9 Εἶχε μὲν οὖν [καὶ] ἡ πρώτη δικαιώματα λατρείας τό τε ἅγιον κοσμικόν.

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "9 Εἶχε μὲν οὖν [καὶ] ἡ πρώτη δικαιώματα λατρείας τό τε ἅγιον κοσμικόν."

Transcript

1 The Earthly Sanctuary 9 Εἶχε μὲν οὖν [καὶ] ἡ πρώτη δικαιώματα λατρείας τό τε ἅγιον κοσμικόν. It was holding indeed then [also] the first 1 regulations of service the and sanctuary earthly. 2 σκηνὴ γὰρ κατεσκευάσθη ἡ πρώτη, ἐν ᾗ ἥ τε λυχνία καὶ ἡ τράπεζα A tabernacle for was prepared the first, in which the both lamp-stand and the table καὶ ἡ πρόθεσις τῶν ἄρτων, ἥτις λέγεται Ἅγια and the purpose of the loaves, 2 which is called Holy; 3 μετὰ δὲ τὸ δεύτερον καταπέτασμα σκηνὴ ἡ λεγομένη Ἅγια Ἁγίων, after and the second curtain a tabernacle the being called Holy of Holies, 4 χρυσοῦν ἔχουσα θυμιατήριον καὶ τὴν κιβωτὸν τῆς διαθήκης a golden having altar 3 and the ark of the covenant περικεκαλυμμένην πάντοθεν χρυσίῳ, ἐν ᾗ στάμνος χρυσῆ ἔχουσα τὸ having been covered all round on all sides 4 with gold, in which a pot 5 golden holding the the μάννα καὶ ἡ ῥάβδος Ἀαρὼν ἡ βλαστήσασα καὶ αἱ πλάκες τῆς διαθήκης, manna and the rod of Aaron the having budded and the tables 6 of the covenant, 5 ὑπεράνω δὲ αὐτῆς Χερουβὶν δόξης κατασκιάζοντα τὸ ἱλαστήριον περὶ ὧν above and it cherubim of glory overshadowing 7 the mercy-seat 8 concerning which οὐκ ἔστιν νῦν λέγειν κατὰ μέρος. not is now to speak in detail. 6 Τούτων δὲ οὕτως κατεσκευασμένων εἰς μὲν τὴν πρώτην σκηνὴν διὰ παντὸς These things and thus having been prepared into indeed the first tabernacle constantly εἰσίασιν οἱ ἱερεῖς τὰς λατρείας ἐπιτελοῦντες, go in the priests the services performing, 7 εἰς δὲ τὴν δευτέραν ἅπαξ τοῦ ἐνιαυτοῦ μόνος ὁ ἀρχιερεύς, οὐ χωρὶς αἵματος ὃ into but the second once of the year alone the chief priest, not without blood which προσφέρει ὑπὲρ ἑαυτοῦ καὶ τῶν τοῦ λαοῦ ἀγνοημάτων, he offers on behalf of himself and the of the people sins of ignorance, 9 8 τοῦτο δηλοῦντος τοῦ πνεύματος τοῦ ἁγίου, μήπω πεφανερῶσθαι τὴν τῶν ἁγίων this making plain of the Spirit of the Holy, not yet to have been revealed the of the holy ὁδὸν ἔτι τῆς πρώτης σκηνῆς ἐχούσης στάσιν, way still the first tabernacle having a standing, 9 ἥτις παραβολὴ εἰς τὸν καιρὸν τὸν ἐνεστηκότα, καθ ἣν δῶρα τε which a parable for the time the having present, according to which gifts both καὶ θυσίαι προσφέρονται μὴ δυνάμεναι κατὰ συνείδησιν τελειῶσαι and sacrifices being offered not being able in regard to conscience to perfect 1 Both 'tabernacle' and 'testament' are feminine so either could be implied here. 2 The 'shewbread'. 3 θυμιατήριον, n.n. altar of incense as used by Philo & Josephus, but 'censer' in earlier works only here in the NT. 4 Also Lk. 19:43. 5 στάμνος, n.n. pot only here in the NT. 6 Also in 2 Cor. 3:3. 7 κατασκιάζοντα, v. overshadow, cover with shade, present, active, participle, - only here in the NT 8 ἱλαστήριον, n.n. mercy-seat, propitiation, the covering of the ark, the place where sins are forgiven, also in Rom. 3:25. 9 ἀγνοημάτων, n.n. sin committed in ignorance only here in the NT. 1

