and [war],13 it is possible to point out a similar phraseological equation for a-o-ri-me-ne as the combination of [rage] and [weapon], which in some instances in Homer can be understood as the personification of war (cf. 4). It is found in Homer with words for ʻspearʼ, ἐγχείη and δόρυ, in (5) Il οὐ γὰρ Τυδεΐδεω Διομήδεος ἐν παλάμῃσι / μαίνεται ἐγχείη Δαναῶν ἀπὸ λοιγὸν ἀμῦναι For not in the hands of Diomedes, son of Tydeus, does the spear rage to ward off ruin from the Danaans. (6) 8.110f. ὄφρα καὶ Ἕκτωρ / εἴσεται εἰ καὶ ἐμὸν δόρυ μαίνεται ἐν παλάμῃσιν so that Hector too may know whether my spear, too, rages in my hands.14 Besides these early expressions of the notion the weapon ragesʼ, making the instrument the agent, in later authors the noun for ʻspearʼ may appear as the instrument or object manipulated by a human agent, hence [agent] rages with weaponʼ: (7) Alcm. frg δουρὶ δὲ ξυστῶι μέμανεν Αἶας αἱματῆι τε Μέμνων Ajax raves with sharpened spear and Memnon is thirsty for blood. (8) Bacch εὖτ / ἐν πεδίῳ κλονέω[ν] μαί/νοιτ Ἀχιλλεύς, / λαοφόνον δόρυ σείων whenever Achilles went on his furious rampage in the plain, brandishing his murderous spear. 13 For μένος alone meaning ʻbattle-rageʼ cf. Il εἰ μὴ νὺξ ἐλθοῦσα διακρινέει μένος ἀνδρῶν until night at its coming shall part the fury of warriors. 14 In later authors ἔγχος is also used in the sense of ʻswordʼ, hence as a synonym of ἄορ, cf. S. Aj. 287 ἄμφηκες λαβὼν / ἐμαίετ ἔγχος ἐξόδους ἕρπειν κενάς he took his two-edged sword and made as though to start out, for no reason, E. El. 696 φρουρήσω δ ἐγὼ πρόχειρον ἔγχος χειρὶ βαστάζουσ ἐμῇ I shall be on guard, a sword at the ready in my hand, etc. ξίφος is used by Hom. as equivalent of ἄορ and φάσγανον, Od αὐτὸς δὲ ξίφος ὀξὺ ἐρυσσάμενος παρὰ μηροῦ (cf ἐγὼ δ ἄορ ὀξὺ ἐρυσσάμενος παρὰ μηροῦ, ἐγὼ μὲν ἄνευθεν ἐφ αἵματι φάσγανον ἴσχων).

