TWO PROBLEMS IN ANCIENT MEDICAL COMMENTARIES
|
|
- Μαριάμ Λαμέρας
- 8 χρόνια πριν
- Προβολές:
Transcript
1 Classical Quarterfy 44 (i) (1994) Printed in Great Britain 270 TWO PROBLEMS IN ANCIENT MEDICAL COMMENTARIES I. AN ANONYMOUS COMMENTARY ON HIPPOCRATES' APHORISMS Thirty years ago, H. Flashar discussed the introduction to an anonymous commentary on the Aphorisms of Hippocrates. 1 The text contains an interesting picture of Hippocrates äs a culture hero, who saved suffering humanity by the introduction of systematic medicine. The first section of this introduction oifers some complicated problems. It ends with an extremely long and difficult sentence, which has not yet been explained quite satisfactorily, and it contains a curious use of the verb σαρκόω, combined with την φύσιν, which has led Flashar to suspect Christian influence. These two points are the subject of the first part of this paper. The text äs printed by Flashar runs thus: 01 μεν πλείστοι των παλαιοτέρων Ιατρών σποράδην έζηΰρόν τίνα της ιατρικής, εκ κληδόνων ή τριόδων σννάγοντες καΐ από τύχης η μαντείας ή κλήσεων δαιμόνων ή αλλά) τινι τρόπω. Ιπποκράτης δε ό των Άσκληπιαδών ταύτην ως αν ε'ίποι τις πλαζομενην συνάξας και τελείως ύφάνας πλήρη και άρτίαν είργάσατο κεφαλήν επιθείς και ουκ αν τις άμάρτοι λέγων, ως ό προνοητικός θεός ελεήσας το άνθρώπινον γένος άλλεπαλλήλοις νόσοις άπολλύμενον αυτήν την φύσιν σαρκώσας Ίπποκράτην κατήγαγε προς άρτίαν ταύτης παράδοσιν. 'ίσως γαρ και τούτο αΐνίττεται κατά το προοίμιον των Αφορισμών λέγων, ως επειδή κατά την πεΐραν ή Ιατρική σχεδόν ακατάληπτος εστίν (ούτε γαρ οτε βουλόμεθα τοις πάθεσιν των ανθρώπων έντυγχάνουσιν οι Ιατροί τύχη γαρ και τω σπανίω της γενέσεως δουλεύει ταύτα, έτι γε μην και έπικίνδυνον (Flashar; επικίνδυνος MSS) τω εν σώματι ρευστώ μεν δια την ύλην και αβεβαίω, κεκτημενω δε και θείαν δύναμιν φυχικήν γυμνάζεσθαι την Ίατρικήν τεχνην και ουκ άφύχω και άτίμω καθάπερ τάς αλλάς τεχνας, προς δε τούτοις και τα πάθη υπό πολλών αίτιων γεννάσθαί τε και αύζάνεσθαι και δια τούτο δυσχεραίνειν την πεΐραν εν τω διακρίνειν το ιτοιητικόν αίτιον) φέρε τω λόγω φιλώ χωρίς υποκείμενης ύλης άσωμάτως πάσαν περιλαβών εν όλίγω χρόνω διδάξω και δια τούτο και έπιστημονικόν λόγον εχειν ποιήσω, και τάς αιτίας πάσας ΰποθήσομαι συν ταίς διαγνώσεσι προς το σε λοιπόν τ-fj πείρα γυμνάζειν τον λόγον, τότε δ' αν σοι κατά τύχην περιπέστ) τι πάθος έφαρμόζειν τον λόγον καΐ γυμνάζειν και αληθή τούτον εύρίσκειν. If I understand Flashar (op. cit. p. 405 n. 2) correctly, he takes the last long sentence äs a paraphrase of Hipp. Aph. 1.1: ή δε πείρα σφαλερή, ή δε κρίσις χαλεπή. Reading έπικίνδυνον, 'wovon dann γυμνάζεσθαι την Ιατρικην τεχνην abhängig ist' (ibid. 406 n. 2), he supposes that the clause beginning with προ? δε τούτοις still 'depends' on it, but in a very loosely construed way. He states that Hippocrates cannot be the subject of the last part of the sentence, because the intervening infinitives would point to an indirect construction, while the first person Singular would mean a reversion to direct speech. 1 H. Flashar, 'Beiträge zur spätantiken Hippokratesdeutung', Hermes 90 (1962), The text may be found äs well in F. R. Dietz, Scholia in Hippocratem et Galenum II 244f. For further references, see Flashar, op. cit. pp. 402ff. I would like to thank Professor Dirk M. Schenkeveld and the anonymous reader of CQ for their helpful criticism on an earlier draft of this paper. Research for this paper was made possible by a fellowship of the Royal Dutch Academy of Arts and Sciences.