2 τὸν λατρεύοντα, the serving, 10 μόνον ἐπὶ βρώμασιν καὶ πόμασιν καὶ διαφόροις βαπτισμοῖς, δικαιώματα σαρκὸς μέχρι only on foods and drinks and various 10 washings, ordinances carnal until καιροῦ διορθώσεως ἐπικείμενα. time of new order 11 imposed upon. 12 The Heavenly Sanctuary 11 Χριστὸς δὲ παραγενόμενος ἀρχιερεὺς τῶν γενομένων ἀγαθῶν, διὰ τῆς Christ and having come chief priest of the having become 13 good things, through the μείζονος καὶ τελειοτέρας σκηνῆς οὐ χειροποιήτου τοῦτ ἔστιν οὐ ταύτης greater and more perfect 14 tabernacle not made with hands this is not of this τῆς κτίσεως, the creation, 12 οὐδὲ δι αἵματος τράγων καὶ μόσχων διὰ δὲ τοῦ ἰδίου αἵματος εἰσῆλθεν nor through blood of goats and of calves through but of the own blood he entered in ἐφάπαξ εἰς τὰ ἅγια, αἰωνίαν λύτρωσιν εὑράμενος. once for all into the holy places, eternal redemption 15 having found. 13 εἰ γὰρ τὸ αἷμα τράγων καὶ ταύρων καὶ σποδὸς δαμάλεως ῥαντίζουσα τοὺς If for the blood of goats and bulls and ashes of a red heifer 16 sprinkling the κεκοινωμένους ἁγιάζει πρὸς τὴν τῆς σαρκὸς καθαρότητα, having been defiled sanctifies unto the of the flesh purifying, πόσῳ μᾶλλον τὸ αἷμα τοῦ Χριστοῦ, ὃς διὰ πνεύματος αἰωνίου ἑαυτὸν προσήνεγκεν how much more the blood of the Christ, who through Spirit Eternal himself offered ἄμωμον τῷ θεῷ, καθαριεῖ τὴν συνείδησιν ἡμῶν ἀπὸ νεκρῶν ἔργων εἰς faultless to the God, will cleanse the conscience of us from dead works for τὸ λατρεύειν θεῷ ζῶντι. the to serve God living Καὶ διὰ τοῦτο διαθήκης καινῆς μεσίτης ἐστίν, ὅπως θανάτου γενομένου εἰς And through this testament of a new a mediator he is, so as a death having taken place for ἀπολύτρωσιν τῶν ἐπὶ τῇ πρώτῃ διαθήκῃ παραβάσεων τὴν ἐπαγγελίαν λάβωσιν redemption of te upon the first testament transgressions the promise may receive οἱ κεκλημένοι τῆς αἰωνίου κληρονομίας. the having been called of the eternal inheritance. 16 ὅπου γὰρ διαθήκη, θάνατον ἀνάγκη φέρεσθαι τοῦ διαθεμένου Where for a testament, death of necessity to be born of the testator; 17 διαθήκη γὰρ ἐπὶ νεκροῖς βεβαῖα, ἐπεὶ μὴ τότε ἰσχύει ὅτε ζῇ ὁ διαθέμενος. a testament for over dead firm, since not at all it has force when lives the testator. 10 A word only found in Hebrews here & 1:4, 8:6. 11 διορθώσεως, n.f., - reformation, new order, - only here in the NT. 12 See end note. 13 Many of the best MSS have μελλόντων 'about to come' 14 'perfect' is a superlative and cannot be qualified, but this is how it is translated. 15 Here & Lk. 1:68, 2: δαμάλεως, n.f. red heifer, whose ashes, mixed with water were to be sprinkled on those defiled or contaminated, - only here in the NT. 17 καθαρότητα, n.f. purifying, - only here in the NT. 18 Can be a question, a statement or an exclamation. 2

