DOES Κεφαλή MEAN "SOURCE" OR "AUTHORITY OVER" IN GREEK LITERATURE? A REBUTTAL RICHARD S. CERVIN URBANA-CHAMPAIGN, ILLINOIS

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "DOES Κεφαλή MEAN "SOURCE" OR "AUTHORITY OVER" IN GREEK LITERATURE? A REBUTTAL RICHARD S. CERVIN URBANA-CHAMPAIGN, ILLINOIS"

Transcript

1 Triti} IONS (1989) DOES Κεφαλή MEAN "SOURCE" OR "AUTHORITY OVER" IN GREEK LITERATURE? A REBUTTAL RICHARD S. CERVIN URBANA-CHAMPAIGN, ILLINOIS In this paper I shall discuss the meaning of one of the Greek words which is at the center of the debate over women's roles in the Church: κεφαλή "head/ 7 This paper is a rebuttal of Wayne Grudem's article on the meaning of κεφαλή. Grudem's article has obviously had some influence because it is often appealed to by traditionalists in support of an interpretation of 1 Corinthians 11 that men have authority over women. INTRODUCTION In his article "Does kephalê('head') Mean 'Source 7 or 'Authority Over' in Greek Literature? A Survey of 2,336 Examples," 1 Wayne Grudem claims to have analyzed 2,336 occurrences of the word in Greek literature in order to determine whether κεφαλή can mean "source" or "authority over." His findings are directly relevant for our understanding of Paul's use of this word in the New Testament. Grudem concludes that (1) κεφαλή never means "source," and (2) "authority over" is a "common and readily understood" meaning of the word, and that the latter meaning "best suits the New Testament" (p. 80). Is Grudem correct in his assessment of the meaning of κεφαλή? My answer is "no." Grudem's article includes some questionable assumptions. I will expose Grudem's assumptions, and I will further demonstrate that many of the 49 passages which Grudem cites as evidence for "authority over" do not mean what Grudem claims they mean, and that Grudem has misrepresented the evidence. The first part of this paper will contain a summary and critique of Grudem's assumptions and methodology. In Part Two I will discuss Grudem's treatment of the argument for the meaning of "source." in Part Three I will discuss each of Grudem's examples at length, and I will demonstrate that most of the examples Grudem cites do not *As an appendix in The Role Relationship of Men and Women, by George W. Knight III (revised ed., Chicago: Moody, 1985). All quotations from Grudem's article are taken from the appendix in this book. The article also appeared in TrinJ 6 (1985)

2 86 TRINITY JOURNAL support his claim. All translations of original texts are my own unless otherwise specified. /. METHODOLOGY There are several problems in the methodology of Grudem's argument. First, he invokes evidence from various léxica. Grudem seems to take a rather disparaging view of Liddell-Scott-Jones's Greek-English Lexicon 2 (henceforth LSJ). He says: "Liddell-Scott is the tool one would use when studying Plato or Aristotle, for example; but it is not the standard lexicon that scholars use for the study of the New Testament" (p. 62). Grudem has a great deal of praise for Bauer-Arndt-Gingrich 3 (henceforth BAG) as the "standard" New Testament lexicon. In making these statements regarding LSJ and BAG, Grudem has demonstrated that he does not really understand the significance of LSJ. Grudem wrongly claims that LSJ "emphasizes Classical Greek" (ibid.). This is not so. LSJ is the only comprehensive Greek-English lexicon of Ancient Greek currently available. While LSJ was originally planned to cover only Classical Greek, 4 it currently covers Homer and other pre-classical authors, the Classical period, the Hellenistic period, and the Graeco- Roman period, including the New Testament and the Septuagint (this amounts to a time span of roughly 1400 years, 800 B.C. to A.D. 600). In order to deal expressly with the New Testament and the Septuagint, the contributors and editors of LSJ included a team of theologians, Milligan among them. 5 The value of BAG lies more in its citations of literature and its bibliography than in the definitions per se. I do not wish to undermine the value of BAG, but it is deficient in certain respects (e.g., it does not treat the idiomatic expressions of prepositions while LSJ does). Insofar as theologians use only BAG, they automatically restrict their understanding of the Greek language, which in turn seriously affects their exegesis. Just as numerous NT léxica have been produced over the years, so also are there léxica for very many individual Greek authors. I have checked the following for any definition of "authority over, leader" for κεφαλή: H. Stephanus, Thesaurus Graecae Linguae (8 vols.; revised by K. Hase, W. and L. Dindorf; Paris: A. Firmin Didot, ); F. W. Sturz, Lexicon Xenophonteum (4 vols.; Leipzig; ); D. F. Ast, Lexicon Platonicum sive vocum Platonicarum (3 vols.; Leipzig: Weidmann, ); E.-A. Bétant, Lexicon Thucydideum (2 vols.; Geneva, ); W. Dindorf, Lexicon Sophocleum (Leipzig: Teubner, 1870); F. Ellendt, Lexicon Sophocleum (2nd ed.; corrected by Hermann Genthe; Berlin, 1872); W. Dindorf, Lexicon Aeschyleum (Leipzig: Teubner, 1876); J. Rum- ^th edition, with Supplement (Oxford: Oxford University Press, 1968). 3 A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 1957). 4 PrefacetoLSJ,p.lO. s Ibid.,p.9.