2 TWO PROBLEMS IN ANCIENT MEDICAL COMMENTARIES 271 I would suggest a different solution: λέγων ως is picked up, after a very long Interruption, by φέρε... διδάξω κτλ., ως functioning äs a colon (äs often) and being left untranslated; Hippocrates is without doubt the subject of διδάξω. The επειδήclause has two predicates: επειδή...ή Ιατρική... ακατάληπτος εστίν..., έτι γ μην επικίνδυνος (with the MSS): 'since medicine is impossible to grasp, and, moreover, 2 risky'. The parenthesis beginning with οίίτε γαρ stops after δουλεύει ταύτα, and does not continue until αίτιον, äs printed by Flashar. Now, the main problem left is the part τω εν σώματι κτλ.: τω does not go with a (substantival) adjective ρευστώ, äs Flashar takes it, but it should be construed with the infinitive γυμνάζεσθαι (and through that with γ ννάσθαίτ και αΰξάνεσθαι και... δυσχεραίνειν). Την Ιατρικήν τεχνην functions äs a subject accusative with γυμνάζεσθαι. The construction τω... γυμνάζεσθαι is difficult because of the many constituents intervening between the article and its 'noun', the infinitive. However, in the very same passage we have an undisputed parallel for this construction: the final part of the sentence contains the substantival infinitives (or rather accusative-and-infinitive-constructions) (προς) το σε...γυμνάζειν, τότε δ'...έφαρμόζειν, where το goes with both infinitives. Here, too, there is a remarkably large number of constituents intervening between the article and the infinitive; perhaps we should consider reading ότε δ' αν instead of τότε δ' αν (so Dietz). The translation of the last sentence might run: ' For maybe he makes an allusion to this, too, in the prooemium of the Aphorisms, saying: " Since medicine is practically impossible to grasp by means of experience (for doctors do not encounter diseases in people when we wish to; 3 for that depends on chance and the rarity of its appearance) and [since medicine is] moreover risky in that medicine is practised on a body, (which is on the one hand in a state of flux and unstable because of its material, but on the other hand possesses a divine spiritual power), and not on something lifeless and worthless, like the other technai, and further [it is risky] in that diseases originate and grow from many causes and therefore experience has trouble in distinguishing the causa efficiens: come on, let me grasp it purely theoretically, without underlying material, bodilessly, and explain it briefly; thereby I shall bring about that medicine has a basis of theory and I shall suggest all the various causes with the diagnoses, in order that you may henceforth train your theoretical knowledge with experience and that you may adapt your theory, whenever you meet a case of disease, and exercise it and find it true."' It is interesting to notice the many parallele between this short explanatory paraphrase of Hippocrates' prooemium and Galen's commentary on the same passage (XVII 2, Kühn). These parallele make it likely that our text is meant äs a commentary on the whole first chapter οι Aphorisms. As Galen explicitly teils us, ancient exegetes were agreed that this whole chapter was meant äs the prooemium (op. cit. 346 K.). Like our commentator, Galen, too, points out that ή πείρα σφαλερά holds good 'δια το της ΰλης αξίωμα', because of the value of the material. And on p. 353 it is added: ό καιρός οξύς δια την της τέχνης ΰλην, λέγω δε το σώμα peov άει 4... επισφαλής δε ή πείρα και αυτή δια την ΰλην, and then Galen adds precisely the difference with the other technai noted in our commentary: they practise safely on lifeless material. Galen explains the second part of Aphorisms 1.1 (δει δε ου μόνον έωντόν παρεχειν 2 For the combination γε μην, see J. D. Denniston, The Greek Partides (Oxford, 1959), p "Οτε βουλόμεθα is, of course, unremarkable in Greek of this date. 4 Cf. ibid. 346: 6 μεν ουν καιρός εστίν o vs δια το της ύλης ρευστόν, ην ή τέχνη μεταχειρίζεται.
