THE MACEDONIAN «ΑΝΑΠΕΣΩΝ» AS MODEL FOR THE FLEMISH «NASCENDO MORIMUR»

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "THE MACEDONIAN «ΑΝΑΠΕΣΩΝ» AS MODEL FOR THE FLEMISH «NASCENDO MORIMUR»"

Transcript

1 THE MACEDONIAN «ΑΝΑΠΕΣΩΝ» AS MODEL FOR THE FLEMISH «NASCENDO MORIMUR» More than ten years ago, in 1966, Professor Otto Demus delivered his significant Wrightsman Lectures under the auspices of the New York University Institute of Fine Arts, in which he demonstrated the role which the art of Byzantium had played in the development of Western art1. His principal consideration was to show the function of Byzantine artists as teachers and pacemakers and to call attention to a number of Byzantine models which have found their way into the art of the West. Accepting the now well established contention of these lectures it is our purpose in this essay to show the intercultural fertilization which is visible not only in the Italian art of the Duecento to the Quattrocento, but also in the art of the 16th century Flemish masters. In an essay on intercultural relations, which have played such an important role in he evolution of art it seems increasingly necessary to maintain a balanced view which should prevent us from falling into the kind of cultural chauvinism that sees either Western influences in Byzatium or Byzantine influences in the West wherever we are : It is this concern that is reflected in the following introductory references in which we want to remind the reader of the contributions of the Byzantine East to the West and of the West to the Byzantine East. Following these remarks we shall discuss the Macedonian «Anapeson» and the Flemish «Nascendo Morimur» and then conclude the essay with an attempt to show certain similarities in the representations of the two themes. With respect to the Byzantine influence upon the Italian schools we are aware of numerous instances of adaptations and copies, both of form and style. Only few illustrations shall be presented in this context. Thus, for example, in the Head of St. Andrew of the Last Judgment (1293) in Santa Cecilia in Trastevere, Rome, by Pietro Cavallini ( ) the earliest great representative of the Roman school we recognize the Byzantine type of «the wise old man» which was resurrected from antique prototypes of the early Paleologan period, and found its most classical formulation in the Byzantine fresco of the third quarter of the 13th century in Sopocani. In other instances, not only form and style but also the theological theme and message of the Byzantine East 1. Otto Demus, «The Wrightsman Lectures 1966» were published under the title «Byzantine Art and the West», London, Weidenfeld and Nicolson, 1970.

2 206 Otto F. A. Meinardus were fully accepted to the point of actually copying the Byzantine prototype. An illustration of this is a mid-13th century head of Christ from a panel now in the Byzantine Museum in Athens, which may represent a high-water mark of this development. The Western triumphant Christ on the Cross was replaced by the Byzantine model of the suffering Christ, thereby emphasizing a religious accent, which, of course, was already expressed in the theology and piety of this period. The contured features of Christ in Giunta Pisano s (13th century) or Cimabue s Crucifixes are copies of the above mentioned Byzantine Crucifixus1. Kurt Weitzmann has devoted serious thought to the iconography of the Threnos, the Lamentation of Christ preceding His burial, which shows the Holy Virgin embracing the dead body of her son and receiving it into her laps and St. John caressing Christ s hand with Nicodemus and Joseph of Arimathea bowing down to hold His feet. One of the earliest representations of the Threnos is a Byzantine wall-painting belonging to the second half of the 12th century in the Church of St. Panteleimon in Gornji Nerezi, Serbia2. The finest copy of the Hellenistic-Byzantine theme3 is the wall-paiting of «The Lamentation of Christ» in the Capella dell Arena in Padua by Giotto di Bondone ( ), the father of Renaissance art. Here, byzantine form, style and theological content are fully accepted and copied. These observations are important to our discussion since they represent milestones in a development which led to our theme of the «Anapeson». It is not surprising that the Flemish master or masters of the 16th century were considerably more flexible and liberated in their adaptations of Byzantine models than the Italian masters of the Quattrocento. Where as the Italian artists had accepted form, style and message of their respective Byzantine models^ the Dutch masters two centuries later projected independent new thoughts on wood or canvas based largely upon the rediscovered liberating humanism of the age. This humanistic conviction, which assigned a prédominent interest to the affairs of mankind as compared to the supernatural and scholastic theological tradition of the medieval Church had also its reflections in the paintings of this period, of which our «Nascendo Morimur» is a good illustration. But before we discuss the relationship of the «Anapeson», which is a strictly Byzantine theme, to the 16th century «Nascendo Morimur», we should not neglect to state also the influence of Western masters on contemporary Byzantine art. Grigore Nandris has pointed out that the wall-painting of the «Ana- 1. Ibid., p. 218, figs. 242 and 253; p. 224, figs Milkjkovic-Pepek Petar, Nerezi, Beograd 1966, pp Weitzmann K., The Origin of the Threnos, «De Artibus Opuscula XL, Essays in Honor of E. Panofsky», New York 1961.