3 CERVDM: Κεφαλή IN GREEK LITERATURE 87 pel, Lexicon Theocriteum (Leipzig: Teubner, 1879); R. J. Cunliffe, A Lexicon of the Homeric Dialect (1924; new edition, Norman: University of Oklahoma Press, 1963); J. E. Powell, A Lexicon to Herodotus (2nd ed.; 1938; reprinted by Georg Olms Verlag, 1977); A. Mauersberger, Polybios-Lexicon (4 parts; Berlin: Acadamie-Verlag, ); J. H. Sleeman and G. Pollet, Lexicon Plotinianum (Leiden: E. J. Brill, 1980); J. I. McDougall, Lexicon in Diodorum Siculum (2 vols.; Georg Olms Verlag, 1983). None of these has any such listing; in fact, the only lexicon I have found which defines κεφαλή as "leader" is D. Dhimitrakou, Μέγα Λεξικον της Ελληνικής Γλώσσης (9 vols.; Athens, privately published, ), and he explicitly states that the meaning of "leader" is medieval (vol. 5, p. 3880). The seventh edition of Liddell and Scott also notes that κεφαλή denoting chief is Byzantine (seventh edition, revised and augmented throughout [New York: Harper and Bros., 1889] 801), but this reference was deleted from the ninth edition along with all other references to Byzantine Greek (see Preface to ninth ed., pp ). Apparently, the only other léxica to include such a definition are the NT léxica. Why is this so? The soil of Greek lexicography has been amply tilled and ploughed over the centuries, and if "leader" is a common understanding of κεφαλή, as Grudem claims, then why is it apparently never so listed in any Greek lexicon outside the purview of the NT? I offer several possible reasons, not the least of which is tradition and a male-dominant world-view. The expertise of theologians is the NT, not Classical, or even Hellenistic, Greek, per se. While it may be true that some theologians have had a grounding in Classical Greek (especially those of the 19th century), they spend their time pondering the NT, not Plato, Herodotus, or Plutarch. And it must never be forgotten that it was philologists like Moulton and Deissmann who exploded the myth that the language of the NT was "special" or "unique," rather than the colloquial Koiné. Another reason stems from Latin a very unlikely source. In the West, Latin has always been more popular than Greek, and until the last century Latin was the lingua franca of the scholarly world. Now the Latin word for "head," caput, does have the metaphorical meaning of "leader" (see the Oxford Latin Dictionary, p. 274f). Thus, for English-speaking theologians, at least, English, Hebrew, and Latin all share "leader" as a common metaphor for head. Thus, the forces of tradition, a male-dominant culture, the identical metaphor in three languages, and a less than familiar understanding of the Greek language as a whole, could, in my mind, very easily lead theologians to assume that the metaphor of "leader" for head must be appropriate for Greek as well. Grudem assumes that if "leader" is a common metaphor for κεφαλή, then there should be several examples of such a usage in Greek authors of the Classical, Hellenistic, and Graeco-Roman periods. Grudem is correct in this assumption. He therefore set about to collect a sampling of the occurrences of the word in several Greek

