fonts The Design and Use of a Multiple- Alphabet Font with Ω

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "fonts The Design and Use of a Multiple- Alphabet Font with Ω"

Transcript

1 The Design and Use of a Multiple- Alphabet Font with Ω abstract The Ω project aims to offer open and flexible means for typesetting different scripts. By working at several different levels, it is possible to offer natural support for different languages and scripts, and strictly respect typographical traditions for each of them. This is illustrated with a large PostScript Type 1 font for the commonly used left-to-right non-cursive alphabets, called omlgc (Ω Latin-Greek-Cyrillic). This font, which more than covers the Unicode sections pertaining to those alphabets, as well as those of IPA, Armenian, Georgian and Tifinagh (Berber), is built virtually out of smaller glyph banks. The Ω typesetting engine, based on that of TEX, is used to print documents using this font. The characters can be accessed either directly, or through the use of filters, called Ω Typesetting Processes (ΩTPs), which are applied to the input stream. Yannis Haralambous Atelier Fluxus Virus, 187, rue Nationale, Lille, France. yannis@fluxus-virus.com John Plaice School of Computer Science and Engineering, The University of New South Wales, Sydney 2052, Australia. plaice@cse.unsw.edu.au 1 Introduction Typesetting in different scripts and languages is a problem that is arguably solved today, either by software giants that adapt their products to what they consider to be local typesetting specifications, or by individual users brewing their own limited, but practical, systems for their own needs. However, typesetting in different scripts and languages while still keeping the quality of traditional typography and having an open system that can be adapted to any language and script, without loss of power or quality, is still an unachieved goal. The Ω project aims to solve precisely this problem. The Ω project consists of the design of fonts for different languages and scripts, as well as of an engine that can be used for all possible situations. As a result, in order to ensure efficiency and openness, one can work at different levels, each one adapted to a specific aspect of multilingual typesetting. These levels correspond to methods used in the Ω system. They will be illustrated through the omlgc font, designed for currently Latin, Greek, Cyrillic, IPA, Armenian, Georgian and Tifinagh. This paper was typeset by the Ω engine, using fonts omlgc and omarab. 2 Ω Methods: an Overview When developing an Ω typesetting convention for a given language, one can work at the following levels: 1. The font level. A font is a container of glyphs needed to typeset in a given language. These glyphs may or may not correspond to the graphical units of a script, whether these are called letters, ideograms or otherwise; these glyphs may be parts of graphicalunits, combinations thereof, or new independent symbols. The glyphs must be provided to the screen previewer and as well as to the printing engine. Ω itself is not concerned with these glyphs: as with TEX, Ω works only with metrics, not necessarily those for the fonts actually containing the glyphs. The font level is the lowest development level, in the sense that glyphs are indivisible units that can be used in other higher level structures but cannot be dynamically modified by Ω itself. Voorjaar

2 Yannis Haralambous and John Plaice 2. The virtual font level. Once we have the glyphs we need, we combine them to form what is normally considered for a language to be a gramatically correct script entity (aring accent alone is of no use, but å is part of the grammar for the Swedish language). A virtual font character is a combination of glyphs taken from several real fonts, or of other virtual font characters. In the å example above, the glyphs of the ring and of the letter a can actually be taken from entirely different fonts, in different formats (bitmap, PostScript, TrueType, etc.). Passing from real fonts to virtual fonts is essentially a matter of optimization of storage and memory: taking the seven Greek vowels, the three accents, two spirits, diaeresis, subscript iota and macron/breve diacritics (that makes 16 glyphs) we produce 336 (!) virtual glyphs. Describing a character in a virtual font is hundreds of times smaller than the PostScript code describing a hypothetical similar glyph. Virtual fonts are directly used by Ω: they can have up to 2 16 positions and 2 32 kerning pairs. The files created by Ω can be processed (previewed, printed) by utilities we have adapted; if the user has to use his/her own utilities which are not 16-bit clean, there is a tool to devirtualize the files, i.e. replace virtual fonts by the underlying real fonts, and make them as standard as TEX output files. 3. The ΩTP level. When Ω reads a document, it first tokenizes it and expands commands and then forwards the data to the typesetting engine. Between these two steps we introduce an arbitrary numberof filters, which we callω Typesetting Processes (ΩTPs). These are written in a Lex-like syntax, are loadedwhile readingthe document, and can be activated and de-activated dynamically. A typicalexamplefor anωtpis contextualanalysisofarabic. Of course, this operation can also been done by macros, say using L A TEX commands; but an ΩTP will do it much faster, it will avoid conflicts with other L A TEX commands, and it will be much easier to create and configure. Contextual analysis of Arabic is a typical example of a script property that is low level and that should not use any macros or other high-level structures. In the case of the Latin script one would not expect a user to constantly think of fi and ff ligatures and place them manually: it would be very bad strategy to use an fi ligature command ; in Arabic this property is of the same nature, and hence should remain completely transparent. 1 But efficiencyshould not only be limited to speed of typesetting: sometimes it is very important to also optimize the configuration and customization time and effort. A typical example is the management of encodings, whether input or output: by using a universal encoding which we call it Unicode++, since it is a superset of Unicode as intermediate step for our ΩTPs, every new input encoding requires only a foo Unicode++ ΩTP and every new font encoding only a Unicode++ foo one; these are significantly easier to make than if one had to rewrite all processing steps, including contextualanalysis and other bells and whistles. 4. The hyphenation and sorting engines. Hyphenation and sorting rules are grammatical properties of a language: they have nothing to do with typographical aspects, input methods, font encodings, etc. They only have to be performed on the Unicode++ level, i.e. the most abstract one. Once defined at this level, they can immediately be used with every input and output encoding. Ω hyphenation and sorting engines are still under development: for the time being, Ω does not sort, and hyphenation takes place as in TEX, using the (virtual) font encoding. 1. And undoubtedly this would be the case if the lingua franca of computer science was Arabic or Urdu, and not English. 28 MAPS