3 18 ὅθεν οὐδὲ ἡ πρώτη χωρὶς αἵματος ἐγκεκαίνισται Whence not even the first without blood has been dedicated; λαληθείσης γὰρ πάσης ἐντολῆς κατὰ τὸν νόμον ὑπὸ Μωϋσέως παντὶ τῷ λαῷ, having been spoken every command according to the law by Moses to all of the people, λαβὼν τὸ αἷμα τῶν μόσχων καὶ τῶν τράγων μετὰ ὕδατος καὶ ἐρίου κοκκίνου καὶ having taken the blood of the calves and of the goats with water and wool 20 scarlet and ὑσσώπου, αὐτό τε τὸ βιβλίον καὶ πάντα τὸν λαὸν ἐρράντισεν, hyssop, self 21 both the book and all the people he sprinkled, 20 λέγων, Τοῦτο τὸ αἷμα τῆς διαθήκης ἧς ἐνετείλατο πρὸς ὑμᾶς ὁ θεός. saying, This the blood of the testament which commanded unto you the God. 21 καὶ τὴν σκηνὴν δὲ καὶ πάντα τὰ σκεύη τῆς λειτουργίας τῷ αἵματι And the tabernacle moreover and all the vessels of the service of the blood ὁμοίως ἐρράντισεν. likewise he sprinkled. 22 καὶ σχεδὸν ἐν αἵματι πάντα καθαρίζεται κατὰ τὸν νόμον καὶ, χωρὶς And almost by blood all things cleansed according to the law and, without αἱματεκχυσίας οὐ γίνεται ἄφεσις. blood-shedding 22 not becomes remission. The Blood of Christ 23 Ἀνάγκη οὖν τὰ μὲν ὑποδείγματα τῶν ἐν τοῖς οὐρανοῖς τούτοις καθαρίζεσθαι, Necessary therefore the indeed patterns of the in the heavens with these things to be cleansed, αὐτὰ δὲ τὰ ἐπουράνια κρείττοσιν θυσίαις παρὰ ταύτας. selves but the heavenly things with better sacrifices than these. 24 οὐ γὰρ εἰς χειροποίητα εἰσῆλθεν ἅγια Χριστὸς, ἀντίτυπα τῶν ἀληθινῶν, ἀλλ εἰς Not for into made with hands entered holy places Christ, paradigms 23 of the true, but into αὐτὸν τὸν οὐρανόν, νῦν ἐμφανισθῆναι τῷ προσώπῳ τοῦ θεοῦ ὑπὲρ ἡμῶν self the heaven, now to appear in the presence of the God on behalf of us; 25 οὐδ ἵνα πολλάκις προσφέρῃ ἑαυτόν, ὥσπερ ὁ ἀρχιερεὺς εἰσέρχεται εἰς τὰ nor that frequently he should offer himself, just as the chief priest enters into the ἅγια κατ ἐνιαυτὸν ἐν αἵματι ἀλλοτρίῳ, holy places year by year with blood belonging to others, 26 ἐπεὶ ἔδει αὐτὸν πολλάκις παθεῖν ἀπὸ καταβολῆς κοσμοῦ νυνὶ δὲ ἅπαξ ἐπὶ for the it behoves him frequently to suffer from foundation of world; now but once upon συντελείᾳ τῶν αἰώνων εἰς ἀθέτησιν τῆς ἁμαρτίας διὰ τῆς θυσίας αὐτοῦ consummation of the ages for rejection of the sin through the sacrifice of him πεφανέρωται. he has been manifested. 27 καὶ καθ ὅσον ἀπόκειται τοῖς ἀνθρώποις ἅπαξ ἀποθανεῖν, μετὰ δὲ And according to how many it is appointed to the men once to die, after and τοῦτο κρίσις, this judgment, 19 Also in 10:20 20 Also in Revelation 1: ie the book. 22 αἱματεκχυσίας, n.f., blood-shedding, - only here in the NT. 23 Also in 1 Peter 3:21. 3