4 88 TRINITY JOURNAL authors ranging from Homer (8th c. B.C.) to Libanius (4th c. A.D.) in order to see if and how often the metaphor of "leader" is used by native Greeks. This is a proper methodological first step. Grudem says that he took a collection of about 2000 occurrences from the Thesaurus Linguae Graecae (TLG). The authors checked and the number of occurrences in each author are listed by Grudem on pp of his article. With respect to the authors listed by Grudem, he claims that "all the extant writings of an author were searched and every instance of kephalë'v/às examined and tabulated with the exception of fragmentary texts and a few other minor works that were unavailable to me" (p. 65, emphasis mine). I myself have access to the TLG here at the University of Illinois, and I have checked several of the authors in Grudem's list as to the frequencies. I have found some rather different figures for the same authors in Grudem's list: Grudem claims that κεφαλή occurs 114 times in Herodotus I found 121 occurrences; Grudem found 56 in Aristophanes I found 59; Grudem found 97 in Plato I found 90; Grudem found 1 in Theocritus I found 15. The discrepancy may be due to our using different "editions" of the TLG database; but his assertion that he has checked every instance may be overstated. 6 Grudem further states that the Loeb editions were used by him "where available; otherwise, standard texts and translations were used" (p. 65, emphasis mine). I find the last phrase of this sentence very disturbing. One cannot conduct a word-study of Greek (or any foreign language) by using translations! One must have the original text! Furthermore, how does one know which Loeb editions were available to Grudem and when he used translations? Nowhere does he identify which text he used for his examples. Grudem notes in passing that his study did not turn up any examples of κεφαλή meaning "source" (p. 68). It must be pointed out, however, that two of his examples (21-22) are cited by Payne for "source." 7 These examples will be dealt with later. Against those who claim that κεφαλή may denote "source," Grudem says that in order to demonstrate that κεφαλή may indeed mean "source," the examples "ought to be cases in which the meaning is unambiguous and not easily explained in terms of other known senses of kephalë". (That is consistent with sound lexical research)" (p. 70, emphasis mine). This is very true in principle, and is equally true of Grudem's study. Unfortunately for Grudem, two of his examples do not exist, and the vast majority of the rest are either ampere have been a number of corrections, additions, and deletions to the TLG databank since Grudem received his printout in One of the drawbacks to the TLG databank, and it is a serious one, is that variant readings are not taken into account. Furthermore, the TLG databank is based on standard Greek texts (Oxford, Teubner, Budé, etc.), and the editors of the TLG have not, as far as I know, practiced textual criticism as they processed the texts. 7 Response to "What does kephalr Mean in the New Testament?" by B. and A. Mickelsen, in Women, Authority, and the Bible (ed. Β. and A. Mickelson; Downers Grove: Inter-Varsity, 1986) 124f.