3 The Macedonian «ΆναπεσΏν» as Model for the Flemish «Nascendo Morimur» 207 peson» at the Mount Athos Monastery of Dochiariou was created under Italian influence, and that the motif of the raising of the cover from the sleeping Child was often employed by the Italian masters, including Raphael ( ) and Sassoferrato ( )1. Indeed, borrowing, adapting and copying of forms, styles and themes from the northern European masters by the Byzantine iconographers was much more widely practised than is often realized. The use of Western woodcuts from the illustrations of the Holy Scriptures as models for wall-paintings is well known to the student of Byzantine paintings of the Last Judgment. Ludwig H. Heydenreich2 and Juliette Renaud3 have convincingly demonstrated that the woodcuts of Lucas Cranach the Elder (1522) and those of Hans Holbein (1523) provided inspiration for the wall-paintings of the cycle of the Apocalypse in the Mount Athos monasteries of Dochiariou, Dionysiou and others4. 5 The Macedonian «A n a p e s ο η» In this context we shall refer to a specifically Macedonian «Anapeson» in order to stress the typological simplicity of the wall-paintings of this theme by the masters of the Macedonian School of the 13th and 14th centuries. These paintings are probably also the oldest extant representations of the «Anapeson». Later developments of the theme have added additional figures and symbols to the prototype. Thus, for example, in the 15th century «Anapeson» of the Manasija Monastery in Resava, Serbia, the Christchild still rests on a kline, but it is attended by the Holy Virgin on the left and by an angel on either side. The angels hold in their hands the instruments of the passion6. A similar painting with the Holy Virgin attenting the resting Christchild is portrayed in 1. Nandris Grigore, Christian Humanism in the Neo-Byzantine Mural-Painting of Eastern Europe, Wiesbaden 1970, p Heydenreich L. H., Der Apokalypsen-Zyklus im Athosgebiet und seine Beziehungen zur deutschen Bibelillustration der Reformation, «Zeitschrift für Kirchengeschichte» VIII (1939), 2 ff. Heydenreich argues that the cycle in the Monastery of Dionysiou is dependent upon the model provided by the woodcuts of Hans Holbein. 3. Renaud Juliette, Le cycle de l'apocalypse de Dionysiou. Interpretation byzantine de gravures occidentales. Bibliothèque de l Ecole des Hautes Études. Sciences religieuses, LIX, Paris, 1943, p. 1 ff. Renaud feels that the cycle in the Monastery of Dionysiou reflects the woodcuts of Lucas Cranach the Elder. 4. Huber Paul, Athos, Leben, Glaube, Kunst, Zürich 1969, pp Huber sees Cranach s woodcuts as models for the maintings in the monasteries of Dionysiou and Xenophontos, Holbein s woodcuts as models for the paintings in the Monastery of Dochiariou. 5. Djuric Vojislav J., Resava, Beograd 1963, p. XIII, fig. 7. Petkovic V., La Peinture serbe du moyen âge, Beograd 1934, pl. CCIV. Stanoyevic St., Mirkovic L., Boskovic Dj., Manastir Manasija, Beograd 1928, pi. XVIII.

4 208 Otto F. A. Meinardus the 15th century «Anapeson» of the Metamorphosis Monastery, the Great Meteoron, Thessaly1. An even further developed form of the «Anapeson» is found in the church of Ljutibrod between Cerepis and Mezdra in Bulgaria. In this wall-painting the Christchild holds in his left hand the instruments of the passion, the spear and the sponge. Moreover, instead of resting on a kline, the Christchild lies on an altar, thereby portraying the idea of the fulfilment of the Old Testament sacrifice2. Other later and even more complex forms of the «Anapeson» are found in my article of «The Place of the Anapeson of Soumela in Byzantine Art»3. For the purpose of establishing a link between the Byzantine representation of the «resting Child» and the Flemish «Nascendo Morimur» we must turn to the Macedonian «Anapeson», of which there are only few examples. The oldest and most typical painting of this subject is found on the western wall of the nave of the Protaton in Karyes on Mount Athos. This «Anapeson» is now generally attributed to Manuel Panselinos, who painted it probably soon after 1300 under the patronage of Andronicus Paleologus ( ). 1. Gytakos Michael, «Θρησκευτική καί Ηθική Εγκυκλοπαίδεια», Athens 1963, vol. II, p Grabar André, La Peinture Religieuse en Bulgarie, Paris 1928, pp , fig M e i n a r d u s O., The Place of the Anapeson of Soumela in Byzantine Art, «Oriens Christianus» LV (1971)

