Orpheus Hymn to Apollo
|
|
- Φίλομενης Ζαφειρόπουλος
- 8 χρόνια πριν
- Προβολές:
Transcript
1 Orpheus Hymn to Apollo Translated and annotated by Thomas Taylor in 1792 Page 1 of 8
2 Απόλλωνος, θυμίαμα μάνναν Ἐλθέ, μάκαρ Παιάν, Τιτυοκτόνε, Φοῖβε Λυκωρεῦ, Μεμφῖτ, ἀγλαότιμε, ἰήιε, ὀλβιοδῶτα, χρυσολύρη, σπερμεῖε, ἀρότριε, Πύθιε, Τιτάν. Γρύνειε, Σμινθεῦ, Πυθοκτόνε, Δελφικέ, μάντι, ἄγριε, φωσφόρε δαῖμον, ἐράσμιε, κύδιμε κοῦρε Μουσαγέτα, χαροποιέ, ἑκηβόλε, τοξοβέλεμνε, Βάγχιε καὶ Διδυμεῦ, ἑκάεργε, Λοξία, ἁγνέ Δήλι ἄναξ, πανδερκὲς ἔχων φαεσίμβροτον ὄμμα, χρυσοκόμα, καθαρὰς φήμας χρησμούς τ ἀναφαίνων κλῦθί μευ εὐχομένου λαῶν ὕπερ εὔφρονι θυμῶι. τόνδε σὺ γὰρ λεύσσεις τὸν ἀπείριτον αἰθέρα πάντα, γαῖαν τ ὀλβιόμοιρον ὕπερθέν τε καὶ δι ἀμολγοῦ νυκτὸς ἐν ἡσυχίαισιν ὑπ ἀστεροόμματου ὄρφνης ῥίζας νέρθε δέδορκας, ἔχεις δέ τε πείρατα κόσμου παντός σοὶ δ ἀρχή τε τελευτή τ ἐστὶ μέλουσα, παντοθαλής σὺ δὲ πάντα πόλον κιθάρηι πολυκρέκτωι ἁρμόζεις, ὁτὲ μὲν νεάτης ἐπὶ τέρματα βαίνων, ἄλλοτε δ αὖθ ὑπάτην, ποτὲ Δώριον εἰς διάκοσμον πάντα πόλον κιρνὰς κρίνεις βιοθρέμμονα φῦλα, ἁρμονίηι κεράσας τὴν παγκόσμιον ἀνδράσι μοῖραν μίξας χειμῶνος θέρεός τ ἴσον ἀμφοτέροισιν, εἰς ὑπάταις χειμῶνα, θέρος νεάταις διακρίνας, Δώριον εἰς ἔαρος πολυηράτου ὥριον ἄνθος ἔνθεν ἐπωνυμίην σε βροτοὶ κλήιζουσιν ἄνακτα Πᾶνα, θεὸν δικέρωτ, ἀνέμων συρίγμαθ ἱέντα οὕνεκα παντὸς ἔχεις κόσμου σφραγῖδα τυπῶτιν. κλῦθι μάκαρ, σώζων μύστας ἱκετηρίδι φωνῇι. To Apollo, fumigation from manna* Blest Pæan, come, propitious to my pray r, Illustrious pow r, whom Memphian tribes revere, Slayer of Tityus, and the God of health, Lycorian Phœbus, fruitful source of wealth. Spermatic, golden-lyr d, the field from thee Receives it s constant, rich fertility. Titanic, Grunian, 1 Smynthian, thee I sing, Python-destroying, 2 hallow d, Delphian king: Rural, light-bearer, and the Muse s head, Noble and lovely, arm d with arrows dread: Far-darting, Bacchian, two-fold, and divine, Pow r far diffused, and course oblique is thine. O, Delian king, whose light-producing eye Views all within, and all beneath the sky: Whose locks are gold, whose oracles are sure, Who, omens good reveal st, and precepts pure: Hear me entreating for the human kind, Hear, and be present with benignant mind; For thou survey st this boundless Æther all, And ev ry part of this terrestrial ball. Abundant, blessed; and thy piercing sight, Extends beneath the gloomy, silent night; Beyond the darkness, starry-ey d, 3 profound, The stable roots, deep fix d by thee are found. The world s wide bounds, all-flourishing are thine, Thyself all the source and end divine: Tis thine all Nature s music to inspire, With various-sounding, harmonising lyre; 4 Now the last string thou tun st to sweet accord, 5 Divinely warbling now the highest chord; Th immortal golden lyre, now touch d by thee, Responsive yields a Dorian melody. All Nature s tribes to thee their diff rence owe, And changing seasons from thy music flow: Hence, mix d by thee in equal parts, advance Summer and Winter in alternate dance; This claims the highest, that the lowest string, The Dorian measure tunes the lovely spring. Hence by mankind, Pan-royal, two-horn d nam d, Emitting whistling winds thro Syrinx fam d; 6 Since to thy care, the figur d seal s consign d, 7 Which stamps the world with forms of ev ry kind. Hear me, blest pow r, and in these rites rejoice, And save thy mystics with a suppliant voice. Page 2 of 8
3 Annotations by Thomas Taylor et al. *Manna is Theosophia, the divine bread of wisdom White Stone [is] the sign of initiation mentioned in St. John s Revelation. It had the word prize engraved on it, and was the symbol of that word given to the neophyte who, in his initiation, had successfully passed through all the trials in the MYSTER- IES. It was the potent white cornelian [or carnelian] of the mediæval Rosicrucians, who took it from the Gnostics. To him that overcometh will I give to eat of the hidden manna (the occult knowledge which descends as divine wisdom front heaven), and will give him a white stone; and in the stone a new name written (the mystery name of the inner man or the EGO of the new Initiate), which no man knoweth saving him that receiveth it. (Revelation ii, 17) 1 As said in the Commentaries: Fruits and grain, unknown to Earth to that day, were brought by the Lords of Wisdom for the benefit of those they ruled front other lokas [spheres].... Wheat has never been found in the wild state: it is not a product of the earth. All the other cereals have been traced to their primogenital forms in various species of wild grasses, but wheat has hitherto defied the efforts of botanists to trace it to its origin. 