3 272 INEKE SLUITER τα δέοντα ποιέοντα, άλλα καί τον νοσΐοντα και τους παρ οντας και τα ΐξωθίν) äs an indication of the τρόπος της διδασκαλία? and of the usefulness of the work (op. cit. 351 K.). He points out (ibid. 352) that aphorisms are the best way to teach and learn a lot in a short compass, which in turn is necessitated by the shortness of life all this is also expressed in the ^>epe-clause of our commentary. Thus, by means of this paraphrase our commentator has put quite an extended version of Aphorisms l. l in the mouth of Hippocrates: a whole programme of medical teaching is hidden in its few words. The actual wording of Aphorisms 1.1 is no more than an allusion (αΐνίττεται) to the task imposed on Hippocrates by the provident god and fully accepted by Hippocrates himself: to provide accurate reports of the art of medicine for generations to corne (άρτίαν ταύτης παράδοσιν). Let us now turn to the other problem of our passage. Flashar rightly calls attention to the phrase αυτήν την φύσιν σαρκώσας. He comments (406, n. 1): 'Nach dieser eigenartigen Formulierung ist...nicht daran gedacht, dass der vorhersehende Gott seine eigene Natur, sondern die Natur selbst in Hippokrates menschliche Gestalt annehmen lässt. Dies lässt sich wohl nur im Sinne der stoischen Lehre verstehen... Der Begriff der Fleischwerdung ist hier ohne den Einfluss christlicher Lehren schwer vorstellbar, dann aber zeugt der Gedanke von einer schon sehr weitgehenden Vermischung stoisch-christlicher Weltanschauung.' Flashar thinks the sentence betrays Christian influence and possibly indicates Christian authorship. This Suggestion has so far gone unchallenged. O.Temkin, in his recent Hippocrates in a World of Pagans and Christians, gives his explicit support to Flashar. 5 Temkin translates: ' and nobody is likely to err who says that the provident god, taking pity on mankind, which was being destroyed by successive diseases, made nature herseif into flesh and led Hippocrates to the perfect transmission of this [art]' (op. cit. 46). I do not, however, feel so confident about the Christian influence, even when taking into account a point Temkin seems to allude to, viz. the difference between incarnation of the φύσις and that of the Λόγος. Although it may be hard to imagine the concept of σάρκωσις outside a Christian context, it seems even harder to imagine it within such a context, but applied to Hippocrates. A second problem is the reference of ταύτης. All agree that this should be the art of medicine (= ταύτην in the preceding sentence). However, with αυτήν την φύσιν, 'nature itself, in between, ταύτης is highly ambiguous and the Greek seems forced, äs appears from Temkin's translation. A definite solution I cannot offer. One might attempt to evade the problem of the representation of Hippocrates äs Christ and to render ταύτης unambiguous by making αυτήν την φύσιν, too, refer to medicine (Ιατρική). With a slight adaptation of Temkin's rendering this would yield: '...the provident god...made the very essence (sc. of medicine) into flesh..." and led Hippocrates to its perfect transmission.' In this Interpretation, Hippocrates is regarded äs medicine incarnate and takes on the Status of a daemon. 1 While one cannot exclude the possibility that the verb σαρκόω 5 Baltimore-London, 1991, 46; Perhaps this necessitates reading αυτής (= ipsius) rather than αυτήν, bearing in mind that αυτή; can stand for ταύτης in later Greek, cf. Blass-Debrunner 277. However, I am not convinced that this is necessary. For φύσις 'essence', see Lampe s.v. II A. 7 Cf. for daemones/heroes helping mankind, because a supreme god took pity, Maximus of Tyrus, Dissert. XV 6: Souls that are freed from their bodies take pity on other souls (ο'ικτΐίρουσα, φιλανθρωπία). Προστετακται δε avrf] ΰπο του eoü έπιφοιτάν την γήν, και άναμίγνυσθαι πάσ η μεν ανδρών φύσει, πάατ] δε ανθρώπων τΰχ-τ) και γνώμύ] και τΐχντ). The daemones show a preference for the occupation they had when they were still human. Thus, Asclepius takes care of medicine. For other parallels see the commentary on Posidon. Fr. 108 by Edelstein and Kidd.