5 The Macedonian «Άναπεσών» as Model for the Flemish «Nascendo Morimur» 209 Panselinos was associated with the Macedonian School on Mount Athos and had also worked in the region of Salonica. As David Talbot Rice1 has pointed out, «though he can hardly have been responsible for all the paintings which have been attributed to him by popular tradition, the suggestion put forward by certain authorities that he was a fictitious personage seems entirely groundless. Indeed his style seems to have been fairly personal if paintings in the Protaton at Karyes which have recently been attributed to him are to be regarded as typical»2. This painting portrays the Christchild with a cross-nimbus resting on a kline. The head of the Christchild is turned towards the right, and with His right hand He provides support for His head. In His left hand He holds a scroll. The left arm is stretched out and rests on His left thigh. The left leg is slightly elevanted. A similar wall-painting of the Anapeson, although with the inscription Άναπεσών exists in the Catholicon of the Monastery of Xenophontos on Mount Athos3. The wellknown «Anapeson» of Mistra, Peloponnesos, which should be assigned to the 14th century, adorns the conch of the diakonikon of the Perivleptos Church. G. Millet, who published this painting, called it «Emmanuel dormant»4. Typologically speaking, the Mistra «Anapeson» falls into the same category as the 13th century «Anapeson» of the Protaton in Karyes. From an interpretative point of view we are able to distinguish three distinct theological themes, which from the 13th to the 16th and 17th centuries have been attached to the «Anapeson». This fact necessarily complicates any assessment not only of the religious but also of the art-historical links or relationships which this theme might have with Western artistic representations. The theologian recognizes that these themes are somewhat interlocked, and yet, it is possible to state certain emphasises. Firstly, there is the Old Testament messianic theme, secondly, there is the idea of the incarnation and redemption, and thirdly, there is the typically Russian theme of the Emmanuel the Watchful, the never-sleeping eye of God. The fathers of the Eastern Church used to interpret the «Anapeson» as a representation of Christ s sleep of the dead in fulfilment of the Old Testament prophecy «Judah, you lion s whelp, you have returned from the kill, my son, and crouch and stretch like a lion ; and like a lion (or a lioness) who dare rouse you?» (Geneis 49:9). 1. Rice David T., Byzantine Art, Munich 1964, p Xyngopoulos Andreas, Manuel Panselinos, Athens 1956, p Brockhaus Heinric h,die Kunst in den Athos-Klöstem, Leipzig 1891, p Fichtner Fritz, Wandmalereien der Athos Klöster, Berlin 1931, p Millet Gabriel, Monuments Byzantins de Mistra, Paris 1910, pl. 115, no

6 210 Otto F. A. Meinardus The Eastern Church has always seen a messianic significance in this passage. The first line of this verse refers to the youthful vigor of the tribe. The last two lines are almost identical with Numbers 24:9a, and may well be a quotation from another poem. In the Numeri quotation of the poem we read that in the preceding oracle Balaam had likened Israel to a hunting lion, here the nation is compared to a lion that has finished his hunting and his meal and is sitting down to a rest from which none dares to rouse him. The Hermeneia explains that the true source of the words is a prophecy of the Patriarch Jacob made to his sons before his death, that they refer to Judah, from whose race Christ came. Iconographically the lion does not appear in the Protaton, Xenophontos or Mistra paintings, although in later Byzantine representations of this theme the lion is included, as for example in the 18th century wall-painting in the catholicon of the Monastery of Philotheou, Mount Athos, where the Christchild is represented asleep, inconjured by the Holy Virgin and two angels. All three kneel and in deep reverence watch over the sleeping Child. At the feet of the Christchild a lion sleeps. The blending of the themes of the Incarnation and the Vicarious Suffering and Death of the Christ is a further development of the «Anapeson» as we find it in the composition of the Φοβερά Προστασία, the Terrible Protection of the Monastery of Koutloumousiou on Mount Athosh Here, the Christchild supported by His mother knows of his divine mandate, and as indicated by the attending angels holding the instruments of the passion and the Cross, the Christchild is aware of His suffering and death. Byzantine artists from the 16th to the 19th century have composed a wide variety of forms and styles with this particular interpretation of the «Anapeson»12. Lastly, the iconography of the «Anapeson» also inspired the Old Testament theme of the Emmanuel the Watchful, known as the Nedremanoe oko gne, the eye of the Lord which does not sleep, based upon the Song of Ascents : «Behold, he who keeps Israel will neither slumber nor sleep» (Ps. 121:4). This interpretation of the theme developed primarily in the iconographie art of 16th and 17th century Russia3. As we shall point out now, it is the second theme, the fusing of the doctrines of the Incarnation and the Vicarious Death of birth and death which may have given the artist of the «Nascendo Morimur» a point of departure for his presentation. 1. Xyngopoulos A., Benaki Museum. Catalogue of Icons, Athens 1936, pp (In Greek). 2. M e i n a r d u s O., loc., cit. 3. Skrobucha Heinz, Merveilles des leones, Paris 1969, pp