2 Grunian, to thee I sing Grynæus, according to Strabo, lib. 13, is a town of Myrinæus, and is likewise a temple of Apollo, and a most ancient oracle and temple, sumptuously built of white stone. Python-destroying king Says Olympiodorus in MS. Comment. in Phædon: Typhon, Echidna, and Python being the progeny of Tartarus and Earth, which is conjoined with Heaven, form, as it were, a certain Chaldaic triad, which is the inspective guardian of the whole of a disordered fabrication [i.e. of the fabrication of the last of things]. Οτι Ταρταρου και Γης της συζυγουσης τω Ουρανω, ο Τυφων, η Εχιδνα, ο Πυθων, οιον Χαλδαικη τις τρια εφοπος της ατακτους πασης δημιουργιας. And in another part of the same Commentary he says that... Typhon is the cause of the all-various subterranean winds and waters, and of the violent motion of the other elements. But Echidna is a cause reveng- 1 Theosophical Glossary, art. White Stone 2 Cf. Secret Doctrine, II pp Full text under the tile Wheat is not a product of the earth, in our Down to Earth Series. ED. PHIL. Page 3 of 8
4 ing and punishing rational and irrational souls; and hence the upper parts of her are those of a virgin, but the lower those of a serpent. And Python is the guardian of the whole of prophetic production. Though it will be better to say, that he is the cause of the disorder and obstruction pertaining to things of this kind. Hence, also, Apollo destroyed Python, in consequence of the latter being adverse [to the prophetic energy of the former]. Ο μεν Τυφων της παντοιας των υπογειων πνευματων και υδατων, και των αλλων στοιχειων βιαιου κινησεως αιτιος η δε Εχιδνα τιμωρος αιτια και κολαστικη λογικων τε και αλογων ψυχων διο τα μεν ανω παρθενος τα δε κατω εστιν οφεωδης ο δε Πυθων φρουρος της μαντικης ολης αναδοσεως αμεινον δε της περι ταυτα αταξιας τε και αντιφραξεως αιτον λεγειν διο και Απολλων αυτον αναιρει εντατιουμενον. Starry-eyed The starry-eyed darkness, beyond which Apollo is here said to fix his roots, is the sphere of the fixed stars, the regions immediately beyond which consists of the ethereal worlds, which according to the Chaldeans are three. For they assert that there are seven corporeal worlds, one empyrean and the first; 3 after this, three ethereal, and then three material worlds, which last consist of the inerratic sphere, the seven planetary spheres, and the sublunary region. 4 But that, according to the Orphic theology, there is an ethereal world beyond the sphere of the fixed stars is evident from the following mystic particulars respecting the Oracle of Night, which are transmitted to us by Proclus in his admirable Commentary on the Timaeus, p. 63e and p. 96a. Says he: The artificer of the universe, prior to his whole fabrication, is said to have betaken himself to the Oracle of Night, to have been there filled with divine conceptions, to have received the principles of fabrication, and, if it be lawful so to speak, to have solved ail his doubts. Night, too, calls upon the father Jupiter to undertake the fabrication of the universe; and Jupiter is said by the theologist [Orpheus] to have thus addressed Night: Μαια Θεων υπατη, Νυξ αμβροτε, πως ταδε φρασεις; Πως δει μ αθανατων αρχην κρατεροφρονα θεσθαι; Πως δε μοι εν τι τα παντ εσται, και χωρις εκαστον; O Nurse supreme of all the pow rs divine, Immortal Night! how with unconquer d mind Must I the source of the Immortals fix? And how will ail things but as one subsist, Yet each its nature separate preserve? To which interrogations the Goddess thus replies: Αιθερι παντα περιξ αφατω λαβε τω δ ενι μεσσω Ουρανον, εν δε τε γαιαν απειριτον, εν δε θαλασσαν, Εν δε τε τειπεα παντα, τα τ ουρανους εστεφανωτο. 3 [Unmanifested, or First Logos] 4 [See The Seven Worlds of Being in our Secret Doctrine s First Proposition Series. ED. PHIL.] Page 4 of 8