5 Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται 35 A corresponding compound ʻraging with the spearʼ is attested lexicographically in Hesychius and the EM with the second element μάργος ʻmad, ragingʼ:15 (9) Hesych.: ἐγχεσίμαργος ἔγχει μαινόμενος (10) EM: Ἐγχείμαργος: Ὁ μαινόμενος τῷ δόρατι παρὰ τὴν ἔγχει δοτικὴν, καὶ μαργαίνω, τὸ μαίνομαι.16 The epic language attests a corresponding present μαργαίνω to rage (in battle)ʼ in (11) Il ἣ νῦν Τυδέος υἱόν, ὑπερφίαλον Διομήδεα, / μαργαίνειν ἀνέηκεν ἐπ ἀθανάτοισι θεοῖσι. / Κύπριδα μὲν πρῶτον σχεδὸν οὔτασε χεῖρ ἐπὶ καρπῷ Now she has incited the son of Tydeus, rash Diomedes, to vent his rage on immortal gods. Cypris first has he wounded in close fight on the hand at the wrist. 17 and from Aeschylus onward there is a present participle *μαργάων raging (in battle), cf. (12) A. Sept. 380 Τυδεὺς δὲ μαργῶν καὶ μάχης λελιμμένος yet Tydeus, raging and eager for battle.18 There is thus sufficient evidence for the combination of the notions [weapon] and [rage] in Greek which makes the assumption of the same combination in the PN a-o-ri-me-ne plausible. 15 Cf. Od μάργε Madman!, Od μάργην σε θεοὶ θέσαν The gods have made you mad. On the etymology, PIE *merǵ-, cf. Massetti forthcoming, especially on the semantic connection between be madʼ and be insatiate, be a glutton, etc.ʼ (ἀατὸς πολέμοιο insatiate for battleʼ, etc.), cf. also fn One might take ἐγχεσίμαργος to be a folk-etymological creation after ἐγχεσίμωρος reinterpreted as spear-crazyʼ, i.e. with μῶρος/μωρός crazy, stupidʼ as second member. But Hesychius himself glosses this differently: ἐγχεσίμωροι περὶ τὰ δόρατα μεμορημένοι, τουτέστιν πεπονημένοι; ἐγχεσίμωρος πολεμικός. On the compounds in μωρος, e.g. ἰόμωρος famous for strengthʼ cf. Heubeck Cf. also Hesych. μαργαίνων μαινόμενος [...]. 18 λελιμμένος hungry, eagerʼ with μάχης may vary the epic μάχης ἄατον (Il ) / ἄατος πολέμοιο (Hes. Th. 714) / μάχης ἀκόρητοι (Il ) / πολέμου ἀκορήτω (Il ), note also Hesych. ἔλιπεν ἐπιθυμητικῶς ἤσθιεν. Taken together, these forms may show the same connection between furyʼ and greed, gluttonyʼ as μάργος, μαργαίνω, μαργάω.

6 36 Daniel Kölligan 4.1. Now, the raging spear (and, as proposed here, also the ʻraging swordʼ) might simply be a rhetorical device, a metonymy describing the warrior s rage by the instrument that makes it most visible, his weapon. It is not unlikely, however, that in the early epic and ex hypothesi in Mycenaean times the weapon could be conceived of as invested with a power of its own, being a temporary embodiment and personification of Ares. It is the war god himself who is said to take out the menos out of the spear in various instances in the Iliad, cf. (13) Il δούπησεν δὲ πεσών, δόρυ δ ἐν κραδίῃ ἐπεπήγει, / ἥ ῥά οἱ ἀσπαίρουσα καὶ οὐρίαχον πελέμιξεν / ἔγχεος: ἔνθα δ ἔπειτ ἀφίει μένος ὄβριμος Ἄρης And he fell with a thud, and the spear was fixed in his heart that still beating made the spear-butt quiver; but there at length did mighty Ares stay its fury. (14) Il Αἰνείας δ ἐπὶ Μηριόνῃ δόρυ χάλκεον ἧκεν / ἔλπετο γὰρ τεύξεσθαι ὑπασπίδια προβιβῶντος. / ἀλλ ὃ μὲν ἄντα ἰδὼν ἠλεύατο χάλκεον ἔγχος / πρόσσω γὰρ κατέκυψε, τὸ δ ἐξόπιθεν δόρυ μακρὸν / οὔδει ἐνισκίμφθη, ἐπὶ δ οὐρίαχος πελεμίχθη / ἔγχεος ἔνθα δ ἔπειτ ἀφίει μένος ὄβριμος Ἄρης. And Aeneas cast his spear of bronze at Meriones, for he hoped to strike him as he advanced under cover of his shield. But Meriones, looking steadily at him, avoided the spear of bronze; he stooped forward, and the long spear fixed itself in the ground behind him, and the butt of the spear quivered; but there at length did mighty Ares stay its fury. (15) Il Ἕκτωρ δ Αὐτομέδοντος ἀκόντισε δουρὶ φαεινῷ / ἀλλ ὃ μὲν ἄντα ἰδὼν ἠλεύατο χάλκεον ἔγχος / πρόσσω γὰρ κατέκυψε, τὸ δ ἐξόπιθεν δόρυ μακρὸν / οὔδει ἐνισκίμφθη, ἐπὶ δ οὐρίαχος πελεμίχθη / ἔγχεος ἔνθα δ ἔπειτ ἀφίει μένος ὄβριμος Ἄρης. But Hector cast at Automedon with his bright spear, but he, looking steadily at him, avoided the spear of bronze, for he stooped forward, and the long spear fixed itself in the ground behind him, and the butt of the spear quivered; but there at length did mighty Ares stay its fury. 4.2 In addition to this, the genitive in the phrase μένος Ἄρηος mentioned in 2 and 3 may be understood as expletive as in ἲς Τηλεμάχοιο ʻthe strength of Telemachos = strong T. 19 If Ares equals μένος and Ares takes the μένος out of the spear, one 19 Cf. with an adjective βίη Ἡρακληείη ʻthe strength of Herakles = mighty Heraklesʼ. Cf. Chantraine 1953: 62.