7 The Macedonian «Άναπεσών» as Model for the Flemish «Nascendo Morimur» 211 The Flemish «N ascendo Morimur» The oil-painting known as the «Nascendo Morimur», with the birth we die, is in the possession of the Mittelrhein Museum, Florinsmarkt, in Koblenz, Germany, where it is registered under No. M 60. The painting is attributed to an anonymous artist belonging to the circle of the 16th century Dutch painter Marten Jacobsz Heemskerk ( ), sometimes called Van Veen, who worked first in Delft and later in Flaarlem. The painting is on wood and measures 49.3 X 63.5 cm1. It portrays a naked male infant resting on a dark floor in front of a black wall. In the upper right hand corner of the wall there is a window. The window-scene includes a mountainous and wooden landscape with a Gothic belltower and a church. On the road leading to the church one recognizes though with much difficulty an old man, apparently on his way to the church2. The child supports his head with his right hand, touching the blond curly hair. The right elbow presses against a human skull. His eyes are down-cast and closed. In his left hand he holds a staff surmounted by a rectangular sign with the inscription «NASCENDO MORIMUR». The idea of interweaving life and death is shown by the colouring of the infant body. The face, especially the cheeks and the ears, as well as the hands and feet are portrayed in pink indicating blood-circulation and life. The body with the arms and legs the knees show some colouring are kept in a greyishwhite, suggesting death or the process of dying. The employment of the emblem which verbalizes the theme of the paining ut pictura poesis was quite common in the 16th and 17th century, and parallels to the «Nascendo Morimur» are «Mors vitae initium» (death is the beginning of life) and «Vive memor lethi» (live in the awareness of death)3. Conclusion Arrangement and form of the child in the «Anapeson» and the «Nascendo Morimur» betray certain similarities. The expressions and the messages of the «Anapeson» and the «Nascendo Morimur» contain certain relationships. With respect to the arrangement and form we notice similarities of posture. There are differences which reflect the theological climate. The «Anapeson» shows the Christchild with a cross-nimbus and clothed in an expensively 1. Schaar Eckhard, Kataloghefte des Mittelrhein-Museums Koblenz. Niederländische Meister, Koblenz 1968, pp. 11, A similar landscape is shown on the painting «Venus» by M. J. Heemskerk in the Wallraf-Richartz Museum, Cologne. 3. Schöne A., Emblemata, Stuttgart 1967, p Also, «Reallexikon zur Deutschen Kunstgeschichte», Vol. V, «Embleme».

8 212 Otto F. A. Meinardus embroidered garment and completely covered, except for the head, hands and feet. On the other hand, the «Nascendo Morimur» is a naked child. His head, hands and feet are the only parts of the body which convey the idea of «life», for they are painted pink. The Christchild of the «Anapeson» holds in Hisleft hand a scroll, the Book of Life; while the «Nascendo Morimur» holds in his left hand the universal message that «with the birth we die». Whether the artist intended to portray with the staff and the inscription a T -Cross, we do not know. Therefore, we should leave open the possibility that this theme may have included the idea of a universal redemption. The message of the «Anapeson» is that the Christchild recognizes already at His birth His messianic ministry of suffering and death for the salvation of mankind. Christ s birth and death, Bethlehem and Golgotha, incarnation and redemption are the theological frame of reference of the «Anapeson». Birth and death, though historically and physically manifested in Jesus Christ, gain spiritual significance for the human race, in terms of rebitrh and everlasting life through faith. On the other hand, the humanistic interpretation of the interrelationship of birth and death, as given in the «Nascendo Morimur» excludes the soteriological aspects of the Christian doctrine of redemption and merely states the physiological truth or law that «with the birth we die». That this truth also bears a spiritual and moral message, no one can deny. For that matter, the universality of this law is underlined by the representation of the human skull, which may even be a borrowing of the medieval portrayal of redemption, namely Adam s skull. In short, the Christian soteriological message of the «Anapeson» has been transformed in the «Nascendo Morimur» into a universalist and humanistic spirituality and morality. It is very likely that the anonymous Flemish artist, embued with the spirit of his age, used the Byzantine model of the «Anapeson» for his message that with our birth we all begin to die, and that we are responsible for the way and the manner in which we spend our time on earth. The church which appears through the open window, may well sympolize the redemption which is promised to him who is aware of and responds to the wisdom and the message of the «Nascendo Morimur»1. Koblenz OTTO F. A. MEINARDUS 1. Attention should be called to the naked Christchild supporting His head with His righ arm andresting on the lap of His mother in «The Madonna and Child with Saints and Frederico da Montrefeltro, Duke of Urbino», the famous Brera Madonna, Brera, Milano, by Pietro della Francesca (1410/ ).