3 The Design and Use of a Multiple-Alphabet Font with Ω fonts 5. The macro-command level. L A TEX is a terrific construction, featuring commandsat multiple levels: TEX primitives, plain TEX commands, internal L A TEX commands, higher L A TEX commands and environments, and user-defined commands and environments. Our goal is to keep all script-dependent processing independent of the L A TEX macro level; because of this goal, every L A TEX package is usable in any script and language, and both Ω packageand L A TEX package developers are able to work independently, while still producing mutually compatible software. By doing this, we also allow ourselves in the future to use a different typesetting engine than that found in TEX, without having to completely redo our system. The ΩTPs might need modification, but the basic structure would remain the same. 3 Level 1: Designing glyphs For us, a glyph is simply a character from a PostScript, TrueType or Metafont font; so whycallitsuch? Because we want to make the distinction between the images and the combinations of images, the latter of which Ω will use as characters for typesetting. For our omlgc font, the glyphs are produced so that the glyphs for different alphabets look similar. With the exception of Tifinagh, the history of the different alphabets making up this font is strongly interrelated; as such, it has been possible to design a single font with a characteristic feel. For Tifinagh, the script has evolved less than the others, so we are doing creative experimentation only time will tell if it is successful, i.e. native speakers accept it. For example, the following two words are Armenian. Although the traditional Armenian script does not look so rounded, it is consideredcommon practice today to typeset books in this manner, and Armenians are not in the least suprised. The omlgc font is a modern looking font. One question that might arise is what to do with classical versions of these alphabets. For ancient Greek, it is common to use modern typefaces, even though there are typefaces specifically made for ancient Greek, such as Porson or Greek Sans Serif. Can this also be done for Coptic and Slavonic excerpts, without breaking the homogeneity of the surrounding modern font? We have tried to do this for Slavonic includingnot only lettersandaccentsfound in Unicode, but also all the other diacritics and symbols needed for Old Church Slavonic and our example looks like this: вѣщ а є и исъ ирєч є є и. The result is not particularly exiting to look at, butthatis the point: it should fit with the surrounding text, not look exotic. Finally, a lot of work has gone into the design of IPA glyphs, which are problematic since many of them are basically font variations (italics, small caps, etc.) of other glyphs, thereby making it a nightmare to design fonts including IPA that have variations. As we will see in the next section, a big part of the omlgc font is obtained by combining a limited number of glyphs. This corresponds to a grammaticalreality: it is quite natural to assume that placing diacritics does not affect the shapes of either the base character or the diacritic itself. Often this is true, but there are times when typographical quality requires special shapes. For example, one can imagine an h with a lower vertical stem, so that the accent of the Esperanto character ĥ is not excessively high. This is really a matter of taste, and we include these kinds of characters as variant forms in our font, and leave the decision whether to use them to the user. Another similar example comes from the use of diaereses over the Greek vowels iota and upsilon, whether lowercase or uppercase; the distance between the two dots of the diaeresis depends on the letter underneath: compare,, Ϊ, Ϋ, while in the Latin script the diaeresis (Umlaut, tréma) always has the same width: ï, ö, ẅ, etc. Voorjaar

4 Yannis Haralambous and John Plaice 4 Level 2: Combining Glyphs into Virtual Fonts A character of a virtual font can be a combination of characters of other fonts (whether real, in which case we actually have glyphs, or again virtual, in which case we have subcombinations etc.), of black boxes of arbitrary height and width, and of PostScript code. (Since the latter feature is not implemented in all DVI drivers, one should refrain from using it, at least for the moment.) We have chosento work intenselywith virtual fonts for two reasons: first because by combining glyphs we can optimize space (and space managementis crucial when you deal with 16-bit fonts), and second, to be able to use 16-bit fonts without re-writing all of the world s drivers for TEX s DVI format (the underlying real fonts are 8-bit only, so that a devirtualized 16-bit font becomes 8-bit and can be processed by any decent DVI driver). Virtual fonts are built using a (portable) Perl script, which reads the configuration file omlgc.cfg. This file contains one entry for each character of the virtual font. This line consists of (a) a 4-digit hexadecimal number, designating the character s position in the font, (b) an operator, (c) one or more character names, depending on the operator, (d) a certain number of optional operators and values. The N operator. The N stands for NAME, and simply means that the string following the operator is the (PostScript) name of one or more characters in some of the fonts loaded. For example, 03A5 N Upsilon means that at position 03A5 of the virtual font, we have placed the Greek letter capital Upsilon Υ. The PostScript names we have used try to be as standard as possible. However, most of the time there is no standard, so we just have to invent names. The same glyph can be used for several font positions: for example, the Croatian Dze Ð has exactly the same shape as the Icelandic Eth Ð: we use the same glyph, and hence the same PostScript name. Nevertheless we try to optimize the use of glyphs so that one can typeset in one script without necessarily loading the PostScript fonts for other scripts: for example, although the capital Latin A has the same shape as the Greek capital Alpha, we use two different glyphs in two different PostScript fonts, so that when typesetting Greek one can avoid loading the Latin PostScript font. These considerations will be irrelevant when we will be able to use 16-bit monobloc PostScript fonts; for the moment, this is not part of the Adobe Type 1 font specification. There are severaloptions we can use: #KRN, #KRNLEFT,#KRNRIGHT,#HOFFSET,#VOFFSET. The first three concern kerning: we can state that a given characterhas the same kerning behavior as some other character, which we give by name. For example, ç will be kerned exactly like the letter c : everytime there is a kerning pair in the kernings file using c a new kerning pair will be defined, using ç instead of c (and if there is a c-c kerning pair, three new kerning pairs will be defined: ç-c, c-ç, ç-ç ). Sometimes we kern a letter like some given letter on the right and like some other letter on the left: this is typically the case of ligatures: æ will be kerned as a on the left, and as e on the right; in that case, we use the #KRNLEFT and #KRNRIGHT operators: 00C6 N AE #KRNLEFT=A #KRNRIGHT=E 00E6 N ae #KRNLEFT=a #KRNRIGHT=e 0110 N Eth #KRN=D The operators #HOFFSET and #VOFFSET will offset the glyph without affectingthe box of the character. The XHAC and CHAC operators. 30 MAPS