4 TWO PROBLEMS IN ANCIENT MEDICAL COMMENTARIES 273 betrays Christian influence, there is no need to assume an application that would surely be blasphemous in Christian eyes. 8 II. JOHN OF ALEXANDRIA ON THE USE OF PREPOSITIONS In a recent article Vivian Nutton comments on the complicated tradition of a commentary on Galen's On Sectsfor Beginners, ascribed to John of Alexandria (7th cent. A.D.). 9 Nutton explains (op. cit. 510f.) that in fact we have to take into account three different texts: the Latin translation of John's commentary, ascribed to Burgundio of Pisa; another Latin version of a commentary on Galen, on the name of Agnellus of Ravenna. This is closely related to 'John' (äs the first translation will henceforth be called). This commentary, too, goes back to a Greek original. And, thirdly, a Greek commentary on Galen's On Sects, ascribed to Archelaus. This commentary bears signs of being closely linked to parts of Agnellus' Latin version. The Greek 'John' is lost. Probably all three texts are either translations or adaptations of John's commentary, but it cannot be excluded that the similarities are accidental in so far äs they are due to the fact that all three texts contain a typical commentary-discourse, provoked by the same source-text. On balance, Nutton rejects this latter possibility for the relationship between John and Agnellus: they are thought to be versions of the same Greek text. On p. 517 Nutton discusses the commentaries on the second sentence of Galen's tract, which runs (in the original Greek, I 64 K.): εξ ων δ' αν τις ή μη παροϋσαν vyeiav έργάζοιτο η παροϋσαν δίαφνλάττοι, γινώσκ σθαι μ ν άναγκαΐον τοις ιατροί?. In Latin, this sentence is translated äs follows (19,28ff. Pritchet): ex quibus autem quis vel non presentem sanitatem operetur vel presentem custodiat, noscendum quidem necessarium medicis. As Nutton has seen, the commentaries notice that Galen seems to use the expression for a material cause (ΰλικόν, materialis), instead of the expected instrumental cause (οργανικόν, Instrumentalis), and they ask themselves why. According to Nutton, neither John's nor Agnellus' answer makes sense. Their comments run äs follows:. John 22, Pritchet: Et quidam opponunt Galieno dicentes ' Quare dixit ex quibus et non "quali organo'"?' dicimus quod hec prothesis accipitur: sie enim in figuris grammaticorum (infantes grammaticorum MSS 10 ). Agnellus 40, 19-21: hie prothesis prothes accipit: sie enim infantes grammaticorum dicent, pro T ponent D. Archelaus' version provides a first key to the solution: 8 Formerly, I entertained the possibility that σαρκόω was used in its medical sense here, i.e. meaning 'to make fleshy or strong' (LSJ, s.v.). The word and its derivatives (like σάρκωσις or σαρκωτικός) are very frequent in all medical writers. For a link with φύσι;, cf. Gal. X 178 K. (no exact parallel). If we take σαρκώσας in this way, it means that the flesh is put back on the bones of emacerated humanity. The provident god himself is the superior doctor who applies this medicine; Hippocrates then consolidates the effect. Αυτήν την φύσιν would then either have to refer to an abstract 'nature itself, or perhaps more probably to the constitution of humanity. In that case one might translate: 'the provident god...gave nature itself a fleshy strength and sent down Hippocrates etc.' However, on balance this solution seems less attractive. 9 V. Nutton, 'John of Alexandria again: Greek Medical Philosophy in Latin Translation', CQ 41 (1991), The article itself is a reaction to an earlier article by R. J. Hankinson, 'Notes on the Text of John of Alexandria', CQ 40 (1990), References to John's commentary are based on the edition by C. D. Pritchet, lohannis Alexandrini Commentaria in librum De sectis Galeni, Leiden, For further references, see Nutton, op. cit., esp. notes 9 and Cg's anonymous referee informs me that in figuris does not appear in any manuscript, but is an emendation of the second printed edition. 12 OCQ 44