9 ΠΕΡΙΛΗΨΗ Otto F. A. M einardus, Ό «Άναπεσών»της Μακεδονίας ώςπρότυπο για τό Φλαμανδικό «Nascendo Morimur». Tò δοκίμιο αύτό για είκονογραφικές σχέσεις ανάμεσα σέ διάφορες πολιτιστικές ένότητες δείχνει τήν έπίδραση της βυζαντινής τέχνης στούς Φλαμανδούς καλλιτέχνες του 16ου αίώνος. Γιά τό σκοπό αυτό διαλέξαμε άπό τή Μακεδονία τό θέμα «Άναπεσών», τοϋ όποιου τό πιό τυπικό παράδειγμα παρουσιάζεται στό δυτικό τοίχο του νάρθηκος τοΰ Πρωτάτου στις Καρυές τοϋ 'Αγίου Όρους, καί τό «Nascendo Morimur» (με τή γέννηση πεθαίνουμε), ζωγραφιά στό Μουσείο του Μέσου Ρήνου στό Κομπλέντς τής Γερμανίας, ή όποια άποδίδεται σέ άνώνυμο καλλιτέχνη, πού άνήκει στον κύκλο τοΰ Ολλανδού ζωγράφου τοϋ 16ου αί. Marten J. Heemskerk. Ή διάταξη καί ή μορφή τοΰ παιδιοΰ στον «Άναπεσόντα» καί στό «Nascendo Morimur» προδίδουν ομοιότητες καί τό μήνυμα των δύο εικόνων σχετίζεται. Σχετικά μέ τή διάταξη καί τή μορφή παρατηρούμε ομοιότητες στάσεως. Υπάρχουν διαφορές στό θεολογικό κλίμα τής καλλιτεχνικής έκφράσεως. Ό «Άναπεσων» δείχνει τό Χριστό-παιδί μέ σταυρωτό φωτοστέφανο καί ντυμένο μέ κεντητό ένδυμα, πού τό καλύπτει τελείως, έκτος άπό τό κεφάλι, τά χέρια καί τά πόδια. Στό «Nascendo Morimur» είναι παιδί γυμνό. Τό κεφάλι, τά χέρια καί τά πόδια του είναι τά μόνα μέρη τοΰ σώματος πού δίνουν τήν ιδέα τής ζωής, καθώς είναι ζωγραφισμένα κοκκινωπά. Ό Χριστός-παιδί κρατάει στό άριστερό Του χέρι ενα είλητάριο, τό Βιβλίο τής Ζωής, ενώ στό «Nascendo Morimur» κρατάει στό άριστερό του χέρι τό κοσμικό μήνυμα ότι «μέ τή γέννηση πεθαίνουμε». Τό μήνυμα τοΰ «Άναπεσόντος» είναι οτι ό Χριστός -παιδί άναγνωρίζει κατά τή γέννησή του τό μεσσιανικό Του λειτούργημα τοΰ πάθους καί τοΰ θανάτου γιά τή σωτηρία τής άνθρωπότητος. Γέννηση καί θάνατος, μολονότι ιστορικά καί φυσικά φανερωμένα στον Ίησου Χριστό, παίρνουν πνευματική σπουδαιότητα γιά τό άνθρώπινο γένος ύπό τήν έννοια τής άναγεννήσεως καί τής αιώνιας ζωής διά τής πίστεως. Έξ άλλου ή ούμανιστική έρμηνεία τής συσχετίσεως ζωής καί θανάτου, δπως εΐκονίζεται στό «Nascendo Morimur», άποκλείει τις σωτηριολογικές άπόψεις τής χριστιανικής διδασκαλίας περί λυτρώσεως καί άπλώς βεβαιώνει τήν άλήθεια ή τό νόμο τής φυ