5 The Design and Use of a Multiple-Alphabet Font with Ω fonts These operators will place diacritics over letters, which are considered to have the same height as the lowercase letter x (x-height) or the one of an LGC uppercase letter (capheight). The idea is that the height of letters can fluctuate: a round letter, like o, is slighty higher than a flat letter, like z, to counterbalance a well-known optical effect. The height of accents over these letters must be the same, even if they aren t exactly of x-height, or cap-height: take for example óandú; if we would take the real height of these letters, the accent on the former would be slightly higher than the one on the latter. How is this accent placed precisely? The Perl utility centers the bounding box of the accent over the bounding box of the letter, with a fixed distance of EPSILON (a global value) between the lower boundary of the accent and either the x-height or cap-height of the font (those two are also global values provided with the utility). Available options are #KRN, #KRNLEFT, #KRNRIGHT, #LETTERLIKE, #ACCENTLIKE and #LETTERREVLIKE. The options #LETTERLIKEand #ACCENTLIKEallow us to use given glyphs, with the metrics of other glyphs. These options are extremely important in certain cases. A typical example is Vietnamese: the letter o with hook ơ is significantly wider than the plain o, nevertheless accents have to be centered on the o part of the letter: ớ, ờ, ở, ỡ. This trick allows us to correctly place an accent on the vertical stem of a b or an h : ḃ, ḣ, ḧ; to obtain this result, we simply ask the accent to be placed as if the letter was an l : 0603 CHAC b dot #LETTERLIKE=l 0623 CHAC h dot #LETTERLIKE=l 0627 CHAC h dieresis #LETTERLIKE=l We have the same functionality with accents: using the #ACCENTLIKE operator we can place an accent as if it was some other accent. The typical example is again Vietnamese, where there are combined accents circumflex + grave, circumflex + acute, which have to be centered with respect to the middle axis of the circumflex (and hence as if there were no acute or grave accent): 06D0 CHAC O circumflexacute #KERN=O #ACCENTLIKE=circumflex 06D1 XHAC o circumflexacute #KERN=o #ACCENTLIKE=circumflex 06D2 CHAC O circumflexgrave #KERN=O #ACCENTLIKE=circumflex 06D3 XHAC o circumflexgrave #KERN=o #ACCENTLIKE=circumflex 06D4 CHAC O circumflexhook #KERN=O #ACCENTLIKE=circumflex 06D5 XHAC o circumflexhook #KERN=o #ACCENTLIKE=circumflex 06D6 CHAC O circumflextilde #KERN=O #ACCENTLIKE=circumflex 06D7 XHAC o circumflextilde #KERN=o #ACCENTLIKE=circumflex Another example is Slavonic, with letters such as, where the accent has to be placed on the right part of the ligature, as in. This means that we should use the metrics of a given character, justified on the right of our box: this is the rôle of the #LETTERREVLIKEoperator. The ADJ operator. This operator allows us to concatenate two characters, using a box with width equal to the sum of the widths of the two boxes, ± the eventual kerning between those characters, and height/depth the maximum height/depth of the two characters. We first wanted to use that operator for the Croatian digraphs Lj, Nj, etc. but then decidedthat the whole idea of having code positions for these digraphs was so silly that we could very well do without. The operator nevertheless proved very useful for cases such as the Greek capital vowels with accent Ά, Ε, Η, Ι, Ο, Υ, Ω. This operator takes a special option, #MOVELEFT, which allows us to overide a kern between two characters and move the second one horizontally, together with its box. Voorjaar