5 All things receive enclos d on ev ry side, In Ether s wide, ineffable embrace; Then in the midst of Ether place the Heav n, In which let Earth of infinite extent, The Sea, and Stars the crown of Heav n be fixt. We also learn from Psellus, that according to the Chaldeans there are two solar worlds; one which is subservient to the ethereal profundity; the other zonaic, being one of the seven spheres. And Proclus, in Tim. p. 264d, informs us that... according to the most mystic assertions, the wholeness of the Sun is in the supermundane order. For there a solar world and a total light subsist, as the Oracles of the Chaldeans affirm. Οι γε μυστικωτατοι των λογων, και την ολοτητα αυτου (solis) την εν τοις υπερκοσμιοις παραδεδωκασιν εκει γαρ ο ηλιακος κοσμος, και ολον το φως, ως αι τε Χαλδαιων φημι λεγουσι. These ethereal worlds pertain to the supermundane order of Gods, in which the wholeness of the Sun subsists. But by the wholeness (ολοτης ) Proclus means the sphere in which the visible orb of the Sun is fixed, and which is called a wholeness, 5 because it has a perpetual subsistence, and comprehends in itself ail the multitude of which it is the cause. Conformably to this, the Emperor Julian (in Orat. v) says: The orb of the Sun revolves in the starless, much above the inerratic sphere. Hence he is not the middle of the planets, but of the three worlds [i.e. of the three ethereal worlds], according to the telestic hypothesis. Ο δισκος επι της αναστρου φερεται, πολυ της απλανους υψηλοτερας, και ουτω δε των μεν πλανωμενων ουκ εξει το μεσον, τριων δε των κοσμων κατα τας τελεστικας υποθεσεις. From all this, therefore, it is evident why Apollo in this hymn is said to fix his roots beyond the starry-eyed darkness; for this signifies that Gods are inserted by him in the ethereal worlds; roots being indicative of summits (ακροτητες ) and such, according to the Orphic and Chaldaic theologists, are the Gods. Hence Proclus (in MS. Comment. in Parmenid.) beautifully observes, As trees by their summits are firmly established in the earth, and all that pertains to them is through this earthly; after the same manner divine natures are by their summits rooted in The One, and each of them is a unity and one, through an unconfused union with The One Itself. καθαπερ γαρ τα δενδρα ταις εαυτων κορυφαις ενιδρυνται τη γη, και εστι γηινα κα τατ εκεινας, τον αυτον τροπον και τα θεια, ταις εαυτων ακροτησιν ενερριζωται τω ενι, και εκαστον αυτων ενας εστι και εν, δια την προς το εν ασυγχυτον ενωσιν. 5 [See Chaos to sense, latent deity to reason in our Secret Doctrine s First Proposition Series. ED. PHIL.] Page 5 of 8
6 Harmonising lyre Gesner 6 well observes, in his notes on this hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras to Plato. The lyre of Apollo, however, is not only indicative of the harmony of the universe, of which this divinity is the source, but particularly adumbrates according to the Orphic and Pythagoric doctrine, the celestial harmony, or the melody caused by the revolutions of the celestial spheres. This harmony of the spheres is admirably unfolded by Simplicius in his Commentary on the second book of Aristotle s Treatise on the Heavens, as follows: The Pythagoreans said, that an harmonic sound is produced from the motion of the celestial bodies; and they scientifically collected this from the analogy of their intervals; since not only the ratios of the intervals of the sun and moon, and Venus and Mercury, but also of the other stars, were discovered by them. Simplicius 7 adds, Perhaps the objection of Aristotle to this assertion of the Pythagoreans may be solved as follows, according to the philosophy of those men: all things are not commensurate with each other, nor is every thing sensible commensurate to every thing, even in the sublunary region. This is evident from dogs, who scent animals at a great distance, and which are not smelt by men. How much more, therefore, in things which are separated by so great an interval as those which are incorruptible from the corruptible, and celestial from terrestrial natures, is it true to say that the sound of divine bodies is not audible by terrestrial ears? But if any