10 214 Otto F. A. Meinardus σιολογίας δτι «μέ τή γέννηση άποθνήσκουμε». Δέν μπορεί κανείς ν άρνηθή ότι καί αυτή ή άλήθεια φέρει ένα πνευματικό καί ήθικό μήνυμα. Ή παγκοσμιότητα του μηνύματος αυτοϋ υπογραμμίζεται άπό τήν παράσταση ανθρωπίνου κρανίου, πού μπορεί να είναι καί δάνειο άπό τή μεσαιωνική παράσταση τής λυτρώσεως, δηλαδή τό κρανίο του Άδάμ. Τό χριστιανικό σωτηριολογικό μήνυμα τού «Άναπεσόντος» μέ τό «Nascendo Morimur» μετασχηματίσθηκε σέ κοσμική καί ουμανιστική πνευματικότητα. Ή εκκλησία, πού φαίνεται στό ανοιχτό παράθυρο, μπορεί κάλλιστα να συμβολίζη τή λύτρωση, μιά υπόσχεση προς εκείνον, πού κατανοεί τό σοφό μήνυμα του «Nascendo Morimmo.

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

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

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

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

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

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

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

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

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

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

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

Διαβάστε περισσότερα

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

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

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

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ

Επιβλέπουσα Καθηγήτρια: ΣΟΦΙΑ ΑΡΑΒΟΥ ΠΑΠΑΔΑΤΟΥ EΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΕΚΠΑΙΔΕΥΤΙΚΟ ΤΕΧΝΟΛΟΓΙΚΟ ΙΔΡΥΜΑ ΤΕΙ ΙΟΝΙΩΝ ΝΗΣΩΝ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ & ΕΠΙΚΟΙΝΩΝΙΑΣ Ταχ. Δ/νση : Λεωφ. Αντ.Τρίτση, Αργοστόλι Κεφαλληνίας Τ.Κ. 28 100 τηλ. : 26710-27311 fax : 26710-27312

Διαβάστε περισσότερα

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

The challenges of non-stable predicates

The 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 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

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

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

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

ΔΗΜΟΚΡΙΤΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΡΑΚΗΣ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΑΓΩΓΗΣ

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

Διαβάστε περισσότερα

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

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

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

F-TF Sum and Difference angle

F-TF Sum and Difference angle F-TF Sum and Difference angle formulas Alignments to Content Standards: F-TF.C.9 Task In this task, you will show how all of the sum and difference angle formulas can be derived from a single formula when

Διαβάστε 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

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

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

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

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

ΙΩΑΝΝΗΣ ΛΑΣΚΑΡΗΣ, ΠΗΓΩΝΙΤΗΣ Ή ΣΥΡΠΑΓΑΝΟΣ, Ο ΚΑΛΟΜΙΣΙΔΗΣ

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

Διαβάστε περισσότερα

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

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

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

Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ

Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ Η ΕΡΕΥΝΑ ΤΗΣ ΓΛΩΣΣΙΚΗΣ ΑΛΛΑΓΗΣ ΣΤΑ ΚΕΙΜΕΝΑ ΤΗΣ ΜΕΣΑΙΩΝΙΚΗΣ ΕΛΛΗΝΙΚΗΣ: ΜΕΘΟΔΟΛΟΓΙΚΗ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ ΠΗΓΩΝ Θεόδωρος Μαρκόπουλος University of Uppsala thodorismark@yahoo.gr Abstract This paper discusses methodological

Διαβάστε περισσότερα

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ»

«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ» I ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΝΟΜΙΚΩΝ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΠΟΛΙΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΟΙΚΟΝΟΜΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ «ΔΙΟΙΚΗΣΗ ΚΑΙ ΟΙΚΟΝΟΜΙΑ» ΚΑΤΕΥΘΥΝΣΗ: ΟΙΚΟΝΟΜΙΚΗ

Διαβάστε περισσότερα

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

CRASH COURSE IN PRECALCULUS

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

Διαβάστε περισσότερα

Section 7.6 Double and Half Angle Formulas

Section 7.6 Double and Half Angle Formulas 09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