4 28 οὕτως καὶ ὁ Χριστὸς ἅπαξ προσενεχθεὶς εἰς τὸ πολλῶν ἀνενεγκεῖν ἁμαρτίας, ἐκ so also the Christ once having been offered for the of many to bear sins, from δευτέρου χωρὶς ἁμαρτίας ὀφθήσεται τοῖς αὐτὸν ἀπεκδεχομένοις εἰς σωτηρίαν. second without sin will appear to the him waiting patiently into salvation. The Tabernacle The description of the Tabernacle given in Hebrews derives mainly from the Book of Exodus:- Verse 2. 26:1-30; 25:31-40; 25: Moreover thou shalt make the tabernacle with ten curtains of fine twined linen, and blue, and purple, and scarlet: with cherubims of cunning work shalt thou make them. 2The length of one curtain shall be eight and twenty cubits, and the breadth of one curtain four cubits: and every one of the curtains shall have one measure. 3The five curtains shall be coupled together one to another; and other five curtains shall be coupled one to another. 4And thou shalt make loops of blue upon the edge of the one curtain from the selvedge in the coupling; and likewise shalt thou make in the uttermost edge of another curtain, in the coupling of the second. 5Fifty loops shalt thou make in the one curtain, and fifty loops shalt thou make in the edge of the curtain that is in the coupling of the second; that the loops may take hold one of another. 6And thou shalt make fifty taches of gold, and couple the curtains together with the taches: and it shall be one tabernacle.7and thou shalt make curtains of goats hair to be a covering upon the tabernacle: eleven curtains shalt thou make. 8The length of one curtain shall be thirty cubits, and the breadth of one curtain four cubits: and the eleven curtains shall be all of one measure. 9And thou shalt couple five curtains by themselves, and six curtains by themselves, and shalt double the sixth curtain in the forefront of the tabernacle. 10And thou shalt make fifty loops on the edge of the one curtain that is outmost in the coupling, and fifty loops in the edge of the curtain which coupleth the second. 11And thou shalt make fifty taches of brass, and put the taches into the loops, and couple the tent together, that it may be one. 12And the remnant that remaineth of the curtains of the tent, the half curtain that remaineth, shall hang over the backside of the tabernacle. 13And a cubit on the one side, and a cubit on the other side of that which remaineth in the length of the curtains of the tent, it shall hang over the sides of the tabernacle on this side and on that side, to cover it. 14And thou shalt make a covering for the tent of rams skins dyed red, and a covering above of badgers skins.15and thou shalt make boards for the tabernacle of shittim wood standing up. 16Ten cubits shall be the length of a board, and a cubit and a half shall be the breadth of one board. 17Two tenons shall there be in one board, set in order one against another: thus shalt thou make for all the boards of the tabernacle. 18And thou shalt make the boards for the tabernacle, twenty boards on the south side southward. 19And thou shalt make forty sockets of silver under the twenty boards; two sockets under one board for his two tenons, and two sockets under another board for his two tenons. 20And for the second side of the tabernacle on the north side there shall be twenty boards: 21And their forty sockets of silver; two sockets under one board, and two sockets under another board. 22And for the sides of the tabernacle westward thou shalt make six boards. 23And two boards shalt thou make for the corners of the tabernacle in the two sides. 24And they shall be coupled together beneath, and they shall be coupled together above the head of it unto one ring: thus shall it be for them both; they shall be for the two corners. 25And they shall be eight boards, and their sockets of silver, sixteen sockets; two sockets under one board, and two sockets under another board.26and thou shalt make bars of shittim wood; five for the boards of the one side of the tabernacle, 27And five bars for the boards of the other side of the tabernacle, and five bars for the boards of the side of the tabernacle, for the two sides westward. 28And the middle bar in the midst of the boards shall reach from end to end. 29And thou shalt overlay the boards with gold, and make their rings of gold for places for the bars: and thou shalt overlay the bars with gold. 30And thou shalt rear up the tabernacle according to the fashion thereof which was shewed thee in the mount. 31And thou shalt make a candlestick of pure gold: of beaten work shall the candlestick be made: his shaft, and his branches, his bowls, his knops, and his flowers, shall be of the same. 32And six branches shall come out of the sides of it; three branches of the candlestick out of the one side, and three branches of the candlestick out of the other side: 33Three bowls made like unto almonds, with a knop and a flower in one branch; and three bowls made like almonds in the other branch, with a knop and a flower: so in the six branches that come out of the candlestick. 34And in the candlestick shall be four bowls made like unto almonds, with their knops and their flowers. 35And there shall be a knop under two branches of the same, and a knop under two branches of the same, and a knop under two branches of the same, according to the six branches that proceed out of the candlestick. 36Their knops and their branches shall be of the same: all it shall be one beaten work of pure gold. 37And thou shalt make the seven lamps thereof: and they shall light the lamps thereof, that they may give light over against it. 38And the tongs thereof, and the snuffdishes thereof, shall be of pure gold. 39Of a talent of pure gold shall he make it, with all these vessels. 40And look that thou make them after their pattern, which was shewed thee in the mount. 23Thou shalt also make a table of shittim wood: two cubits shall be the length thereof, and a cubit the breadth thereof, and a cubit and a half the height thereof. 24And thou shalt overlay it with pure gold, and make thereto a crown of gold round about. 25And thou shalt make unto it a border of an hand breadth round about, and thou shalt make a golden crown to the border thereof round about. 26And thou shalt make for it four rings of gold, and put the rings in the four corners that are on the four feet thereof. 27Over against the border shall the rings be for places of the staves to bear the table. 28And thou shalt make the staves of shittim wood, and overlay them with gold, that the table may be borne with them. 29And thou shalt make the dishes thereof, and spoons thereof, and covers thereof, and bowls thereof, to cover withal: of pure gold shalt thou make them. 30And thou shalt set upon the table shewbread before me alway. Verse 3. 26: And thou shalt make a vail of blue, and purple, and scarlet, and fine twined linen of cunning work: with cherubims shall it be made: 32And thou shalt hang it upon four pillars of shittim wood overlaid with gold: their hooks shall be of gold, upon the four sockets of silver. 33And thou shalt hang up the vail under the taches, that thou mayest bring in thither within the vail the ark of the testimony: and the vail shall divide unto you between the holy place and the most holy. Verse 4. 30:1-6; 25:10-16; 16:33; 25:16; and Numbers 17:8-10; Deuteronomy 10:3-5. And thou shalt make an altar to burn incense upon: of shittim wood shalt thou make it. 2A cubit shall be the length thereof, and a cubit the breadth thereof; foursquare shall it be: and two cubits shall be the height thereof: the horns thereof shall be of the same. 3And thou shalt overlay it with pure 4