5 274 INEKE SLUITER Archelaus 64, 4-5: το γαρ εξ ων αντί του δι ων παρέλαβε. 'The original question must...have focused on Galen's choice of one Greek preposition over another' (op. cit. 517), and, I may add, the answer will duly have dealt with this. Obviously, this brought John into difficulties. His translation quali organo correctly indicates the different type of cause involved, but it obscures the fact that the cause required is usually expressed by a different preposition. Agnellus' translation seems to have adhered closer to the Greek. Therefore, Nutton Claims (rightly, I think) that the problems in John are due to mistranslation from the Greek, while those in Agnellus should be considered corruptions in the transmission of the Latin Version (op. cit. 517). I do not agree with Nutton, however, in his evaluation of the appeal to the grammarians. According to him, the comment found in Agnellus tries to defend Galen by criticizing 'grammarians' slovenly habits'. But, he adds, in the rest of the commentary (both in John's and Agnellus' versions) the role played by grammarians is generally a positive one. He considers this problematic; in fact, we shall see that this problem is based on a misunderstanding. All in all, Nutton believes that the grammatical remark was indeed part of the Greek original (p. 518); he thinks John's text (with the reading of the MSS infantes) is in order; the Latin translates the Greek periphrastic παίδες γραμματικών, for γραμματικοί). The incoherence is the translator's fault. Agnellus he emends äs follows: hie prothesis \prothes] accipitur: sie enim et infantes grammaticorum dicent, pro EX ponent DIA - 'Here the preposition is used (is acceptable?), for, the grammarians say, they will put εξ instead of δια'. The problem with this emendation is the unpleasant stress on the fact that a preposition is used (and not another part of speech?). To my mind, all versions of the commentary try to defend Galen's text by appealing to a rule which the grammarians had deduced from their literary experience. The commentaries definitely do not appeal to sloppy Greek or Latin supposedly used by the grammarians themselves that would be an intrinsically weak argument. The only possibility open to the commentators is, of course, to refer to a usage which is established äs correct Greek or Latin by the grammarians in their role of guardians of linguistic correctness and in fact that is exactly what they are doing. For in both Greek and Roman grammar the phenomenon of one preposition being used instead of another is well known, and although it is regularly mentioned among the causes of solecism, many instances from respected and authoritative older writers are quoted only to be explained by the rule of hand 'πρόθεσις αντί προθέσεως', or 'praepositio pro praepositione\ Some examples must suffice: Greek examples: Herodianus Gramm. Graec. III ii 38,3ff.: περί πάντων (Hom. //. B 831): οϋτως φυλακτέον τον τόνον της προθέσεως, είτε γαρ πρόθεαίς εστίν αντί προθέσεως παρειλ-ημμένη, αντί τον υπέρ πάντων, φυλάσσεται ό τόνος καθότι πάσα πρόθεσις αντί ετέρας προθέσεως λαμβανομένη φυλάσσει τον αυτόν τόνον. Gramer Anecd.Oxon. Ι (Epimerismi) 354, loff.: (ττερί instead of υπέρ): πρόθεσις αντί προθέσεως παραλαμβανομένη τον 'ίδιον τόνον φυλάττουσα. Choeroboscus, Psalm-epimerisms (ed. Goetting) 49,lf : πολλοί δε και προθέσει κέχρηνται αντί προθέσεως. Gregorius Corinthius (ed. Donnet) 48, 305ff., e.g. 50, 320 κατά instead of εν. See also 202 in the context of barbarisms/solecisms. Eustathius Comm. in II. I 317,44 vd Valk: το δε 'ΰπο "Ιλιον ήλθεν''. αντί του κατά το "Ιλιον, Ινα και νυν ληφθείη αντί προθέσεως πρόθεσις. Etymologicum Orion s.v. Μέτωπον: πρόθεσις αντί προθέσεως (νίζ. μετά instead of υπέρ). Maxinms Planudes (in Bachmann Anecd. Graeca II) 30f. προθέσεις αντί προθέσεων.