5 CERVIN: Κεφαλή IN GREEK LITERATURE 89 biguous, false, or illegitimate on other grounds. This will become clear by examining Grudem's examples in their context. II. ON THE MEANING "SOURCE" Grudem critiques the various arguments which have been put forth in recent years by those who advocate ''source" as the meaning of κεφαλή in the NT. Grudem condemns, and rightly so, the Mickelsens and some NT commentators (e.g., F. F. Bruce and C. K. Burkett, among others), who have claimed that the meaning of "source" is "common" for κεφαλή. Grudem points out that the alleged "common meaning" of "source" was propounded by Bedale in the 1950s, and Grudem proceeds briefly to criticize Bedale. I have not seen Bedale's article, so I will not comment on it, except to say that some of Grudem's criticisms appear valid. Grudem points out that the actual attestation for the meaning of "source" rests on two citations from the ancient literature: Herodotus 4.91 and the Orphic Fragment 21A. Grudem points out, again rightly, that two examples do not constitute "common," especially when both examples are from the Classical and pre-classical periods (respectively). (However, it must be pointed out that, out of 2,336 occurrences, Grudem claims to have found 49 examples of head meaning "leader"; that is 2.1%, a figure which hardly deserves the epithet "common" by anyone's standards.) Grudem further proceeds to dismiss the translation of "source" for both of these passages, and in this he is wrong. Grudem dismisses the Herodotus passage by quoting the several meanings cited in LSJ for κεφαλή denoting "end, top, brim," etc., and concludes that when Herodotus speaks of the κεφαλαί of the river, he means "the many 'ends' of a river where tributaries begin to flow toward the main stream" (p. 58). He goes on to state: "Those who cite Herodotus or the Tiead of a river' examples to show that kephale could have meant 'source' at the time of the New Testament have not been careful enough in their use of Herodotus or Liddell- Scott" (ibid.). These words are equally true of Grudem himself because he has failed to comprehend Herodotus. The entire passage, , is rather long to be cited in full, but I will cite enough to show that Grudem's explanation is wrong: Δαρείος δϊ ώς διέβη τον Βόσπορον κατά τήν σχεδίην, έπορεύετο δια της θρηίκης, άπικόμενος δϊ έπί Τεάρου ποταμού τάς πηγάς έστρατοπεδεύσατο ημέρας τρεις, ό δέ Τέαρος λέγεται ύπο των περιοίκων ávaí ποταμών άριστος τά τε άλλα (τα) ές άκεσιν φέροντα και δή και άνδράσι και ΐπποισι ψώρην άκέσασθαι. ειστ δϊ αύτοΰ ai πηγαι δυών δέουσαι τεσσεράκοντα, έκ πέτρης της αυτής ^έουσαι- και ai μ^ν αύτέων είσί ψυχραί, ai & θερμαί. ( ) 8 8 Ι have used the Oxford Classical Text of Herodotus.

6 90 TRINITY JOURNAL Now when Darius had crossed the Bosporus on the pontoon bridge, he proceeded through Thrace and, arriving at the source [lit. "springs"] of the Teams river, he camped [there] for three days. The Tearus is said by the locals to be the best river, in that it is curative in many respects, and it especially cures scurvy in both men and horses. There are 38 springs flowing out of the same rock, some cold and some hot. In context, it is clear that Herodotus is discussing the "source" (πηγαί) of the Tearus river. There are 38 springs, some hot, some cold, which form the source of the river. Darius camped by these springs for three days, and was so impressed with the springs that he ordered a stele erected at the spot which began: Τεάρου ποταμού κεφαλαΐ ΰδωρ άριστον τε και κάλλιστον παρέχονται πάντων ποταμών... (4.91.2) The source [lit. "heads"] of the Tearus river, provides the best and most beautiful water of all rivers... The context of this passage should make it abundantly clear that Herodotus is using κεφαλαί as a synonym of πηγαί, referring to the source of the Tearus. Regarding the Orphic Fragment, Grudem contends that "source" is an inappropriate meaning for κεφαλή as an epithet of Zeus. There are two problems with this fragment, however. First, there is a variant text. Grudem notes the presence of the variant, but he downplays its significance. Secondly, and more importantly, this entire fragment is ambiguous. Following are the two fragments as found in Kern: 9 Fragment 21: Ζευς αρχή, Ζευς μέσσα, Διός δ' εκ πάντα τέτυκται. Ζευς πυθμήν γαίης τε και ουρανού άστερόεντος. Zeus is the beginning, Zeus is the middle, and by Zeus everything is accomplished. Zeus is the foundation both of earth and of sparkling heaven. Fragment 21A: Ζευς πρώτος γένετο, Ζευς ύστατος άργικέραυνος' Ζευς κεφαλή, Ζευς μέσσα- Διός δ* έκ πάντα τελείται Ζευς πυθμήν ναίης τε και ουρανού άστερόεντος Ζευς άρσην γένετο, Ζευς άμβροτος άπλετο νύμφη Ζευς πνοιή πάντων, Ζευς ακαμάτου πυρός ορμή. Ζευς πόντου όίζα Ζευς ήλιος ήδέ σελήνη Ζευς βασιλεύς, Ζευς αρχός απάντων άργικέραυνος' 9 Otto Kern, Orphicorum Fragmenta (Berlin: Weidmann, 1922) 91f.