5 gold, the top thereof, and the sides thereof round about, and the horns thereof; and thou shalt make unto it a crown of gold round about. 4And two golden rings shalt thou make to it under the crown of it, by the two corners thereof, upon the two sides of it shalt thou make it; and they shall be for places for the staves to bear it withal. 5And thou shalt make the staves of shittim wood, and overlay them with gold. 6And thou shalt put it before the vail that is by the ark of the testimony, before the mercy seat that is over the testimony, where I will meet with thee. 10And they shall make an ark of shittim wood: two cubits and a half shall be the length thereof, and a cubit and a half the breadth thereof, and a cubit and a half the height thereof. 11And thou shalt overlay it with pure gold, within and without shalt thou overlay it, and shalt make upon it a crown of gold round about. 12And thou shalt cast four rings of gold for it, and put them in the four corners thereof; and two rings shall be in the one side of it, and two rings in the other side of it. 13And thou shalt make staves of shittim wood, and overlay them with gold. 14And thou shalt put the staves into the rings by the sides of the ark, that the ark may be borne with them. 15The staves shall be in the rings of the ark: they shall not be taken from it. 16And thou shalt put into the ark the testimony which I shall give thee. Verse 5. 25: And thou shalt make two cherubims of gold, of beaten work shalt thou make them, in the two ends of the mercy seat. 19And make one cherub on the one end, and the other cherub on the other end: even of the mercy seat shall ye make the cherubims on the two ends thereof. 20And the cherubim shall stretch forth their wings on high, covering the mercy seat with their wings, and their faces shall look one to another; toward the mercy seat shall the faces of the cherubims be. 21And thou shalt put the mercy seat above upon the ark; and in the ark thou shalt put the testimony that I shall give thee. 22And there I will meet with thee, and I will commune with thee from above the mercy seat, from between the two cherubims which are upon the ark of the testimony, of all things which I will give thee in commandment unto the children of Israel. Verse 6. Numbers 18:2-6. Verse 7. 30:10 and Leviticus 16:2, Ground Plan of the Tabernacle 5

6 The events of the Exodus took place when Egypt was in its late bronze age ie. circa 1490BC (although a number of commentators consider that the Exodus took place in the previous century), however, the story was not recorded until after the death of Solomon circa 975BC and then edited on a number of occasions through to circa 620 BC. It is not surprising, therefore, that the historical accuracy of Exodus has been questioned. It seems likely that the editors of the first millennium BC attributed to Moses the instructions for the construction of a tabernacle of a type that existed immediately before Solomon and in doing gave some authority to the design of Solomon's temple. There are a number of inconsistencies which make the Exodus design an unlikely proposition for a nomadic tribe, of relatively meager means, to have effected. 1. As described the Tabernacle was a structure of considerable size, containing objects requiring skill and wealth to possess. It appears unlikely that, in the time of Moses, the Israelites had the skill, the manpower or the wealth to firstly, construct such an edifice, secondly to constantly dismantle and reassemble it. As described it would have been too heavy, complex and expensive to have been a temporary temple. 2. In all probability the first tabernacle was a tent of the Bedouin style such a tent would have been erected on a sandy or earth with minimal green spread and would have had a floor covering (probably of carpet as do Bedouin tents). It would appear that by the late to middle 1 st millennium BC the necessity for such a covering had been forgotten since no such flooring was mentioned. 3. Similarly the mechanics of tent construction had been forgotten, and, as described, the roof would have sagged down since there was no provision for a ridge pole or supports for it. 4. The editors of the story gave precise measurements for the Holy of Holies, but failed to say if these measurements were internal or external measurements or gave the thickness of the walls, nor make explicit the arrangements at the corners and appear to think that the curtains would hang absolutely vertically and with out any tucks or folds. No doubt the Israelites of the Exodus did have a specially adapted tent which was used for the ordinances of divine worship, but the Exodus design, being as it is, after the manner of the plans of the temples of Solomon, Zerubbabel, when he relaid the temple foundations after the destruction of Nebuchadnezzar and again by Herod, appears to have been written, and used, to justify those later designs. Bennett, W.H., 'Exodus', The Century Book of the Bible, c Porter, J.R. The Illustrated Guide to the Bible, Duncan Baird Publications, An excellent explanation of the atonement is given by Canon Leon Morris (1983) in 'The Atonement', IVP, Leicester, UK. 6

Hebrews 9. LGNT - May 31, 2019 Page 1. Book Chapter Verse. Word # Hebrews. # of words. Interlinear English. Numeric value. # of letters.