one, like Pythagoras, who is reported to have heard this harmony, should have his terrestrial body exempt from him, and his luminous and celestial vehicle, and the senses which it contains, purified, either through a good allotment, or through probity of life, or through a perfection arising from sacred operations, such a one will perceive things invisible to others, and will hear things inaudible by others. With respect to divine and immaterial bodies, however, if any sound is produced by them, it is neither percussive nor destructive, but it excites the powers and energies of sublunary sounds, and perfects the sense which is coordinate with them. It has also a certain analogy to the sound which concurs with the motion of terrestrial bodies. But the sound which is with us, in consequence of the sonorific nature of the air, is a certain energy of the motion of their impassive sound. If then, air is not passive there, it is evident that neither will the sound which is there be passive. Pythagoras, however, seems to have said that he heard the celestial harmony, as understanding the harmonic proportions in numbers, of the heavenly bodies, and that which is audible in them. Some one, however, may very properly doubt why the stars are 6 [Conrad Gessner, a.k.a. Konrad Gesner, , Swiss naturalist, bibliographer, botanist, physician, and classical linguist.] 7 [Σιμπλίκιος of Cilicia, c , disciple of Ammonius Hermiae and Damascius, and one of the last Neo- Platonists. He was among the pagan philosophers persecuted by Justinian in the early 6 th century, and was forced for a time to seek refuge in the Persian court, before being allowed back into the empire. He wrote extensively on the works of Aristotle. Although his writings are all commentaries on Aristotle and other authors, rather than original compositions, his intelligent and prodigious learning makes him the last great philosopher of pagan antiquity. His works have preserved much information about earlier philosophers which would have otherwise been lost.] Page 6 of 8
7 seen by our visive sense, but the sound of them is not heard by our ears? To this we reply, that neither do we see the stars themselves; for we do not see their magnitudes, or their figures, or their surpassing beauty. Neither do we see the motion through which the sound is produced; but we see, as it were, such an illumination of them as that of the light of the sun about the earth, the sun himself not being seen by us. Perhaps too, neither will it be wonderful, that the visive sense, as being more immaterial, subsisting rather according to energy than according to passion, and very much transcending the other senses, should be thought worthy to receive the splendour and illumination of the celestial bodies, but that the other senses should not be adapted for this purpose. The last string thou tunest to sweet accord The following quotations from Nicomachus (Harm. lib. i, p. 6) 8 illustrates the meaning of the hypate and nete, or the highest and lowest string, in the lyre of Apollo. Says he: From the motion of Saturn the most remote of the planets, the appellation of the gravest sound, hypate, is derived; but from the lunar motion, which is the lowest of ail, the most acute sound is called nete, or the lowest. But Gesner observes, that a more ancient, and as it were archetypal, appellation is derived from the ancient triangular lyre, 9 a copy of which was found among the pictures lately dug out of the ruins of Herculaneum; in which the highest chord next to the chin of the musician is the longest, and consequently (says he) the sound is the most grave. Gesner proceeds in observing, that three seasons of the year are so compared together in a musical ratio, that hypate signifies the Winter, nete the Summer, and the Dorian measure represents the intermediate seasons, Spring and Autumn. Now the reason why the Dorian melody is assigned to the Spring, is because that measure wholly consists in temperament and moderation, as we learn from Plutarch in his Treatise De Musica. Hence it is with great propriety attributed to the Spring, considered as placed between Winter and Summer; and gratefully tempering the fervent heat of the one, and the intense cold of the other. 