7 CERVIN: ΚεφαλήEM GREEK LITERATURE 91 πάντας γαρ κρύψας αύθις φάος ές πολυγηθές εξ ίερής κραδίης άνενέγκατο, μέρμερα βέζων. Zeus is first,lightning-flashing Zeus is last; Zeus is head, Zeus the middle, and by Zeus everything is accomplished; Zeus is the foundation both of earth and of sparkling heaven; Zeus is male, Zeus is the bride immortal; Zeus is the breath of everything, Zeus is the rage of unresting fire; Zeus is the root of the sea, Zeus is the sun and the moon; Zeus is king, Zeus is the lightning-flashing leader of all; for having covered everyone, he who does baneful things once again brings [them] to delightful light out of his sacred heart. Fragment 21A has κεφαλή whereas Fragment 21 has αρχή, which may mean "source'' or, as Grudem notes, "beginning/ 7 Grudem's understanding of "beginning" for this fragment is quite valid. However, the understanding of "source" is also quite valid, and can be supported in two ways: (1) the scholiast (cited by Kern) has this comment regarding Frag. 21: και αρχή μεν ούτος ώς ποιητικον αίτιον, τελευτή δε ώς τελικόν, μέσα δε ώς εξ ίσου πάσι παρών, καν πάντα διαφόρως αύτοΰ μετέχηι ("And he is the beginning, as the producing cause, and he is the end as the final cause, and he is the middle, as being present in everything equally, and everything partakes of him in a variety of ways.") The idea of "source" is clear; Zeus is the source of everything, he is the first cause. (2) The understanding of "source" can be found in the clause Διός δ' εκ πάντα τελείται/τέτυκται. This clause is itself ambiguous, and may be taken in two ways. Έκ may be in tmesis and go with the verb, in which case the genitive Διός depends on πάντα and can be construed as a "genitive of source" 10 thus: Διός πάντα εκτελείται /έκτέτυκται "everything from Zeus has been accomplished" (the hyperbaton involved in this reading is not difficult as far as Greek poetry is concerned). Alternatively, έκ may be in anastrophe and thus go with Διός, making Διός the agent of the passive: 11 έκ Διός πάντα τελείται/τέτυκται "everything is done by Zeus." Either reading is possible. Grudem's assertion that "source" is "doubtful" in this passage (p. 60) is erroneous. Zeus as the "head/beginning/source/origin/cause" are all plausible readings. This fragment contains a series of epithets of Zeus. Otherwise, there is really no context which can be appealed to in order to settle which meaning(s) were intended by the author, or if all of the possible meanings were intended. As an additional note to this fragment, it may very well be the case that the word "head" is used as a sort of technical term within the Orphic Cult. If this were so, then this fragment would not be relevant for the NT at all. It would 10 Herbert Weir Smyth, Greek Grammar (rev. Gordon M. Messing, Cambridge Press: Harvard University Press, 1956) u Ibid.,

8 92 TRINITY JOURNAL take a specialist in Orphic religion to determine if this word is a technical term or not. As for other examples of κεφαλή meaning "source/' Payne cites two passages from Philo, and Àree from Artemidorus Daldianus. 12 In Preliminary Studies 61, Philo writes of Esau: κεφαλή δϊ ώς ζφου πάντων των λεχθέντων μερών ό γενάρχης έστχν Ήσαΰ, δς τότε μ ν ποίημα, τοτέ δε δρυς ερμηνεύεται... (Loeb). Like the head of a living creature, Esau is the progenitor of all the clans mentioned so far; [his name] is sometimes interpreted as "product" and sometimes as "oak"... Note, however, that Philo does not call Esau the "head" of his clans. Philo is using a simile, "like the head of a living creature," to describe Esau. This simile (like many of the examples Grudem cites, which will be discussed later) has nothing to do with "source" or "authority." It is simply a head-body metaphor which shows that Esau is the "topmost" or "preeminent" part of his clan, just like the head of an animal is the topmost or preeminent part of the animal's body. Philo's On Rewards and Punishments 125 is cited by Payne as meaning "source" while Grudem cites this same passage as an example of "authority" (his examples 21-22). This passage will be dealt with later. Payne also cites six occurrences of κεφαλή meaning "source" from Artemidorus Daldianus (2nd c. A.D.), whose Onirocriticon 12 is a collection of dreams and their interpretations. In Book 1 of his collection, Daldianus sets up a system for the interpretation of dreams whereby parts of the human body represent members of the household: the head represents the father; the feet represent the slaves; the right hand represents a male member while the left hand represents a female member; and so forth (Onirocriticon 1.2). Daldianus uses this system throughout his book. Several of the passages cited by Payne do not warrant the interpretation of "source," however. The passages cited by Payne (with more context than he gives), are as follows: 1.2 (in Pack's text: p. 7, p. 8,1.1): και πάλιν έδοξέ τις τετραχηλοκοπήσθαι. συνέβη και τούτου τον πατέρα άποθανειν, δς και του ζην και του φωτός αίτιος ήν, ώσπερ και ή κεφαλή του παντός σώματος, οίον [δέ] έστι και το τετυφλώσθαι τέκνοις δλεθρον και ούχι τφ ίδόντι σημαίνον και πολλά άλλα δσα τοιαύτα ειποι τις άν. 12 "Response," 124f. n Artemidori Daldiani, Onirocriticon Libri V (ed. Roger A. Pack; Leipzig: Teubner, 1963).