Διαβάστε περισσότερα

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

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

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

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

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

ΑΝΑΠΤΥΞΗ ΤΟΥ ΠΤΥΧΙΑΚΗ. Λεμεσός

ΑΝΑΠΤΥΞΗ ΤΟΥ ΠΤΥΧΙΑΚΗ. Λεμεσός ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΤΟ ΚΑΠΝΙΣΜΑ ΚΑΤΑ ΤΗΝ ΕΓΚΥΜΟΣΥΝΗ ΚΑΙ Η ΒΛΑΠΤΙΚΗ ΕΠΙΔΡΑ ΑΣΗ ΤΟΥ ΣΤΗΝ ΑΝΑΠΤΥΞΗ ΤΟΥ ΝΕΟΓΝΟΥ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ Ονοματεπώνυμο Αγγελική Παπαπαύλου Αριθμός Φοιτητικής

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

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

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

Κατανοώντας και στηρίζοντας τα παιδιά που πενθούν στο σχολικό πλαίσιο

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

Διαβάστε περισσότερα

ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ ΤΣΗΜΑΣΟ ΨΗΦΙΑΚΗ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ

ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ ΤΣΗΜΑΣΟ ΨΗΦΙΑΚΗ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ ΕΘΝΙΚΟ ΜΕΣΟΒΙΟ ΠΟΛΤΣΕΧΝΕΙΟ ΣΜΗΜΑ ΑΓΡΟΝΟΜΩΝ-ΣΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΣΟΜΕΑ ΣΟΠΟΓΡΑΦΙΑ ΕΡΓΑΣΗΡΙΟ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ

Διαβάστε περισσότερα

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ Μελέτη των υλικών των προετοιμασιών σε υφασμάτινο υπόστρωμα, φορητών έργων τέχνης (17ος-20ος αιώνας). Διερεύνηση της χρήσης της τεχνικής της Ηλεκτρονικής Μικροσκοπίας

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

Congruence Classes of Invertible Matrices of Order 3 over F 2

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

Διαβάστε περισσότερα

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

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education www.xtremepapers.com UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *6301456813* GREEK 0543/03 Paper 3 Speaking Role Play Card One 1 March 30

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

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

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

Συντακτικές λειτουργίες

Συντακτικές λειτουργίες 2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

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

Chapter 29. Adjectival Participle

Chapter 29. Adjectival Participle Chapter 29 Adjectival Participle Overview (29.3-5) Definition: Verbal adjective Function: they may function adverbially or adjectivally Forms: No new forms because adverbial and adjectival participles

Διαβάστε περισσότερα

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

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

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

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ

ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ανοικτά Ακαδημαϊκά Μαθήματα στο ΤΕΙ Ιονίων Νήσων ΕΠΙΧΕΙΡΗΣΙΑΚΗ ΑΛΛΗΛΟΓΡΑΦΙΑ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑ ΣΤΗΝ ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ Ενότητα 1: Elements of Syntactic Structure Το περιεχόμενο του μαθήματος διατίθεται με άδεια

Διαβάστε περισσότερα

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

Adjectives. Describing the Qualities of Things. A lesson for the Paideia web-app Ian W. Scott, 2015

Adjectives. Describing the Qualities of Things. A lesson for the Paideia web-app Ian W. Scott, 2015 Adjectives Describing the Qualities of Things A lesson for the Paideia web-app Ian W. Scott, 2015 Getting Started with Adjectives It's hard to say much using only nouns and pronouns Simon is a father.

Διαβάστε περισσότερα

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

ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΚΑΙ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΛΟΓΙΣΤΙΚΗΣ

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

Διαβάστε περισσότερα

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

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

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

Διαβάστε περισσότερα

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

ΔΘΝΗΚΖ ΥΟΛΖ ΓΖΜΟΗΑ ΓΗΟΗΚΖΖ

ΔΘΝΗΚΖ ΥΟΛΖ ΓΖΜΟΗΑ ΓΗΟΗΚΖΖ Ε ΔΘΝΗΚΖ ΥΟΛΖ ΓΖΜΟΗΑ ΓΗΟΗΚΖΖ Κ ΔΚΠΑΗΓΔΤΣΗΚΖ ΔΗΡΑ ΣΜΖΜΑ : Σνπξηζηηθήο Οηθνλνκίαο θαη Αλάπηπμεο (ΣΟΑ) ΣΔΛΗΚΖ ΔΡΓΑΗΑ Θέκα: Σνπξηζκφο θαη Οηθνλνκηθή Κξίζε Δπηβιέπσλ : Νηνχβαο Λνπθάο πνπδάζηξηα : Σζαγθαξάθε

Διαβάστε περισσότερα

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ. Πτυχιακή εργασία Η ΚΑΤΑΘΛΙΨΗ ΣΕ ΕΦΗΒΟΥΣ ΜΕ ΣΑΚΧΑΡΩΔΗ ΔΙΑΒΗΤΗ ΤΥΠΟΥ 1