8 [Νικόμαχος of Gerasa, , important mathematician and neo-pythagorean of the ancient world, best known for his works Introduction to Arithmetic and Manual of Harmonics in Greek. See translation of the former in our Hellenic and Hellenistic Papers. ED. PHIL.] 9 [Cf. The Epigoneion, a 40-string string instrument: This instrument was invented, or at least introduced into Hellas, by Epigonus, a Greek musician of Ambracia, who was admitted to a citizenship at Sicyone. The Epigoneion had 40 strings (Pollux, Onomasticon, Lib. iv, cap. 9, sect. 2). It was probably a harp or psaltery, since in an instrument of so many strings some must have been of different lengths, for tension and thickness only could hardly have produced forty different sounds; strings of varying lengths require a frame, like that of the harp or of the Egyptian Kithara (see Fig. 165); or, in the case of the psaltery, a harp-shaped arrangement of the bridges. Juba, or Jobas, the learned king of Mauritania, who flourished in A.D. 63, declares that Epigonus brought the instrument from Alexandria, and played upon it with the fingers of both hands, not using it only as an accompaniment to the voice, but introducing chromatic passages, and a chorus of other stringed instruments, probably citharas, to accompany the voice. Epigonus was also a skilled citharista, and played it with his bare hands without plectrum (Athen. iv., p. 183d, and xiv, p. 638a). Kathleen Schlesinger, The Instruments of the Modern Orchestra & Early Records of the Precursors of the Violin Family. Vol. 2 of 2, Archaeological Records. Researches into the Remote Origin of the Violin Family: a Bibliography of Music and Archaeology, English and Foreign. London: William Reeves, 1910; p. 315.] Page 7 of 8
8 Emitting whistling winds through Syrinx famed According to the Pythagoric and Platonic theology, which is perfectly conformable to that of Orpheus, Apollo is in the supermundane what Jupiter is in the intellectual order. For as the former illuminates mundane natures with supermundane light, so the latter illuminates the supermundane order with intellectual light. Indeed, there is such a wonderful agreement between these two divinities, that the Cyprian priests, as we are informed by the Emperor Julian, in his most excellent Oration to the Sovereign Sun, raised common altars to Jupiter and the Sun. Hence we cannot wonder that the same thing is here asserted of Apollo which Orpheus elsewhere asserts of Jupiter. For Johan. Diaco. in Hesiod. Theog. 10 quotes the following lines from Orpheus: Ζευς δε τε παντων εστι θεος, παντων τε κεραστης, Πνευμασι συριζων, φωναισι τε αερομικτοις. Jupiter is the God of all, and the mingler of all things, Emitting shrill sounds from winds and air-mingled voices. The figured seal s consigned In the preceding note we have mentioned the profound union which subsists between Apollo and Jupiter. As Jupiter, therefore, considered as the Demiurgus, comprehends in himself the archetypal ideas of ail sensible forms, and what these forms are intellectually in the Demiurgus they are according to a supermundane characteristic in Apollo; hence the latter divinity, as well as the former, may be said to possess the figured seal, of which every visible species is nothing more than an impression. Modern example of a 25-string psalterion (57 x 21 cm) made by Jean Charles Monzani in [Johannes Diaconus, Allegories on Hesiod s Theogony ] Page 8 of 8
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 περισσότεραCHAPTER 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 περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραRight Rear Door. Let's now finish the door hinge saga with the right rear door
Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ
ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΨΥΧΟΛΟΓΙΚΕΣ ΕΠΙΠΤΩΣΕΙΣ ΣΕ ΓΥΝΑΙΚΕΣ ΜΕΤΑ ΑΠΟ ΜΑΣΤΕΚΤΟΜΗ ΓΕΩΡΓΙΑ ΤΡΙΣΟΚΚΑ Λευκωσία 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότερα14 Lesson 2: The Omega Verb - Present Tense
Lesson 2: The Omega Verb - Present Tense Day one I. Word Study and Grammar 1. Most Greek verbs end in in the first person singular. 2. The present tense is formed by adding endings to the present stem.