7 Myc. a-o-ri-me-ne and Hom. δόρυ μαίνεται 37 may conclude that it is the war god himself who is temporarily present in the ʻraging weaponʼ when it is brandished, thrown and made to quiver Further evidence may be seen in the epithets common to Ares and weapons such as θοῦρος ʻimpetuous, furious, ragingʼ, which may imply the same idea, and ὀξύς ʻsharpʼ which, if the concrete meaning is primary, may show the transferal of meaning in the opposite direction: (16) Hom. θοῦρον Ἄρηα (9x), θοῦρος Ἄρης (2x): Il ἀσπίδα θοῦριν, E. Rh. 492 θοῦρον... δόρυ (17) ὀξὺν Ἄρηα (7x), ὀξὺς Ἄρης (1x): Il ὀξὺν ἄκοντα, ἄκων Il , ἄορ , βέλος 4.185, etc. 4.4 Finally, one may point out that Enyalios (: Myc. KN V bis e-nuwa-ri-jo), the god of close combat, who already in Homer is sometimes identified with Ares,21 was invoked by the hoplites in the front line when they adjusted their spears for the first onslaught.22 His shaking spear is mentioned in a Pindaric fragment: (18) Pi. fr. 70b.15 7 ἐν δ ὁ παγκρατὴς κεραυνὸς ἀμπνέων πῦρ κεκίνη[ται τό τ ] Ἐνυαλίου ἔγχος There too the all-powerful, fire-breathing thunderbolt is shaken, as is Enyalius spear... Again, the close relation between the war god and his weapon or even their identification makes it seem likely that the μένος of the weapon as expressed in a-o-rime-ne is actually the war god himself Cf. Mader in Snell 1979: III.485: [zeitweilige?] Verkörperung des Gottes, scil. in the ἔγχος, and on Il : Dies gilt am ehesten dann, wenn Ares u. Waffe hier ident. sind. Cf. also Diller-Sellschopp 1967: 29: Bei Homer haben auch Lanzen und andere Gegenstände, deren Herankommen als ein ungestümer Angriff empfunden wurde, dies Epitheton, für Hesiod liegt in den Gegenständen nicht mehr solche persönlich wirkende Kraft. Cf. also Sideras 1971: 67 fn Cf. Il Ἄρης / δεινὸς ἐνυάλιος. 22 Cf. Gordon Cf. also Il where Ares dives into the warrior: Ἕκτορι δ ἥρμοσε τεύχε ἐπὶ χροΐ δῦ δέ μιν Ἄρης / δεινὸς ἐνυάλιος πλῆσθεν δ ἄρα οἱ μέλε ἐντὸς / ἀλκῆς καὶ σθένεος and on Hector s body he made the armor fit, and there entered into him Ares, the terrible Enyalius, and his limbs were filled within with valor and with strength. As a weapon of gods, ἄορ is said of Ares (Hes. Sc. 457 ἄορ ὀξύ) and Poseidon (Il Ποσειδάων ἐνοσίχθων δεινὸν ἄορ τανύηκες ἔχων ἐν χειρὶ παχείῃ), in the former case probably as a folk-etymology to explain the god s name, which would again show that there was a popular belief identifying the two, cf. Buchholz 1980:

8 38 Daniel Kölligan Bibliography Aura Jorro, Francisco Diccionario Micénico: (DMic.). 2 vols. Madrid. Baumbach, Lydia The Mycenaean Greek Vocabulary II. Glotta 49 (3 4): Bechtel, Friedrich Die historischen Personennamen des Griechischen bis zur Kaiserzeit. Halle/S. Beekes, Robert Stephen Paul Etymological Dictionary of Greek. 2 vols. Leiden u.a. Buchholz, Hans-Günter Archaeologia Homerica: Die Denkmäler und das frühgriechische Epos. Kapitel E, Teil 2: Angriffswaffen: Schwert, Dolch, Lanze, Speer, Keule. Edited by Friedrich Matz and Hans-Günter Buchholz. Göttingen. Chadwick, John Documents in Mycenaean Greek. 2 nd ed. Cambridge. Chadwick, John, and Lydia Baumbach The Mycenaean Greek Vocabulary. Glotta 41 (3 4): Chantraine, Pierre Grammaire Homérique. Tome II: Syntaxe. Paris. Diller-Sellschopp, Inez Stilistische Untersuchungen zu Hesiod. Darmstadt. Gallavotti, Carlo Note sul lessico miceneo. Rivista di filologia e di istruzione classica 39: García Ramón, José Luis Mykenische Personennamen und griechische Dichtung und Phraseologie: i-su-ku-wo-do-to und a-re-me-ne, a-re-i-me-ne. In Colloquium Romanum. Atti del XII Colloquio Internazionale di Micenologia. Roma Febbraio 2006, edited by A. Sacconi, M. Del Freo, L. Godard, and M. Negri. Pisa/Roma, Gordon, Richard L Enyalios. In Der Neue Pauly. entries/brill-s-new-pauly/enyalius-e Gulizio, Joann A-re in the Linear B Tablets and the Continuity of the Cult of Ares in the Historical Period. Journal of Prehistoric Religion 15: Hajnal, Ivo Der mykenische Personenname a-e-ri-qo-ta. In Mykenaïka. Actes du IXe Colloque International sur les Textes Mycéniens et Egéens (Athènes, ), edited by Jean-Pierre Olivier. Athènes (BCH, Suppl. 25). Heubeck, Alfred Iolaos und Verwandtes. MSS 48: Jamison, Stephanie W., and Joel P. Brereton The Rigveda: the earliest religious poetry of India. 3 vols. New York. Lejeune, Michel Mémoires de philologie mycénienne. Troisième série Paris/ Roma. Massetti, Laura. Forthcoming. The Belly of an Indo-European: Some Greek and Iranian Cognates of PIE *merǵ- to divide, cut. In Proceedings of the 27th Annual UCLA Indo-European Conference, 2015, edited by David Goldstein, Stephanie Jamison, and Brent Vine. Bremen. Peters, Martin Untersuchungen zur Vertretung der indogermanischen Laryngale im Griechischen. Wien. Ruijgh, Cornelis Jord Problèmes de philologie mycénienne. Minos: Revista de Filologia Egea 19: Sideras, Alexander Aeschylus Homericus: Untersuchungen zu den Homerismen der aischyleischen Sprache. Hypomnemata 31. Göttingen. Snell, Bruno Lexikon des frühgriechischen Epos. Göttingen. Willi, Andreas Ares the Ripper. Indogermanische Forschungen 119:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

( y) Partial Differential Equations

( y) Partial Differential Equations Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

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

ΑΓΓΛΙΚΑ IV. Ενότητα 6: Analysis of Greece: Your Strategic Partner in Southeast Europe. Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών

ΑΓΓΛΙΚΑ IV. Ενότητα 6: Analysis of Greece: Your Strategic Partner in Southeast Europe. Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Ενότητα 6: Analysis of Greece: Your Strategic Partner in Southeast Europe Ιφιγένεια Μαχίλη Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό,

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

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

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

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

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

ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ

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

Διαβάστε περισσότερα

ΑΛΕΞΑΝΔΡΟΣ ΠΑΛΛΗΣ SCHOOLTIME E-BOOKS

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

Διαβάστε περισσότερα

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

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

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

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

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

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

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ Π Τ Υ Χ Ι Α Κ Η Ε Ρ Γ Α Σ Ι Α: Ο ΡΟΛΟΣ ΤΗΣ ΣΥΝΑΙΣΘΗΜΑΤΙΚΗΣ ΝΟΗΜΟΣΥΝΗΣ ΣΤΗΝ ΑΠΟΤΕΛΕΣΜΑΤΙΚΗ ΗΓΕΣΙΑ ΕΠΙΜΕΛΕΙΑ

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

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

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις

Διαβάστε περισσότερα

Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο

Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο Δρ. Παναγιώτης Πεντάρης - University of Greenwich - Association for the Study of Death and Society (ASDS) Περιεχόµενα Εννοιολογικές

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

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

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

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

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

«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

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

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

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

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

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

Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς

Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς Review 4n.1: Vowel stems of the third declension: πόλις, πρέσβυς We review side by side a model of stems ending in ι: πόλις, πόλεως, ἡ = city-state and a masculine model of stems ending in υ: πρέσβυς,

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

English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based

English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based exclusively on safe, managed code. PDFsharp offers two powerful

Διαβάστε περισσότερα

English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based

English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based English PDFsharp is a.net library for creating and processing PDF documents 'on the fly'. The library is completely written in C# and based exclusively on safe, managed code. PDFsharp offers two powerful

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

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

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

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

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

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Τμήμα Δημοσίων Σχέσεων & Επικοινωνίας Τεχνολογικό Εκπαιδευτικό Ίδρυμα Ιονίων

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

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

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

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

1 String with massive end-points

1 String with massive end-points 1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε

Διαβάστε περισσότερα

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee

Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset

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

Derivations of Useful Trigonometric Identities

Derivations of Useful Trigonometric Identities Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine

Διαβάστε περισσότερα

Η αλληλεπίδραση ανάμεσα στην καθημερινή γλώσσα και την επιστημονική ορολογία: παράδειγμα από το πεδίο της Κοσμολογίας

Η αλληλεπίδραση ανάμεσα στην καθημερινή γλώσσα και την επιστημονική ορολογία: παράδειγμα από το πεδίο της Κοσμολογίας Η αλληλεπίδραση ανάμεσα στην καθημερινή γλώσσα και την επιστημονική ορολογία: παράδειγμα από το πεδίο της Κοσμολογίας ΠΕΡΙΛΗΨΗ Αριστείδης Κοσιονίδης Η κατανόηση των εννοιών ενός επιστημονικού πεδίου απαιτεί

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

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

Πτυχιακή Εργασία. Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους

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

Διαβάστε περισσότερα

John Mavrikakis ENGLISH MULTIBOOK

John Mavrikakis ENGLISH MULTIBOOK units 201 John Mavrikakis ENGLISH MULTIBOOK e-learning for language students (grammar, vocabulary, reading) level 2 (Junior A) DEMO STUDENT S UNIT 10 The alphabet, a, b, c, d, e, f, g, h, i, j, k, l, A,

Διαβάστε περισσότερα

Calculating the propagation delay of coaxial cable

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

Διαβάστε περισσότερα

Partial Trace and Partial Transpose

Partial Trace and Partial Transpose Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This

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

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1. Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

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

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

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

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

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

Διαβάστε περισσότερα

( ) 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