Hebrews 9. LGNT - May 31, 2019 Page 1. Book Chapter Verse. Word # Hebrews. # of words. Interlinear English. Numeric value. # of letters. Book Chapter Verse Interlinear English Word # Hebrews Greek # of letters # of words Numeric value 190901 02401 ΕΙΧΕ 4 620 190901 indeed 02402 ΜΕΝ 3 95 190901 then 02403 ΟΥΝ 3 520 190901 the 02404 Η 1 8

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

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

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

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1

WEEK 21. The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 WEEK 21 The Feast of Firstfruits, the Feast of Pentecost, and the Feast of Tabernacles OUTLINE DAY 1 I. The Feast of Firstfruits signifies the resurrected Christ as the firstfruits for our enjoyment as

Διαβάστε περισσότερα

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

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

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

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

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

Διαβάστε περισσότερα

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

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

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

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

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

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

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

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

Διαβάστε περισσότερα

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

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

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

2007 Classical Greek. Intermediate 2 Translation. Finalised Marking Instructions

2007 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 περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 11: The Unreal Past Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

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

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

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

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

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

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

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

ΦωτίστεΤονΚόσμο. ΠΡΩΤΟΒΟΥΛΙΑ ΧΡΙΣΤΟΥΓΕΝΝΩΝ Νοεμβρίου Δεκεμβρίου 2017

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

Διαβάστε περισσότερα

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

I haven t fully accepted the idea of growing older

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

Διαβάστε περισσότερα

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

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

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

Συστήματα Διαχείρισης Βάσεων Δεδομένων

Συστήματα Διαχείρισης Βάσεων Δεδομένων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo

Διαβάστε περισσότερα

Code Breaker. TEACHER s NOTES

Code Breaker. TEACHER s NOTES TEACHER s NOTES Time: 50 minutes Learning Outcomes: To relate the genetic code to the assembly of proteins To summarize factors that lead to different types of mutations To distinguish among positive,

Διαβάστε περισσότερα

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

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

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

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

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr 9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

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

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

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

[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

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

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

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

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

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

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

ΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ ΕΛΛΗΝΙΔΕΣ

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

Διαβάστε περισσότερα

Hebrews 8. LGNT - September 5, 2019 Page 1. Book Chapter Verse. Word # Hebrews. # of words. Interlinear English. Numeric value. # of letters.

Hebrews 8. LGNT - September 5, 2019 Page 1. Book Chapter Verse. Word # Hebrews. # of words. Interlinear English. Numeric value. # of letters. Book Chapter Verse Interlinear English Word # Hebrews Greek # of letters # of words Numeric value 190801 [and the] main point 02127 ΚΕΦΑΛΑΙΟΝ 9 687 190801 02128 ΔΕ 2 9 190801 of 02129 ΕΠΙ 3 95 190801 the

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

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

Revelation 10. LGNT - September 7, 2019 Page 1. Book Chapter Verse. Word # Rev. Interlinear English. Numeric value. # of words. # of letters.

Revelation 10. LGNT - September 7, 2019 Page 1. Book Chapter Verse. Word # Rev. Interlinear English. Numeric value. # of words. # of letters. Book Chapter Verse Interlinear English Word # Rev Greek # of letters # of words Numeric value 271001 and 03876 ΚΑΙ 3 31 271001 I saw 03877 ΕΙΔΟΝ 5 139 271001 another 03878 ΑΛΛΟΝ 5 181 271001 [mighty] angel

Διαβάστε περισσότερα

ΟΙ ΑΞΙΕΣ ΤΗΣ ΖΩΗΣ THE VALUES OF LIFE Η ΥΠΕΥΘΥΝΟΤΗΤΑ..THE RESPONSIBILITY ΔΗΜΗΤΡΑ ΚΩΝΣΤΑΝΤΙΝΟΥ

ΟΙ ΑΞΙΕΣ ΤΗΣ ΖΩΗΣ THE VALUES OF LIFE Η ΥΠΕΥΘΥΝΟΤΗΤΑ..THE RESPONSIBILITY ΔΗΜΗΤΡΑ ΚΩΝΣΤΑΝΤΙΝΟΥ ΟΙ ΑΞΙΕΣ ΤΗΣ ΖΩΗΣ THE VALUES OF LIFE Η ΥΠΕΥΘΥΝΟΤΗΤΑ..THE RESPONSIBILITY ΔΗΜΗΤΡΑ ΚΩΝΣΤΑΝΤΙΝΟΥ ΜΑΘΗΜΑΤΑ ΥΠΕΥΘΥΝΟΤΗΤΑΣ/ LESSONS ABOUT RESPONSIBILITY Μάθημα 1: Νιώθω υπερήφανος όταν.../ I feel proud when.