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΑ ΤΜΗΜΑ ΝΑΥΤΙΛΙΑΚΩΝ ΣΠΟΥΔΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ ΝΑΥΤΙΛΙΑ ΝΟΜΙΚΟ ΚΑΙ ΘΕΣΜΙΚΟ ΦΟΡΟΛΟΓΙΚΟ ΠΛΑΙΣΙΟ ΚΤΗΣΗΣ ΚΑΙ ΕΚΜΕΤΑΛΛΕΥΣΗΣ ΠΛΟΙΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που υποβλήθηκε στο
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότερα(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)
Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραTMA4115 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 περισσότεραΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ
ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ Ενότητα 1β: Principles of PS Ιφιγένεια Μαχίλη Τμήμα Οικονομικών Επιστημών Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης
Διαβάστε περισσότεραLESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014
LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV 18 February 2014 Slowly/quietly Clear/clearly Clean Quickly/quick/fast Hurry (in a hurry) Driver Attention/caution/notice/care Dance Σιγά Καθαρά Καθαρός/η/ο
Διαβάστε περισσότεραSection 1: Listening and responding. Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016
Section 1: Listening and responding Presenter: Niki Farfara MGTAV VCE Seminar 7 August 2016 Section 1: Listening and responding Section 1: Listening and Responding/ Aκουστική εξέταση Στο πρώτο μέρος της
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραMain 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 περισσότερα7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple
A/ Ονόματα και ένα παράδειγμα 1 Present Simple 7 Present PERFECT Simple 2 Present Continuous 8 Present PERFECT Continuous 3 Past Simple (+ used to) 9 Past PERFECT Simple she eats she is eating she ate
Διαβάστε περισσότεραVerklarte 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 περισσότεραChapter 2 * * * * * * * Introduction to Verbs * * * * * * *
Chapter 2 * * * * * * * Introduction to Verbs * * * * * * * In the first chapter, we practiced the skill of reading Greek words. Now we want to try to understand some parts of what we read. There are a
Διαβάστε περισσότεραAreas 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 περισσότεραΦωτίστεΤονΚόσμο. 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 περισσότεραΚάθε 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 περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραAreas 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 περισσότεραLESSON 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 περισσότεραWEEK 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 περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραΕπιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ
EΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΕΚΠΑΙΔΕΥΤΙΚΟ ΤΕΧΝΟΛΟΓΙΚΟ ΙΔΡΥΜΑ ΤΕΙ ΙΟΝΙΩΝ ΝΗΣΩΝ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : Λεωφ. Αντ.Τρίτση, Αργοστόλι Κεφαλληνίας Τ.Κ. 28 100 τηλ. : 26710-27311 fax : 26710-27312
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011
Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι
Διαβάστε περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ
Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 1: Elements of Syntactic Structure Το περιεχόμενο του μαθήματος διατίθεται με άδεια
Διαβάστε περισσότεραAdjectives. Describing the Qualities of Things. A lesson for the Paideia web-app Ian W. Scott, 2015
Adjectives Describing the Qualities of Things A lesson for the Paideia web-app Ian W. Scott, 2015 Getting Started with Adjectives It's hard to say much using only nouns and pronouns Simon is a father.