6 TWO PROBLEMS IN ANCIENT MEDICAL COMMENTARIES 275 A similar view of the respective roles of ex and δια is provided by Plutarch, Quaest.Plat. X 1009f: Plutarch discusses Plato's claim that τον λόγον εξ ονομάτων και ρημάτων κ ράννυσθαι. In so saying, Plato did not necessarily forget about the other parts of speech. For in that case he would have used the wrong preposition: ου γαρ e/c τούτων ό λόγος, αλλ ίΐττερ αρά, δια τούτων και ουκ avev τούτων κ ράνννσθαι Latin grammarians, too, are perfectly familiär with this phenomenon: 11 they consider the interchange of prepositions äs one of the causes of solecism äs early äs Quintilian , see e.g. Donatus (Gramm. Lat. IV 394, 16) (soloecismus per praepositionem) cum alia pro alia ponitur auf necessaria subtrahitur. 12 But the same usage also occurs (and is perfectly acceptable) in the auctores. In Quintilian (9.3.2) this kind of figures of speech form a specific type, called the genus grammaticum (äs opposed to the genus rhetoricum). His point is that there is an exact correspondence between these figures of speech and the vitia sermonis. 13 Servius often calls attention to such cases, äs e.g. on Verg. Ed. 1 15: conixapro eo quodest enixa: nam hiatus causa mutavit praepositionem. Here, the art of the poet has avoided hiatus by substituting one preposition for another. 14 In his commentary on Verg. A. II 52 Servius even uses the formulaic praepositio pro praepositione. 15 The form of the rule just asks for accidents of transmission : Pompeius (Gramm. Lat. V 268) shows what may happen quite clearly : nam invenimus praepositionem (pro praepositione om. ABC) positam e.q.s. It would seem that at least Agnellus has suifered somewhat along the same lines. This was facilitated even more by the use of the Greek word prothesis in a Latin context, yielding a perfect homoearcton, three consecutive words beginning with pro. Archelaus does not give any problems whatsoever. He states quite clearly what happened, viz. that one preposition group was exchanged for another. Agnellus has preserved clear traces of the same intention. With some minor changes his text yields (Agnellus 40, 19-21): hie prothesis < pro) prothes(f) accipit(ur}: sie enim infantes grammaticorum dicent, pro EX ponent DIA ('Here one preposition is used instead of another: for this is what the grammarians will say, they [i.e. 'auctores'] will use EX instead of DIA. ') John's text, reading infantes, derives from the same Greek model äs Agnellus. Hec is the last trace of the fact that the original context opposed two different prepositions to each other: hec prothesis accipitur: sie enim infantes grammaticorum ('(The use of) this preposition is acceptable: thus the grammarians [say].') One might envisage something like the following äs the Greek original : Πρόθΐσις αντί προθέσ ως παραλαμβάν ται. Οντως γαρ παΐδε? γραμματικών, το ζζ αντί τον δια. Amsterdam, Vrije Universiteit INEKE SLUITER 11 E.g. Prise. Gramm. Lat. III 294, 5; 302, 25ff.; 371, 7f. 18 Cf. ibid. 393, 21ff.; Charisius 268, 29 Barwick; Sergius Explan, in Don., Gramm Lat. IV 564, 14f.; Marius Plotius Sacerdos, Gramm. Lat. VI 450, 17ff.; Isid. Etym. I 33, Cf. the way Dionysius of Halicarnassus uses the word σολοικοφανής: this is used for locutions that might seem to be faulty but for the authority of the writer who uses them: see e.g. DH Thuc. 29; 55; Serv. in Verg. A Notice that according to ancient theory the fact that we are dealing with a prepositional prefix here, is irrelevant. Prepositions can occur single and in compounds per defimtionem. 15 See further Serv. on Verg. E. 8.66; G ; A. 2.2, 52; 3.446; Augustine points out that prepositions can be 'translated' by other prepositions, de Mag. II 4 (about Verg. A ): praepositio est ex pro qua de_possumus, ut arbiträr, dicere. 12-2
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 περισσότεραPhys460.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 περισσότεραEvery 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 περισσότεραHOMEWORK 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 περισσότεραΣυντακτικές 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 περισσότερα2 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 περισσότεραC.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 περισσότεραChapter 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 περισσότεραSection 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 περισσότεραConcrete 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 περισσότεραFinite 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 περισσότεραderivation 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 περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ 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 περισσότεραEE512: 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 περισσότεραΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ
ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ Ενότητα 1β: Principles of PS Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης
Διαβάστε περισσότεραExample 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 περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ
Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 7: More about gerunds Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons
Διαβάστε περισσότεραLESSON 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 περισσότεραChapter 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 περισσότεραHomework 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 περισσότεραCode 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 περισσότεραAdjectives. 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 περισσότεραΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α. Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:.
ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α 2 ειδήσεις από ελληνικές εφημερίδες: 1. Τα Νέα, 13-4-2010, Σε ανθρώπινο λάθος αποδίδουν τη συντριβή του αεροσκάφους, http://www.tanea.gr/default.asp?pid=2&artid=4569526&ct=2 2. Τα Νέα,
Διαβάστε περισσότεραΤΕΧΝΟΛΟΓΙΚΟ 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 περισσότερα4.6 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 περισσότεραLECTURE 2 CONTEXT FREE GRAMMARS CONTENTS
LECTURE 2 CONTEXT FREE GRAMMARS CONTENTS 1. Developing a grammar fragment...1 2. A formalism that is too strong and too weak at the same time...3 3. References...4 1. Developing a grammar fragment The
Διαβάστε περισσότεραInstruction 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 περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ 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 περισσότεραΚΥΠΡΙΑΚΗ 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 περισσότερα3.4 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 περισσότεραΚΥΠΡΙΑΚΗ 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 περισσότεραSection 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 περισσότεραMath 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 περισσότεραPotential 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 περισσότεραMatrices 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 περισσότερα9.09. # 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 περισσότεραSection 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 περισσότεραΕπιβλέπουσα 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 περισσότεραΠΑΝΕΠΙΣΤΗΜΙΟ 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 περισσότεραΑΓΓΛΙΚΑ 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 περισσότεραSection 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 περισσότεραΟι 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 περισσότεραST5224: 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 περισσότερα6.1. 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 περισσότεραdepartment 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 περισσότεραThe 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 περισσότεραAssalamu `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 περισσότεραLESSON 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 περισσότεραPARTIAL 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 περισσότεραLecture 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 περισσότεραEcon 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 περισσότεραthe 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 περισσότεραDémographie spatiale/spatial Demography
ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης
Διαβάστε περισσότεραDynamic types, Lambda calculus machines Section and Practice Problems Apr 21 22, 2016
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Dynamic types, Lambda calculus machines Apr 21 22, 2016 1 Dynamic types and contracts (a) To make sure you understand the
Διαβάστε περισσότερα[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 περισσότεραMean bond enthalpy Standard enthalpy of formation Bond N H N N N N H O O O
Q1. (a) Explain the meaning of the terms mean bond enthalpy and standard enthalpy of formation. Mean bond enthalpy... Standard enthalpy of formation... (5) (b) Some mean bond enthalpies are given below.