6 Yannis Haralambous and John Plaice 5 Levels 3 and 4: ΩTPs and L A T E XMacros Once the structure of fonts is well organized, we use ΩTPs for low level script- or languagedependentoperations and macros for higher level operations. To give an idea of the power of ΩTPs, consider the following Latin transcription for Berber: Tifinagh, d--tira timezwura n.imazighen. Llant di tmurt--nnegh dat tira n ta.erabt d--tla.tinit. Nnulfant--edd dat.imir n ugellid Masinisen..Imazighen n.imir--en, ttarun--tent ghefi.zra, degg.ifran, ghef.igduren, maca tiggti ghef i.zekwan : ttarun fell--asen.isem n umettin, d wi--t--ilan, d wayen yexdem di tudert--is akken ur t ttettun.ina.tfaren. This piece of text can be printed using the Latin script, as in: Tifina, d_tira timezwura n imazi en. Llant di tmurt_nne dat tira n ta rabt d_tlaṭinit. Nnulfant_edd dat imir n ugellid Masinisen. Imazi en n imir_en, ttarun_tent efiẓra, degg ifran, ef igduren, maca tiggti ef iẓekwan : ttarun fell_asen isem n umettin, dwi_t_ilan, d wayen yexdem di tudert_is akken ur t ttettun inaṭfaren. or in the original Tifinagh script, as in:, _. _ _. _. _, _,,, : _, _ _, _. In fact, Berber can also be written right-to-left in the Arabic script (font omarab):... :. In the latter case, a font and direction change were necessary. Otherwise, only the output ΩTP needed to be changed for the three cases. Of course, L A TEX macros can be used to encapsulate these changes. 6 Discussion As this document shows, the omlgc font is now usable for typesetting, using Ω, any language that uses the Latin, Greek, Cyrillic and Tifinagh alphabets, and will soon be ready for the other alphabets in this group. The omlgc font is not the only multi-script font that has been developed. For example, PLAN 9[1],Bitstream [2], and Windows [13] all have their own fonts, basedontheunicode encoding. What does distinguish omlgc from the others is the emphasis that has been placed on typesetting. It is assumed right from the beginning that the set of font positions is much greater than the set of positions in, say, the Unicode definition. This can best be visualized in Figures 3 and 4, given after the references. For the Greek page, we show the complete set of vowels, with all spirits and breathings, with the macron diacritic, used to denote a long vowel in classical Greek. None of these presentation forms appear in the Unicode specification---hence do not appear in the above fonts---but they are absolutely necessary for typesetting Greek. 32 MAPS

7 The Design and Use of a Multiple-Alphabet Font with Ω fonts Similarly, the Cyrillic page shows all the vowels, with both grave (old style) and acute (new style) accents to place emphasis. Once again, none of these forms are in the Unicode specification, but they are needed for typesetting texts in Russian. But having these extra forms is of no cost, since they are all virtual characters (hence no extra memory), which can be accessed as needed in a text using the ΩTPs. We completethe paper with four examples. We begin with three versions (roman, italic, bold) of a Greek text ΠΟΛΛΕΣ ΦΟΡΕΣ ΤΗΝ ΝΥΚΤΑ, by Ανδρ εας Εµπειρ ικος. We follow with three pages of the omlgc font. References 1. C. A. Bigelow and K. Holmes. The design of a Unicode font. Electronic Publishing 6(3): , Bitstream Inc. Bitstream Cyberbit Y. Haralambous and J. Plaice. First applications of Ω: Greek, Arabic, Khmer, Poetica, ISO 10646/Unicode, etc. TUGboat, 15(3): , Y. Haralambous and J. Plaice. Ω, atex extension including Unicode and featuring lex-like filtering processes. In Wlodek Bzyl and Tomasz Plata-Przechlewski, editors, Proceedings of the European TEX Conference, pages , Gdansk, Poland, GUST. 5. Y. HaralambousandJ. Plaice. ΩTimes and ΩHelvetica Fonts underdevelopment: Step One. TUGboat, 17(3): , Y. Haralambous and J. Plaice. A font for typesetting large mathematical symbols in Ω. EuroTEX'98, Saint-Malo, France, March J. Plaice and Y. Haralambous. Typesetting French, German and English in Ω. EuroTEX'98, Saint-Malo, France, March Y. Haralambous, J. Plaice, and J. Braams. Never again active characters! Ω-Babel. TUGboat, 16(4): , Y. Haralambous, J. Plaice, and A. Picheral (trans.). Ω, une extension de TEX incluant UNICODE et des filtres de type Lex. Cahiers GUTenberg, 20:55--80, Translation of refereed conference publication [4]. 10. J. Plaice. Language-dependent ligatures. TUGboat, 14(3): , J. Plaice. Progress in the Omega project. TUGboat, 15(3): , J. Plaice and Y. Haralambous. The latest developments in Ω. TUGboat, 17(3): , Microsoft Inc. True Type in Windows tt/win95tt.htm. Voorjaar