Διαβάστε περισσότεραΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α. Διαβάστε τις ειδήσεις και εν συνεχεία σημειώστε. Οπτική γωνία είδησης 1:.
ΦΥΛΛΟ ΕΡΓΑΣΙΑΣ Α 2 ειδήσεις από ελληνικές εφημερίδες: 1. Τα Νέα, 13-4-2010, Σε ανθρώπινο λάθος αποδίδουν τη συντριβή του αεροσκάφους, http://www.tanea.gr/default.asp?pid=2&artid=4569526&ct=2 2. Τα Νέα,
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ 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 περισσότεραΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ
Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 11: The Unreal Past Το περιεχόμενο του μαθήματος διατίθεται με άδεια Creative Commons
Διαβάστε περισσότεραΣυστήματα Διαχείρισης Βάσεων Δεδομένων
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo
Διαβάστε περισσότεραΟι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού)
Οι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού) Προσδοκώμενα αποτελέσματα Περιεχόμενο Ενδεικτικές δραστηριότητες
Διαβάστε περισσότεραPotential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραWriting for A class. Describe yourself Topic 1: Write your name, your nationality, your hobby, your pet. Write where you live.
Topic 1: Describe yourself Write your name, your nationality, your hobby, your pet. Write where you live. Χρησιμοποίησε το and. WRITE your paragraph in 40-60 words... 1 Topic 2: Describe your room Χρησιμοποίησε
Διαβάστε περισσότεραΑΛΕΞΑΝΔΡΟΣ 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 περισσότεραΘέμα 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 περισσότεραΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ
ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ Ἀναπάντεχες λεπτομέρειες γιὰ τὸν βίο καὶ τὴν πολιτεία τοῦ Ἰωάννη Λάσκαρη μᾶς εἶναι γνωστὲς ἀπὸ μιὰ σειρὰ ἐγγράφων (ποὺ ἀνακάλυψε στὰ κρατικὰ ἀρχεῖα
Διαβάστε περισσότεραΟΔΗΓΙΕΣ 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 περισσότεραFractional 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 περισσότεραdepartment listing department name αχχουντσ ϕανε βαλικτ δδσϕηασδδη σδηφγ ασκϕηλκ τεχηνιχαλ αλαν ϕουν διξ τεχηνιχαλ ϕοην µαριανι
She selects the option. Jenny starts with the al listing. This has employees listed within She drills down through the employee. The inferred ER sttricture relates this to the redcords in the databasee
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραExample 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 περισσότεραΣΟΡΟΠΤΙΜΙΣΤΡΙΕΣ 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 περισσότεραLESSON 26 (ΜΑΘΗΜΑ ΕΙΚΟΣΙ ΕΞΙ) REF : 102/030/26. 18 November 2014
LESSON 26 (ΜΑΘΗΜΑ ΕΙΚΟΣΙ ΕΞΙ) REF : 102/030/26 18 November 2014 But Weekend I love The song I sing I smile I laugh Greek (thing) Greek(people) Greek (man) αλλά (το) Σαββατοκύριακο αγαπώ (το) τραγούδι τραγουδώ
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραΠτυχιακή 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 περισσότεραInstruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραFourier 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 περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ ΣΥΓΚΡΙΤΙΚΗ ΜΕΛΕΤΗ ΤΗΣ ΣΥΓΚΡΑΤΗΤΙΚΗΣ ΙΚΑΝΟΤΗΤΑΣ ΟΡΙΣΜΕΝΩΝ ΠΡΟΚΑΤΑΣΚΕΥΑΣΜΕΝΩΝ ΣΥΝΔΕΣΜΩΝ ΑΚΡΙΒΕΙΑΣ