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

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

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

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

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

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

Math 6 SL Probability Distributions Practice Test Mark Scheme

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

Διαβάστε περισσότερα

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

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

Διαβάστε περισσότερα

Ephesians. Wayne Stewart

Ephesians. Wayne Stewart Ephesians Wayne Stewart 1 Shadows of a ministry independent of Jerusalem and the 12 Taken from CH Welch 2 Acts 28:25 25 (AV) And when they agreed not among themselves, they departed, after that Paul had

Διαβάστε περισσότερα

Volume of a Cuboid. Volume = length x breadth x height. V = l x b x h. The formula for the volume of a cuboid is

Volume of a Cuboid. Volume = length x breadth x height. V = l x b x h. The formula for the volume of a cuboid is Volume of a Cuboid The formula for the volume of a cuboid is Volume = length x breadth x height V = l x b x h Example Work out the volume of this cuboid 10 cm 15 cm V = l x b x h V = 15 x 6 x 10 V = 900cm³

Διαβάστε περισσότερα

Policy Coherence. JEL Classification : J12, J13, J21 Key words :

Policy 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 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

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

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

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

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

Démographie spatiale/spatial Demography

Démographie 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 περισσότερα

ΑΚΑ ΗΜΙΑ ΕΜΠΟΡΙΚΟΥ ΝΑΥΤΙΚΟΥ ΜΑΚΕ ΟΝΙΑΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ

ΑΚΑ ΗΜΙΑ ΕΜΠΟΡΙΚΟΥ ΝΑΥΤΙΚΟΥ ΜΑΚΕ ΟΝΙΑΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΑΚΑ ΗΜΙΑ ΕΜΠΟΡΙΚΟΥ ΝΑΥΤΙΚΟΥ ΜΑΚΕ ΟΝΙΑΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΘΕΜΑ :ΤΥΠΟΙ ΑΕΡΟΣΥΜΠΙΕΣΤΩΝ ΚΑΙ ΤΡΟΠΟΙ ΛΕΙΤΟΥΡΓΙΑΣ ΣΠΟΥ ΑΣΤΡΙΑ: ΕΥΘΥΜΙΑ ΟΥ ΣΩΣΑΝΝΑ ΕΠΙΒΛΕΠΩΝ ΚΑΘΗΓΗΤΗΣ : ΓΟΥΛΟΠΟΥΛΟΣ ΑΘΑΝΑΣΙΟΣ 1 ΑΚΑ

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

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

LESSON 9 (ΜΑΘΗΜΑ ΕΝΝΙΑ) REF : 101/011/9-BEG. 14 January 2013

LESSON 9 (ΜΑΘΗΜΑ ΕΝΝΙΑ) REF : 101/011/9-BEG. 14 January 2013 LESSON 9 (ΜΑΘΗΜΑ ΕΝΝΙΑ) REF : 101/011/9-BEG 14 January 2013 Up πάνω Down κάτω In μέσα Out/outside έξω (exo) In front μπροστά (brosta) Behind πίσω (piso) Put! Βάλε! (vale) From *** από Few λίγα (liga) Many

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

Door Hinge replacement (Rear Left Door)

Door Hinge replacement (Rear Left Door) Door Hinge replacement (Rear Left Door) We will continue the previous article by replacing the hinges of the rear left hand side door. I will use again the same procedure and means I employed during the

Διαβάστε περισσότερα

ΑΝΑΠΤΥΞΗ ΠΡΟΓΡΑΜΜΑΤΩΝ ΕΚΠΑΙΔΕΥΣΗΣ ΜΕ ΣΤΟΧΟ ΤΗΝ ΠΕΡΙΒΑΛΛΟΝΤΙΚΗ ΕΥΑΙΣΘΗΤΟΠΟΙΗΣΗ ΑΤΟΜΩΝ ΜΕ ΕΙΔΙΚΕΣ ΑΝΑΓΚΕΣ ΚΑΙ ΤΗΝ ΚΟΙΝΩΝΙΚΗ ΤΟΥΣ ΕΝΣΩΜΑΤΩΣΗ