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ. Πτυχιακή εργασία Η ΚΑΤΑΘΛΙΨΗ ΣΕ ΕΦΗΒΟΥΣ ΜΕ ΣΑΚΧΑΡΩΔΗ ΔΙΑΒΗΤΗ ΤΥΠΟΥ 1 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ Πτυχιακή εργασία Η ΚΑΤΑΘΛΙΨΗ ΣΕ ΕΦΗΒΟΥΣ ΜΕ ΣΑΚΧΑΡΩΔΗ ΔΙΑΒΗΤΗ ΤΥΠΟΥ 1 ΑΝΔΡΕΑΣ ΑΝΔΡΕΟΥ Φ.Τ:2008670839 Λεμεσός 2014 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ

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

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ ΥΓΕΙΑΣ Πτυχιακή Εργασία "Η ΣΗΜΑΝΤΙΚΟΤΗΤΑ ΤΟΥ ΜΗΤΡΙΚΟΥ ΘΗΛΑΣΜΟΥ ΣΤΗ ΠΡΟΛΗΨΗ ΤΗΣ ΠΑΙΔΙΚΗΣ ΠΑΧΥΣΑΡΚΙΑΣ" Ειρήνη Σωτηρίου Λεμεσός 2014 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ

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

Notes on the Open Economy

Notes on the Open Economy Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.

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

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

Final Test Grammar. Term C'

Final Test Grammar. Term C' Final Test Grammar Term C' Book: Starting Steps 1 & Extra and Friends Vocabulary and Grammar Practice Class: Junior AB Name: /43 Date: E xercise 1 L ook at the example and do the same. ( Κξίηα ηξ παοάδειγμα

Διαβάστε περισσότερα

About these lecture notes. Simply Typed λ-calculus. Types

About these lecture notes. Simply Typed λ-calculus. Types About these lecture notes Simply Typed λ-calculus Akim Demaille akim@lrde.epita.fr EPITA École Pour l Informatique et les Techniques Avancées Many of these slides are largely inspired from Andrew D. Ker

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

[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

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

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

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

Διαβάστε περισσότερα

Physical DB Design. B-Trees Index files can become quite large for large main files Indices on index files are possible.

Physical DB Design. B-Trees Index files can become quite large for large main files Indices on index files are possible. B-Trees Index files can become quite large for large main files Indices on index files are possible 3 rd -level index 2 nd -level index 1 st -level index Main file 1 The 1 st -level index consists of pairs

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

ΣΥΓΧΡΟΝΕΣ ΤΑΣΕΙΣ ΣΤΗΝ ΕΚΤΙΜΗΣΗ ΚΑΙ ΧΑΡΤΟΓΡΑΦΗΣΗ ΤΩΝ ΚΙΝΔΥΝΩΝ

ΣΥΓΧΡΟΝΕΣ ΤΑΣΕΙΣ ΣΤΗΝ ΕΚΤΙΜΗΣΗ ΚΑΙ ΧΑΡΤΟΓΡΑΦΗΣΗ ΤΩΝ ΚΙΝΔΥΝΩΝ ΕΘΝΙΚΗ ΣΧΟΛΗ ΤΟΠΙΚΗΣ ΑΥΤΟΔΙΟΙΚΗΣΗΣ Δ ΕΚΠΑΙΔΕΥΤΙΚΗ ΣΕΙΡΑ ΤΜΗΜΑ ΠΟΛΙΤΙΚΗΣ ΠΡΟΣΤΑΣΙΑΣ ΣΥΓΧΡΟΝΕΣ ΤΑΣΕΙΣ ΣΤΗΝ ΕΚΤΙΜΗΣΗ ΚΑΙ ΧΑΡΤΟΓΡΑΦΗΣΗ ΤΩΝ ΚΙΝΔΥΝΩΝ Σπουδάστρια: Διαούρτη Ειρήνη Δήμητρα Επιβλέπων καθηγητής:

Διαβάστε περισσότερα

Ενημέρωση και ευαισθητοποίηση μαθητών/τριών γενικών τάξεων σχετικά με τα ΑμεΑ How can a teacher inform and sensitize his/her students about disability

Ενημέρωση και ευαισθητοποίηση μαθητών/τριών γενικών τάξεων σχετικά με τα ΑμεΑ How can a teacher inform and sensitize his/her students about disability Ενημέρωση και ευαισθητοποίηση μαθητών/τριών γενικών τάξεων σχετικά με τα ΑμεΑ How can a teacher inform and sensitize his/her students about disability Δημοτικό Σχολείο Πευκοχωρίου Χαλκιδικής 1 ος μήνας

Διαβάστε περισσότερα