9 CERVEM: Κεφαλή IN GREEK LITERATURE 93 And again, someone thought that he had been decapitated. It turned out that this man's father had died, who [the father] was the source of both life and light, just as the head is [the source] of the entire body. For example, to be blind is destructive for children, and not just for the one who sees a vision [lit. sign], but [who sees] many other things which one would speak of (in Pack's text: p. 9,2.6-11): οίον κεφαλή εις πατέρα, πους ας δοΰλον, δεξιά χάρ εις πατέρα υΐον φίλον άδελφόν, αριστερά χάρ άς γυναίκα κοά μητέρα και φίλην και θυγατέρα και άδελφήν, αίδοΐον εις γονείς και γυναίκα και τέκνα, κνήμη εις γυναίκα κα\ φίλην. των δϊ άλλων έκαστον, τίνα μή μακρολογωμεν, οΰτω σκοπητέον. For example, the head represents the father; the foot represents the slave; the right hand represents the father, son, male-friend, brother; the left hand represents the wife, mother, female-friend, daughter, and sister; the genitals represent the parents, wife and children; the shin represents a woman and female-friend. Thus, each of the other matters must be considered, so that we may not be long-winded (in Pack's text: p. 43, ): Αφηρήσθαι δέ δοκειν της κεφαλής είτε καταδίκης είτε ύπο ληστών ατε έν μονομαχία Ατε οΐφδήποτε τρόπφ (ού γαρ διαφέρει) πονηρον τφ γονείς Ιχοντι κάί τφ τέκνα γονεΰσι μέν γαρ έοικεν ή κεφαλή δια το του ζην αίτίαν ávar τέκνοις ok δια το πρόσωπον και τήν εικόνα. And it seems that to deprive [one] of his head, either by legal judgement, or by thieves, or by single combat, or by any other means (for it makes no difference), is an evil deed as far as the one who has parents since they are the cause of life; and to the children because of the face and image (in Pack's text: p. 234, ): έν μέν (ούν) τφ πρώτφ βιβλίφ Ιφην είναι τήν κεφαλήν πατέρα του ίδόντος έν δϊ τφ δευτέρφ λέοντα ávaí βασιλέα ή νόσον, και έν τφ περί θανάτου τοις πενομένοις το άποθανειν χρηστον είναι και λυσιτελές επέδειξα, έπειδάν ούν πένης άνήρ πατέρα έχων πλούσιον όναρ ύπο λέοντος τήν κεφαλήν άφηρήσθαι δόξη κώ άποθανειν, είκός έστι τον πατέρα αύτοϋ αποθανόντα κληρονόμον αυτόν κατακλείψειν, και τούτον τον τρόπον άλυπος &ν γένοιτο και εύπορος, ούτε φορτικον έτι έχων τον πατέρα ούτε ύπο της πενίας θλιβόμενος έστι γαρ ή μέν κεφαλή ό πατήρ, ή δϊ άφαίρεσις ή στέρησις του πατρός ό δέ λέων ή νόσος ήν νοσήσας ό πατήρ άποθάνοι άν ό δϊ θάνατος ή μεταβολή του βίου και το δια τον πλούτον άνενδεές.