Διαβάστε περισσότεραOn 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 περισσότερα14 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 περισσότεραStrain 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 περισσότεραΠώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your
Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your GP practice in Islington Σε όλα τα Ιατρεία Οικογενειακού
Διαβάστε περισσότερα7 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 περισσότεραΚατανοώντας 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 περισσότεραOrdinal 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 περισσότεραMath221: 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 περισσότεραΣυστήματα 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 περισσότεραInverse 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 περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ 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 περισσότεραANSWERSHEET (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 περισσότεραModern 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 περισσότερα2007 Classical Greek. Intermediate 2 Translation. Finalised Marking Instructions
2007 Classical Greek Intermediate 2 Translation Finalised Marking Instructions Scottish Qualifications Authority 2007 The information in this publication may be reproduced to support SQA qualifications
Διαβάστε περισσότεραΗ 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 περισσότεραΑΓΓΛΙΚΑ ΙΙΙ. Ενότητα 12b: The Little Prince. Ζωή Κανταρίδου Τμήμα Εφαρμοσμένης Πληροφορικής
Ενότητα 12b: The Little Prince Τμήμα Εφαρμοσμένης Πληροφορικής Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπως εικόνες, που υπόκειται
Διαβάστε περισσότεραOther 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 περισσότεραΑναερόβια 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 περισσότεραSecond 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 περισσότεραPaper 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 περισσότεραI haven t fully accepted the idea of growing older
I haven t fully accepted the idea of growing older 1 Peter Authorship Questions Πέτρος ἀπόστολος Ἰησοῦ Χριστοῦ ἐκλεκτοῖς παρεπιδήμοις διασπορᾶς Πόντου, Γαλατίας, Καππαδοκίας, Ἀσίας 1 Peter Authorship
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραUniform 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 περισσότεραBECAUSE WE REALLY WANT TO KNOW WHAT YOU THINK ABOUT SCHOOL AND YOUR GARDEN. Fairly true If I decide to learn something hard, I can.
BECAUSE WE REALLY WANT TO KNOW WHAT YOU THINK ABOUT SCHOOL AND YOUR GARDEN Name GRADE Science Teacher A. What do I think about School? bit I try hard to do well in school. I look forward to coming to school.
Διαβάστε περισσότεραCalculating 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 περισσότεραSCHOOL 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 περισσότεραΛέξεις, φράσεις και προτάσεις
1 Λέξεις, φράσεις και προτάσεις (Words, phrases and clauses) The Greek language, like all human languages, has a Lexicon and a Grammar that are used to create sentences. The Lexicon consists of the words
Διαβάστε περισσότεραΙΠΛΩΜΑΤΙΚΗ 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 περισσότεραGalatia SIL Keyboard Information
Galatia SIL Keyboard Information Keyboard ssignments The main purpose of the keyboards is to provide a wide range of keying options, so many characters can be entered in multiple ways. If you are typing
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραNowhere-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 περισσότεραΚΥΠΡΙΑΚΟΣ 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 περισσότεραBlock Ciphers Modes. Ramki Thurimella
Block Ciphers Modes Ramki Thurimella Only Encryption I.e. messages could be modified Should not assume that nonsensical messages do no harm Always must be combined with authentication 2 Padding Must be
Διαβάστε περισσότεραThe 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 περισσότεραRight 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 περισσότεραΣτο εστιατόριο «ToDokimasesPrinToBgaleisStonKosmo?» έξω από τους δακτυλίους του Κρόνου, οι παραγγελίες γίνονται ηλεκτρονικά.
Διαστημικό εστιατόριο του (Μ)ΑστροΈκτορα Στο εστιατόριο «ToDokimasesPrinToBgaleisStonKosmo?» έξω από τους δακτυλίους του Κρόνου, οι παραγγελίες γίνονται ηλεκτρονικά. Μόλις μια παρέα πελατών κάτσει σε ένα
Διαβάστε περισσότεραHomework 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 περισσότεραΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE ΑΠΟ ΑΤΟΜΑ ΜΕ ΤΥΦΛΩΣΗ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΜΑΚΕΔΟΝΙΑΣ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΚΟΙΝΩΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΚΠΑΙΔΕΥΤΙΚΗΣ ΚΑΙ ΚΟΙΝΩΝΙΚΗΣ ΠΟΛΙΤΙΚΗΣ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE
Διαβάστε περισσότεραWriting 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 περισσότεραStatistical 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 περισσότεραPolicy Coherence. JEL Classification : J12, J13, J21 Key words :
** 80%1.89 2005 7 35 Policy Coherence JEL Classification : J12, J13, J21 Key words : ** Family Life and Family Policy in France and Germany: Implications for Japan By Tomoko Hayashi and Rieko Tamefuji
Διαβάστε περισσότεραF-TF Sum and Difference angle
F-TF Sum and Difference angle formulas Alignments to Content Standards: F-TF.C.9 Task In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when
Διαβάστε περισσότεραPartial 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