8 Yannis Haralambous and John Plaice ΠΟΛΛΕΣ ΦΟΡΕΣ ΤΗΝ ΝΥΚΤΑ Ανδρ εας Εµπειρ ικος Οσοι απ ο σ ας γυρ ιζετε τ ην ν υκτα µ εσ στο υς δρ οµους, αµ εριµνοι η σκεπτικο ι, τ ην ανοιξι, κατ ατ ην εποχ ητο υ Επιταφ ιου Θρ ηνου, η εκε ικοντ αστ ις ωρες τ ις χαρο υµενες πο υ οδηγο υν στ ην θριαµβευτικ ην τ ην ανωσιν πο υ π αει ν α γ ιν Π ασχα, πρ ιν ακουσθο υν ο ι αναστ ασιµες καµπ ανες, κα ι, ακ οµη περισσ οτερο, τ ις ν υκτες το υ καλοκαιριο υ στο υς δρ οµους το υς ονειρικο υς το υ σκοτεινο υλονδ ινου, στο υς αλλους το υς πλατε ις η το υς στενο υς, πο υ εκτε ινονται γ υρω απ οτ ο νμ οσχοβα στ η Μ οσχα, η στ ας οδο υς τ ης κ ατασπρης Αθ ηνας, σ ε δορυ αλωτες στιγµ ες τ ης θλ ιψεως, η σ ε αφρ οεσσες στιγ- µ ες ε υδαιµον ιας, οταν παρ αθυρα κα ι εξ ωφυλλα χα ινουν δι απλατα ανοικτ αγι αν αδεχθο υν δροσι α κα ιµ υρα, οσοι απ ο σ ας ν υκτωρ γυρ ιζετε στο υς δρ οµους πανευτυχε ις πο υ εκσπερµατ ισατε, η δυστυχε ις πο υ κ αποια γυνα ικα δ εν εστερξε ν ασ ας δεχθ κα ι δ εν εστ αθη, λ ιγο αν προσ εξετε, θ α ακο υσετε πολλ α, οσα στ ην τ υρβη τ ης ηµ ερας δ υσκολον ε ιναι ν α ακουσθο υν. Οσοι απ ο σ ας γυρ ιζετε τ ην ν υκτα µ εσ στο υς δρ οµους, αµ εριµνοι η σκεπτικο ι, τ ην ανοιξι, κατ ατ ην εποχ ητο υ Επιταφ ιου Θρ ηνου, η εκε ι κοντ αστ ις ωρες τ ις χαρο υµενες πο υ οδηγο υν στ ην θριαµβευτικ ην τ ην ανωσιν πο υ π αει ν αγ ιν Π ασχα, πρ ιν ακουσθο υν ο ι αναστ ασιµες καµπ ανες, κα ι, ακ οµη περισσ οτερο, τ ις ν υκτες το υ καλοκαιριο υ στο υς δρ οµους το υς ονειρικο υς το υ σκοτεινο υ Λονδ ινου, στο υς αλλους το υς πλατε ις η το υς στενο υς, πο υ εκτε ινονται γ υρω απ ο τ ον Μ οσχοβα στ η Μ οσχα, η στ ας οδο υς τ ης κ ατασπρης Αθ ηνας, σ ε δορυ αλωτες στιγµ ες τ ης θλ ιψεως, η σ ε αφρ οεσσες στιγ- µ ες ε υδαιµον ιας, οταν παρ αθυρα κα ι εξ ωφυλλα χα ινουν δι απλατα ανοικτ α γι αν α δεχθο υν δροσι α κα ιµ υρα, οσοι απ ο σ ας ν υκτωρ γυρ ιζετε στο υς δρ οµους πανευτυχε ις πο υ εκσπερµατ ισατε, η δυστυχε ις πο υκ αποια γυνα ικα δ εν εστερξε ν ασ ας δεχθ κα ιδ εν εστ αθη, λ ιγο αν προσ εξετε, θ α ακο υσετε πολλ α, οσα στ ην τ υρβη τ ης ηµ ερας δ υσκολον ε ιναι ν α ακουσθο υν. Οσοι απ ο σ ας γυρ ιζετε τ ην ν υκτα µ εσ στο υς δρ οµους, αµ εριµνοι η σκεπτικο ι, τ ην ανοιξι, κατ α τ ην εποχ η το υ Επιταφ ιου Θρ ηνου, η εκε ι κοντ αστ ις ωρες τ ις χαρο υµενες πο υ οδηγο υν στ ην θριαµβευτικ ην τ ην ανωσιν πο υπ αει ν αγ ιν Π ασχα, πρ ιν ακουσθο υν ο ι αναστ ασιµες καµπ ανες, κα ι, ακ οµη περισσ οτερο, τ ις ν υκτες το υ καλοκαιριο υ στο υς δρ οµους το υς ονειρικο υς το υ σκοτεινο υ Λονδ ινου, στο υς αλλους το υς πλατε ις η το υς στενο υς, πο υ εκτε ινονται γ υρω απ ο τ ον Μ οσχοβα στ η Μ οσχα, η στ ας οδο υς τ ης κ ατασπρης Αθ ηνας, σ ε δορυ αλωτες στιγµ ες τ ης θλ ιψεως, ησ ε αφρ οεσσες στιγµ ες ε υδαιµον ιας, οταν παρ αθυρα κα ι εξ ωφυλλα χα ινουνδι απλατα ανοικτ αγι αν α δεχθο υν δροσι ακα ιµ υρα, οσοι απ οσ ας ν υκτωρ γυρ ιζετε στο υς δρ οµους πανευτυχε ις πο υ εκσπερµατ ισατε, η δυστυχε ις πο υκ αποια γυνα ικα δ εν εστερξε ν α σ ας δεχθ κα ιδ εν εστ αθη, λ ιγο αν προσ εξετε, θ α ακο υσετε πολλ α, οσα στ ην τ υρβη τ ης ηµ ερας δ υσκολον ε ιναι ν α ακουσθο υν. Figure 1. Roman, italic and bold versions of a Greek text 34 MAPS