10 94 TRINITY JOURNAL In the first book I said that the head was [represented] the father of the dreamer [lit. the one who sees], and in the second book the lion was [represented] a king or a disease, and in the book about death I demonstrated that it is good and beneficial for the poor to die. Now whenever a poor man who has a wealthy father dreams that he is deprived of his head by a lion and dies, it is likely that when his father dies, he will leave [him] an inheritance, and in this manner he will be without grief and [will be] well-off, neither having his father as a burden, nor suffering by poverty; for the head is the father, and the deprivation [of the head] is the loss of the father; the lion is the disease which the father contracts and dies from; and the death [of the father] is the change of livelihood due to the wealthy man's abundance. It should be apparent that Artemidorus Daldianus's use of head is directly related to his theory of dream interpretation. He uses head more as a representation of one's father than as a metaphor for "source/ 7 Furthermore, only two of these passages, 1.2 and 1.35, mention anything about the head being the "source" or "cause" of life. III. GRUDEM'S EXAMPLES Grudem has cited 49 examples of what he claims are occurrences of κεφαλή*meaning "authority over" or "leader." Let us examine each passage in detail to see if Grudem is correct. A few of the passages Grudem cites are incorrectly referenced, and shall be noted. First of all, 12 of these passages (nos ) are from the NT, and are therefore illegitimate as evidence, since they are disputed texts. In citing these NT passages, Grudem commits the logical fallacy of assuming what he sets out to prove. The whole purpose of Grudem's study is to determine whether or not κεφαλή can denote "authority over " or "leader" in Paul's epistles. He cannot therefore cite Paul as supporting evidence. This brings his count down to 37. What then of the rest of his examples? (1-2) The first two come from Herodotus (5th c. B.C.) and the second example is not even the word κεφαλή, it is κάρη! By failing to cite the Greek text of Herodotus, Grudem leads his readers to conclude that κεφαλή is used twice in this passage, a conclusion which is patently false. In the context of this passage, the Argives, a Greek tribe, send to the Delphic oracle for advice as to their best course of action in view of the pending invasion of Greece by Persia. The Argives had just lost 6,000 soldiers in battle with the Spartans. The oracle answers: εχθρέ περικτιόνεσσι, φίλ' άθανάτοισι θεοΐσι, ασω τον προβόλαιον έχων πεφυλαγμένος ήσο και κεφαλήν πεφύλαξο- κάρη δε το σώμα σαώσα Ι have used the Oxford Classical Text of Herodotus.

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

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

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

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

C.S. 430 Assignment 6, Sample Solutions

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

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

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

Διαβάστε περισσότερα

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

Math 6 SL Probability Distributions Practice Test Mark Scheme

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

Διαβάστε περισσότερα

Section 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

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

Lecture 2. Soundness and completeness of propositional logic

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

Διαβάστε περισσότερα

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

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

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

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Διαβάστε περισσότερα

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

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

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

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

Διαβάστε περισσότερα

Ακαδημαϊκός Λόγος Εισαγωγή

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

5.4 The Poisson Distribution.

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

Διαβάστε περισσότερα

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

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

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

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΠΑΡΑΓΟΝΤΕΣ ΠΟΥ ΕΠΗΡΕΑΖΟΥΝ ΤΗ ΖΩΗ ΤΟΥ ΠΑΙΔΙΟΥ ΚΑΙ ΕΦΗΒΟΥ ΜΕ ΔΙΑΒΗΤΗ ΤΥΠΟΥ 1 ΠΟΥ ΧΡΗΣΙΜΟΠΟΙΟΥΝ ΑΝΤΛΙΕΣ ΣΥΝΕΧΟΥΣ ΕΚΧΥΣΗΣ ΙΝΣΟΥΛΙΝΗΣ