Διαβάστε περισσότεραΜηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
Διαβάστε περισσότεραDémographie spatiale/spatial Demography
ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης
Διαβάστε περισσότεραLecture 15 - Root System Axiomatics
Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the
Διαβάστε περισσότεραFINAL 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 περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραΤΟ ΣΤΑΥΡΟΔΡΟΜΙ ΤΟΥ ΝΟΤΟΥ ΤΟ ΛΙΜΑΝΙ ΤΗΣ ΚΑΛΑΜΑΤΑΣ
ΤΟ ΣΤΑΥΡΟΔΡΟΜΙ ΤΟΥ ΝΟΤΟΥ ΤΟ ΛΙΜΑΝΙ ΤΗΣ ΚΑΛΑΜΑΤΑΣ ΖΑΡΑΒΕΛΑ Δ. 1, και ΒΡΥΩΝΗΣ Δ. 1 1 4ο Τ.Ε.Ε. Καλαμάτας, Δ/νση Δευτεροβάθμιας Εκ/σης Μεσσηνίας e-mail: dzaravela@yahoo.qr ΕΚΤΕΝΗΣ ΠΕΡΙΛΗΨΗ Από τις κύριες
Διαβάστε περισσότεραΕγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα
[ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη
Διαβάστε περισσότεραΑναερόβια Φυσική Κατάσταση
Αναερόβια Φυσική Κατάσταση Γιάννης Κουτεντάκης, BSc, MA. PhD Αναπληρωτής Καθηγητής ΤΕΦΑΑ, Πανεπιστήµιο Θεσσαλίας Περιεχόµενο Μαθήµατος Ορισµός της αναερόβιας φυσικής κατάστασης Σχέσης µε µηχανισµούς παραγωγής
Διαβάστε περισσότεραAdvanced 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 περισσότεραΣυντακτικές λειτουργίες
2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.
Διαβάστε περισσότεραCode Breaker. TEACHER s NOTES
TEACHER s NOTES Time: 50 minutes Learning Outcomes: To relate the genetic code to the assembly of proteins To summarize factors that lead to different types of mutations To distinguish among positive,
Διαβάστε περισσότερα1999 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 περισσότεραΑπόκριση 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 περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραthe total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
Διαβάστε περισσότεραΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας»
ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ ΣΧΟΛΗ ΑΝΘΡΩΠΙΣΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΕΠΙΣΤΗΜΩΝ ΤΗΣ ΠΡΟΣΧΟΛΙΚΗΣ ΑΓΩΓΗΣ ΚΑΙ ΤΟΥ ΕΚΠΑΙ ΕΥΤΙΚΟΥ ΣΧΕ ΙΑΣΜΟΥ «ΠΑΙ ΙΚΟ ΒΙΒΛΙΟ ΚΑΙ ΠΑΙ ΑΓΩΓΙΚΟ ΥΛΙΚΟ» ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ που εκπονήθηκε για τη
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραChapter 29. Adjectival Participle
Chapter 29 Adjectival Participle Overview (29.3-5) Definition: Verbal adjective Function: they may function adverbially or adjectivally Forms: No new forms because adverbial and adjectival participles
Διαβάστε περισσότερα«ΨΥΧΙΚΗ ΥΓΕΙΑ ΚΑΙ ΣΕΞΟΥΑΛΙΚΗ» ΠΑΝΕΥΡΩΠΑΪΚΗ ΕΡΕΥΝΑ ΤΗΣ GAMIAN- EUROPE
«ΨΥΧΙΚΗ ΥΓΕΙΑ ΚΑΙ ΣΕΞΟΥΑΛΙΚΗ» ΠΑΝΕΥΡΩΠΑΪΚΗ ΕΡΕΥΝΑ ΤΗΣ GAMIAN- EUROPE We would like to invite you to participate in GAMIAN- Europe research project. You should only participate if you want to and choosing
Διαβάστε περισσότεραΠτυχιακή Εργασία. Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους
ΑΛΕΞΑΝΔΡΕΙΟ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΣΧΟΛΗ ΤΕΧΝΟΛΟΓΙΑΣ ΤΡΟΦΙΜΩΝ ΚΑΙ ΔΙΑΤΡΟΦΗΣ ΤΜΗΜΑ ΔΙΑΤΡΟΦΗΣ ΚΑΙ ΔΙΑΙΤΟΛΟΓΙΑΣ Πτυχιακή Εργασία Παραδοσιακά Προϊόντα Διατροφική Αξία και η Πιστοποίηση τους Εκπόνηση:
Διαβάστε περισσότερα( ) 2 and compare to M.
Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραΤ.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