9 The Design and Use of a Multiple-Alphabet Font with Ω fonts A B C D E F 0600 Ḁ ḁ Ḃ ḃ Ḅ ḅ Ḇ ḇ Ć ć Ḋ ḋ Ḍ ḍ Ḏ ḏ 0610 Ḑ ḑ Ḓ ḓ È è É é Ḙ ḙ Ḛ ḛ C c Ḟ ḟ 0620 Ḡ ḡ Ḣ ḣ Ḥ ḥ Ḧ ḧ Ḩ ḩ Ḫ ḫ Ḭ ḭ Í í 0630 Ḱ ḱ Ḳ ḳ Ḵ ḵ Ḷ ḷ L l Ḻ ḻ Ḽ ḽ Ḿ ḿ 0640 Ṁ ṁ Ṃ ṃ Ṅ ṅ Ṇ ṇ Ṉ ṉ Ṋ ṋ Ó ó Ö ö 0650 Ò ò Ó ó Ṕ ṕ Ṗ ṗ Ŕ ŕ Ṛ ṛ R r Ṟ ṟ 0660 Ṡ ṡ Ṣ ṣ Ṡ ṡ Ṡ ṡ Ṡ ṡ Ṫ ṫ Ṭ ṭ Ṯ ṯ 0670 Ṱ ṱ Ṳ ṳ Ṵ ṵ Ṷ ṷ Ú ú Ü ü Ṽ ṽ Ṿ ṿ 0680 Ẁ ẁ Ẃ ẃ Ẅ ẅ Ẇ ẇ Ẉ ẉ Ẋ ẋ Ẍ ẍ Ẏ ẏ 0690 Ẑ ẑ Ẓ ẓ Ẕ ẕ ẖ ẗ ẘ ẙ a ṡ 06A0 Ạ ạ Ả ả A a A a A a A a  â A a 06B0 A a A a A a Ă ă Ẹ ẹ Ẻ ẻ Ẽ ẽ E e 06C0 E e E e E e Ê ê Ỉ ỉ Ị ị Ọ ọ Ỏ ỏ 06D0 O o O o O o O o Ô ô Ớ ớ Ờ ờ Ở ở 06E0 Ỡ ỡ Ợ ợ Ụ ụ Ủ ủ Ứ ứ Ừ ừ Ử ử Ữ ữ 06F0 Ự ự Ỳ ỳ Ỵ Ỷ ỷ Ỹ ỹ Figure 2. Unicode Row 1E: Latin Extended Additional Voorjaar

10 Yannis Haralambous and John Plaice A B C D E F α α α α α α α α α α α α α α ε ε ε ε ε ε ε ε ε ε ε ε ε ε 0840 η η η η η η η η η η η η η η ι ι ι ι ι ι ι ι ι ι ι ι ι ι 0870 ο ο ο ο ο ο ο ο ο ο ο ο ο ο 0880 υ υ υ υ υ υ υ υ υ υ υ υ υ υ 0890 ω ω ω ω ω ω ω ω ω ω ω ω ω ω 08A0 08B0 08C0 08D0 08E0 08F0 Figure 3. Greek vowels with macron, for long syllables (not in Unicode) 36 MAPS

11 The Design and Use of a Multiple-Alphabet Font with Ω fonts A B C D E F 0A00 0A10 Л л Н н Т т Ч ч А а О о О о 0A20 У у У у У у У у Э э Э э Ю ю Я я 0A30 І і Ү ү А а Е е Е е Є є 0A40 И и І І О о У у Ы ы Ъ ъ Э э 0A50 Ю ю Я я Ѣ Ѫ ѫ 0A60 Ү ү Æ æ 0A70 Ə ə Ү ү Ђ ђ 0A80 А Є И І Ю Ѣ 0A90 а є и о ю 0AA0 А Є И І Ю Ѣ 0AB0 а є и ю 0AC0 А Є И І Ю Ѣ 0AD0 а є и ю 0AE0 А Є И І Ю Ѣ 0AF0 а є и ю Figure 4. Accented Cyrillic characters, for emphasis (not in Unicode) Voorjaar

The Design and Use of a Multiple-Alphabet Font with Ω

The Design and Use of a Multiple-Alphabet Font with Ω The Design and Use of a Multiple-Alphabet Font with Ω Yannis Haralambous 1 and John Plaice 2 1 Atelier Fluxus Virus, 187, rue Nationale, 59800 Lille, France. Email: yannis@pobox.com 2 School of Computer

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

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

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

Lecture 2. Soundness and completeness of propositional logic

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

Διαβάστε περισσότερα

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

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

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

Galatia SIL Keyboard Information

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

Διαβάστε περισσότερα

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 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

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

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

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

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

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

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

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

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

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11 Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and

Διαβάστε περισσότερα

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

Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

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

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

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

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

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

Block Ciphers Modes. Ramki Thurimella

Block Ciphers Modes. Ramki Thurimella Block Ciphers Modes Ramki Thurimella Only Encryption I.e. messages could be modified Should not assume that nonsensical messages do no harm Always must be combined with authentication 2 Padding Must be

Διαβάστε περισσότερα

BRAND MANUAL AND USER GUIDELINES

BRAND MANUAL AND USER GUIDELINES ΑΠΟΓΡΑΦΗ ΠΛΗΘΥΣΜΟΥ-ΚΑΤΟΙΚΙΩΝ 2021 ΕΛΛΗΝΙΚΗ ΣΤΑΤΙΣΤΙΚΗ ΑΡΧΗ BRAND MANUAL AND USER GUIDELINES BRIEF THE SCOPE Το 2021 θα πραγματοποιηθεί στην Ελλάδα η Γενική Απογραφή Πληθυσμού Κατοικιών που διενεργείται

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

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

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

Διαβάστε περισσότερα

Terabyte Technology Ltd

Terabyte Technology Ltd Terabyte Technology Ltd is a Web and Graphic design company in Limassol with dedicated staff who will endeavour to deliver the highest quality of work in our field. We offer a range of services such as

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

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Τέλος Ενότητας Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί

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

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

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

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

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

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

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

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

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

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

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

Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής

Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Να γραφεί πρόγραμμα το οποίο δέχεται ως είσοδο μια ακολουθία S από n (n 40) ακέραιους αριθμούς και επιστρέφει ως έξοδο δύο ακολουθίες από θετικούς ακέραιους

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

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

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

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

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

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

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΩΣ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΠΜΣ «ΠΡΟΗΓΜΕΝΑ ΣΥΣΤΗΜΑΤΑ ΠΛΗΡΟΦΟΡΙΚΗΣ» ΚΑΤΕΥΘΥΝΣΗ «ΕΥΦΥΕΙΣ ΤΕΧΝΟΛΟΓΙΕΣ ΕΠΙΚΟΙΝΩΝΙΑΣ ΑΝΘΡΩΠΟΥ - ΥΠΟΛΟΓΙΣΤΗ»

ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΩΣ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΠΜΣ «ΠΡΟΗΓΜΕΝΑ ΣΥΣΤΗΜΑΤΑ ΠΛΗΡΟΦΟΡΙΚΗΣ» ΚΑΤΕΥΘΥΝΣΗ «ΕΥΦΥΕΙΣ ΤΕΧΝΟΛΟΓΙΕΣ ΕΠΙΚΟΙΝΩΝΙΑΣ ΑΝΘΡΩΠΟΥ - ΥΠΟΛΟΓΙΣΤΗ» ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΩΣ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΠΜΣ «ΠΡΟΗΓΜΕΝΑ ΣΥΣΤΗΜΑΤΑ ΠΛΗΡΟΦΟΡΙΚΗΣ» ΚΑΤΕΥΘΥΝΣΗ «ΕΥΦΥΕΙΣ ΤΕΧΝΟΛΟΓΙΕΣ ΕΠΙΚΟΙΝΩΝΙΑΣ ΑΝΘΡΩΠΟΥ - ΥΠΟΛΟΓΙΣΤΗ» ΜΕΤΑΠΤΥΧΙΑΚΗ ΙΑΤΡΙΒΗ ΤΟΥ ΕΥΘΥΜΙΟΥ ΘΕΜΕΛΗ ΤΙΤΛΟΣ Ανάλυση

Διαβάστε περισσότερα

Problem Set 3: Solutions

Problem Set 3: Solutions CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

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

[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

Πανεπιστήμιο Κρήτης, Τμήμα Επιστήμης Υπολογιστών Άνοιξη 2009. HΥ463 - Συστήματα Ανάκτησης Πληροφοριών Information Retrieval (IR) Systems

Πανεπιστήμιο Κρήτης, Τμήμα Επιστήμης Υπολογιστών Άνοιξη 2009. HΥ463 - Συστήματα Ανάκτησης Πληροφοριών Information Retrieval (IR) Systems Πανεπιστήμιο Κρήτης, Τμήμα Επιστήμης Υπολογιστών Άνοιξη 2009 HΥ463 - Συστήματα Ανάκτησης Πληροφοριών Information Retrieval (IR) Systems Στατιστικά Κειμένου Text Statistics Γιάννης Τζίτζικας άλ ιάλεξη :

Διαβάστε περισσότερα

Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο

Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL

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

ΓΡΑΜΜΙΚΟΣ & ΔΙΚΤΥΑΚΟΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ

ΓΡΑΜΜΙΚΟΣ & ΔΙΚΤΥΑΚΟΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ ΓΡΑΜΜΙΚΟΣ & ΔΙΚΤΥΑΚΟΣ ΠΡΟΓΡΑΜΜΑΤΙΣΜΟΣ Ενότητα 12: Συνοπτική Παρουσίαση Ανάπτυξης Κώδικα με το Matlab Σαμαράς Νικόλαος Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons.

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

«Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων. Η μεταξύ τους σχέση και εξέλιξη.»

«Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων. Η μεταξύ τους σχέση και εξέλιξη.» ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΑΓΡΟΝΟΜΩΝ ΚΑΙ ΤΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΤΟΜΕΑΣ ΓΕΩΓΡΑΦΙΑΣ ΚΑΙ ΠΕΡΙΦΕΡΕΙΑΚΟΥ ΣΧΕΔΙΑΣΜΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ: «Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων.

Διαβάστε περισσότερα

Business English. Ενότητα # 9: Financial Planning. Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων

Business English. Ενότητα # 9: Financial Planning. Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ Ανώτατο Εκπαιδευτικό Ίδρυμα Πειραιά Τεχνολογικού Τομέα Business English Ενότητα # 9: Financial Planning Ευαγγελία Κουτσογιάννη Τμήμα Διοίκησης Επιχειρήσεων Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

ΜΟΝΤΕΛΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ

ΜΟΝΤΕΛΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ ΜΟΝΤΕΛΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ Ενότητα 12 Τμήμα Εφαρμοσμένης Πληροφορικής Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons. Για εκπαιδευτικό υλικό, όπως εικόνες, που υπόκειται

Διαβάστε περισσότερα

Overview. Transition Semantics. Configurations and the transition relation. Executions and computation

Overview. Transition Semantics. Configurations and the transition relation. Executions and computation Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition

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

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

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ

Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Η ΔΙΑΣΤΡΕΥΛΩΣΗ ΤΗΣ ΕΛΛΗΝΙΚΗΣ ΓΛΩΣΣΑΣ ΜΕΣΩ ΤΩΝ SOCIAL MEDIA ΤΗΝ ΤΕΛΕΥΤΑΙΑ ΠΕΝΤΑΕΤΙΑ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΤΗΣ ΑΝΑΣΤΑΣΙΑΣ-ΜΑΡΙΝΑΣ ΔΑΦΝΗ Τμήμα Δημοσίων Σχέσεων & Επικοινωνίας Τεχνολογικό Εκπαιδευτικό Ίδρυμα Ιονίων

Διαβάστε περισσότερα

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ "

ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ ΥΓΕΙΑΣ ΤΕΧΝΟΛΟΓΙΚΟ ΕΚΠΑΙΔΕΥΤΙΚΟ ΙΔΡΥΜΑ ΚΑΛΑΜΑΤΑΣ ΣΧΟΛΗ ΔΙΟΙΚΗΣΗΣ ΟΙΚΟΝΟΜΙΑΣ ΤΜΗΜΑ ΜΟΝΑΔΩΝ ΥΓΕΙΑΣ ΚΑΙ ΠΡΟΝΟΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ "ΠΟΛΥΚΡΙΤΗΡΙΑ ΣΥΣΤΗΜΑΤΑ ΛΗΨΗΣ ΑΠΟΦΑΣΕΩΝ. Η ΠΕΡΙΠΤΩΣΗ ΤΗΣ ΕΠΙΛΟΓΗΣ ΑΣΦΑΛΙΣΤΗΡΙΟΥ ΣΥΜΒΟΛΑΙΟΥ

Διαβάστε περισσότερα

Ρηματική άποψη. (Aspect of the verb) Α. Θέματα και άποψη του ρήματος (Verb stems and aspect)

Ρηματική άποψη. (Aspect of the verb) Α. Θέματα και άποψη του ρήματος (Verb stems and aspect) 15 Ρηματική άποψη (Aspect of the verb) Α. Θέματα και άποψη του ρήματος (Verb stems and aspect) imperfective perfective Verb forms in Modern Greek are based either on the imperfective or the perfective

Διαβάστε περισσότερα

TMA4115 Matematikk 3

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

Διαβάστε περισσότερα

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

Στο εστιατόριο «ToDokimasesPrinToBgaleisStonKosmo?» έξω από τους δακτυλίους του Κρόνου, οι παραγγελίες γίνονται ηλεκτρονικά. Διαστημικό εστιατόριο του (Μ)ΑστροΈκτορα Στο εστιατόριο «ToDokimasesPrinToBgaleisStonKosmo?» έξω από τους δακτυλίους του Κρόνου, οι παραγγελίες γίνονται ηλεκτρονικά. Μόλις μια παρέα πελατών κάτσει σε ένα

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

Υλοποίηση Δικτυακών Υποδομών και Υπηρεσιών: OSPF Cost

Υλοποίηση Δικτυακών Υποδομών και Υπηρεσιών: OSPF Cost Υλοποίηση Δικτυακών Υποδομών και Υπηρεσιών: OSPF Cost Πανεπιστήμιο Πελοποννήσου Τμήμα Επιστήμης & Τεχνολογίας Τηλεπικοινωνιών Ευάγγελος Α. Κοσμάτος Basic OSPF Configuration Υλοποίηση Δικτυακών Υποδομών

Διαβάστε περισσότερα

ω 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

ω 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 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

TaxiCounter Android App. Περδίκης Ανδρέας ME10069

TaxiCounter Android App. Περδίκης Ανδρέας ME10069 TaxiCounter Android App Περδίκης Ανδρέας ME10069 Content Android Operating System Development Tools Taxi Counter Algorithm Design Development Process Android Operating System Android is a Linux-based operating

Διαβάστε περισσότερα

Partial Trace and Partial Transpose

Partial Trace and Partial Transpose Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This

Διαβάστε περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/2006

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/2006 ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 11/3/26 Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι το 1 εκτός αν ορίζεται διαφορετικά στη διατύπωση

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 10η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 0η: Basics of Game Theory part 2 Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Best Response Curves Used to solve for equilibria in games

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

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

Διαβάστε περισσότερα

(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)

(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 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

Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

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

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

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

Διαβάστε περισσότερα

Second Order RLC Filters

Second Order RLC Filters ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1. Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

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

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

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Dynamic types, Lambda calculus machines Section and Practice Problems Apr 21 22, 2016

Dynamic types, Lambda calculus machines Section and Practice Problems Apr 21 22, 2016 Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Dynamic types, Lambda calculus machines Apr 21 22, 2016 1 Dynamic types and contracts (a) To make sure you understand the

Διαβάστε περισσότερα

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

Πανεπιστήμιο Πειραιώς Τμήμα Πληροφορικής Πρόγραμμα Μεταπτυχιακών Σπουδών «Πληροφορική»

Πανεπιστήμιο Πειραιώς Τμήμα Πληροφορικής Πρόγραμμα Μεταπτυχιακών Σπουδών «Πληροφορική» Πανεπιστήμιο Πειραιώς Τμήμα Πληροφορικής Πρόγραμμα Μεταπτυχιακών Σπουδών «Πληροφορική» Μεταπτυχιακή Διατριβή Τίτλος Διατριβής Επίκαιρα Θέματα Ηλεκτρονικής Διακυβέρνησης Ονοματεπώνυμο Φοιτητή Σταμάτιος

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

5.4 The Poisson Distribution.

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

Διαβάστε περισσότερα