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

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ

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

Διαβάστε περισσότερα

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

Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ

Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ Πτυχιακή Εργασία Η ΠΟΙΟΤΗΤΑ ΖΩΗΣ ΤΩΝ ΑΣΘΕΝΩΝ ΜΕ ΣΤΗΘΑΓΧΗ Νικόλας Χριστοδούλου Λευκωσία, 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ

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

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

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

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

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

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

Διαβάστε περισσότερα

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

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

TMA4115 Matematikk 3

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

Διαβάστε περισσότερα

Verklarte Nacht, Op.4 (Εξαϋλωμένη Νύχτα, Έργο 4) Arnold Schoenberg (1874-1951)

Verklarte Nacht, Op.4 (Εξαϋλωμένη Νύχτα, Έργο 4) Arnold Schoenberg (1874-1951) 1 Verklarte Nacht, Op.4 (Εξαϋλωμένη Νύχτα, Έργο 4) Arnold Schoenberg (1874-1951) Αναγνώσματα από το βιβλίο Η Απόλαυση της Μουσικής (Machlis, Forney), για τους μαθητές που θα μελετήσουν το έργο: «Ο Σαίνμπεργκ

Διαβάστε περισσότερα

Living and Nonliving Created by: Maria Okraska

Living and Nonliving Created by: Maria Okraska Living and Nonliving Created by: Maria Okraska http://enchantingclassroom.blogspot.com Living Living things grow, change, and reproduce. They need air, water, food, and a place to live in order to survive.

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

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

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

«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

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

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

Διαβάστε περισσότερα

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

Μηχανική Μάθηση Hypothesis Testing

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

[1] P Q. Fig. 3.1

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

Διαβάστε περισσότερα

ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ

ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΕΚΠΑΙΔΕΥΣΗΣ ΣΤΗΝ ΠΡΟΣΧΟΛΙΚΗ ΗΛΙΚΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Διαπολιτισμική Εκπαίδευση και Θρησκευτική Ετερότητα: εθνικές και θρησκευτικές

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

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

1999 MODERN GREEK 2 UNIT Z

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

Διαβάστε περισσότερα

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

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

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

Modern Greek Extension

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

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

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

Διαβάστε περισσότερα

Θέμα διπλωματικής εργασίας: «Από το «φρενοκομείο» στη Λέρο και την Ψυχιατρική Μεταρρύθμιση: νομικό πλαίσιο και ηθικοκοινωνικές διαστάσεις»

Θέμα διπλωματικής εργασίας: «Από το «φρενοκομείο» στη Λέρο και την Ψυχιατρική Μεταρρύθμιση: νομικό πλαίσιο και ηθικοκοινωνικές διαστάσεις» ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΕΣ ΙΑΤΡΙΚΗΣ & ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΤΜΗΜΑΤΑ ΝΟΜΙΚΗΣ & ΘΕΟΛΟΓΙΑΣ ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ «ΣΥΓΧΡΟΝΕΣ ΙΑΤΡΙΚΕΣ ΠΡΑΞΕΙΣ: ΔΙΚΑΙΙΚΗ ΡΥΘΜΙΣΗ ΚΑΙ ΒΙΟΗΘΙΚΗ

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

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

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

Διαβάστε περισσότερα

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

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

Test Data Management in Practice

Test Data Management in Practice Problems, Concepts, and the Swisscom Test Data Organizer Do you have issues with your legal and compliance department because test environments contain sensitive data outsourcing partners must not see?

Διαβάστε περισσότερα

Galatia SIL Keyboard Information

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

Διαβάστε περισσότερα

ΠΑΡΑΜΕΤΡΟΙ ΕΠΗΡΕΑΣΜΟΥ ΤΗΣ ΑΝΑΓΝΩΣΗΣ- ΑΠΟΚΩΔΙΚΟΠΟΙΗΣΗΣ ΤΗΣ BRAILLE ΑΠΟ ΑΤΟΜΑ ΜΕ ΤΥΦΛΩΣΗ

ΠΑΡΑΜΕΤΡΟΙ 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 περισσότερα

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

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

Διαβάστε περισσότερα

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

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

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

Διαβάστε περισσότερα

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

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

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

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

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

Διαβάστε περισσότερα

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

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

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

Διαβάστε περισσότερα

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

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