ΑΝΑΠΤΥΞΗ ΠΡΟΓΡΑΜΜΑΤΩΝ ΕΚΠΑΙΔΕΥΣΗΣ ΜΕ ΣΤΟΧΟ ΤΗΝ ΠΕΡΙΒΑΛΛΟΝΤΙΚΗ ΕΥΑΙΣΘΗΤΟΠΟΙΗΣΗ ΑΤΟΜΩΝ ΜΕ ΕΙΔΙΚΕΣ ΑΝΑΓΚΕΣ ΚΑΙ ΤΗΝ ΚΟΙΝΩΝΙΚΗ ΤΟΥΣ ΕΝΣΩΜΑΤΩΣΗ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ ΠΑΙΔΑΓΩΓΙΚΟ ΤΜΗΜΑ ΕΙΔΙΚΗΣ ΑΓΩΓΗΣ ΧΡΙΣΤΙΝΑ Σ. ΛΑΠΠΑ ΑΝΑΠΤΥΞΗ ΠΡΟΓΡΑΜΜΑΤΩΝ ΕΚΠΑΙΔΕΥΣΗΣ ΜΕ ΣΤΟΧΟ ΤΗΝ ΠΕΡΙΒΑΛΛΟΝΤΙΚΗ ΕΥΑΙΣΘΗΤΟΠΟΙΗΣΗ ΑΤΟΜΩΝ ΜΕ ΕΙΔΙΚΕΣ ΑΝΑΓΚΕΣ ΚΑΙ ΤΗΝ ΚΟΙΝΩΝΙΚΗ ΤΟΥΣ

Διαβάστε περισσότερα

Αναερόβια Φυσική Κατάσταση

Αναερόβια Φυσική Κατάσταση Αναερόβια Φυσική Κατάσταση Γιάννης Κουτεντάκης, BSc, MA. PhD Αναπληρωτής Καθηγητής ΤΕΦΑΑ, Πανεπιστήµιο Θεσσαλίας Περιεχόµενο Μαθήµατος Ορισµός της αναερόβιας φυσικής κατάστασης Σχέσης µε µηχανισµούς παραγωγής

Διαβάστε περισσότερα

Στο εστιατόριο «ToDokimasesPrinToBgaleisStonKosmo?» έξω από τους δακτυλίους του Κρόνου, οι παραγγελίες γίνονται ηλεκτρονικά.

Στο 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 περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8  questions or comments to Dan Fetter 1 Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

Notes are available 1

Notes are available 1 Notes are available 1 Galatians 2:15 4.12 2 3:15, 3:19 interpretational contradiction LAW cannot be and was added Promise 3 Gal.3:19 4 Mt 6:27 Which of you by taking thought can add one cubit unto

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

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

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

Διαβάστε περισσότερα

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 9: Inversion Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons εκτός

Διαβάστε περισσότερα

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

NOTES: OT Reference. Who offers atonement? Who seeks atonement? Text from NASB:1995

NOTES: OT Reference. Who offers atonement? Who seeks atonement? Text from NASB:1995 NES: 1. 2. 3. 4. 5. 6. 7. 8. Gen 6:14 Gen 32:20 Exod 29:33 Exod 29:36 Exod 29:37 Exod 30:10 Exod 30:15 Exod 30:16 Exod 32:30 Lev 1:4 and shall cover [ἀσφαλτόω [כפר it inside and out with pitch. I will

Διαβάστε περισσότερα

ΟΔΗΓΙΕΣ ΕΓΚΑΤΑΣTΑΣΗΣ ΓΙΑ ΠΑΤΩΜΑ WPC INSTALLATION GUIDE FOR WPC DECKING

ΟΔΗΓΙΕΣ ΕΓΚΑΤΑΣTΑΣΗΣ ΓΙΑ ΠΑΤΩΜΑ WPC INSTALLATION GUIDE FOR WPC DECKING 1/12 ΟΔΗΓΙΕΣ ΕΓΚΑΤΑΣTΑΣΗΣ ΓΙΑ ΠΑΤΩΜΑ WPC INSTALLATION GUIDE FOR WPC DECKING Ανοίγουμε τρύπες Ø8 x 80mm στο σημείο κατασκευής, με τρυπάνι. To προτεινόμενο πλάτος και μήκος μεταξύ των 2 οπών να είναι 30-35εκ.,

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

Δημιουργία Λογαριασμού Διαχείρισης Business Telephony Create a Management Account for Business Telephony

Δημιουργία Λογαριασμού Διαχείρισης Business Telephony Create a Management Account for Business Telephony Δημιουργία Λογαριασμού Διαχείρισης Business Telephony Create a Management Account for Business Telephony Ελληνικά Ι English 1/7 Δημιουργία Λογαριασμού Διαχείρισης Επιχειρηματικής Τηλεφωνίας μέσω της ιστοσελίδας

Διαβάστε περισσότερα

Example of the Baum-Welch Algorithm

Example of the Baum-Welch Algorithm Example of the Baum-Welch Algorithm Larry Moss Q520, Spring 2008 1 Our corpus c We start with a very simple corpus. We take the set Y of unanalyzed words to be {ABBA, BAB}, and c to be given by c(abba)

Διαβάστε περισσότερα

Homework 8 Model Solution Section

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα