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            "1435": {
                "pageid": 1435,
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                "title": "References",
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                        "*": "'''1. Problem and history'''\n\n\u201cReference\u201d is any sign joining a sentence with any preceding sentence. References are the means for making the text a compact entity. They are the basis for evaluating text cohesion. In qualitative linguistics there is an ample literature on different kinds of references and their behavior. The sentences joined by a commoh reference are called ''hrebs''.\n\nThe only law, known in the literature as \u201cH\u0159eb\u00ed\u010dek\u00b4s reference law\u201d, originates from H\u0159eb\u00ed\u010dek\u00b4s (1985) derivation. Altmann (1988: 81-85) proposed merely some further problems for investigation. The law was corroborated on many Turkish texts. Formula (4) supports Herdan\u00b4s version of the type-token ratio (<math>\\rightarrow</math>). \n\n'''2. Hypothesis''' \n\n''The number of references in text depends on the number of words and the number of sentences''.\n\n\u201cWord\u201d is every word-like entity (token) in the text.\n\u201cSentence\u201d in written texts is demarcated by orthographic signs.\n\n'''3. Derivation'''\n\nLet\n\n<math>r</math> = number of references in text\n\n<math>s</math> = number of sentences in text\n\n<math>n</math> = number of word tokens in text\n\n<math>v</math> = number of word types in text (vocabulary of the text)\n\n<math>w</math> = vocabulary richness\n\n<math>a, b, c, A, B</math>   = constants\n\nH\u0159eb\u00ed\u010dek\u00b4s assumptions: \n\n(i)\tthe richer the vocabulary, the smaller the number of references,\n\n(ii)\tthe more sentences in the text, the greater the number of references.\n\nThe change in the number of references relative to the change in the vocabulary richness is proportional to the number of sentences,  \n\n<math>\\frac{\\partial r}{\\partial w}=As</math>,\t \n\t\nand, at the same time, the change in the number of references relative to the change in the number of sentences is proportional to the vocabulary richness of the text,\n\n<math>\\frac{\\partial r}{\\partial s}=Aw</math>.\n\nThis yields the following solution<:\n\n(1) <math>r=csw\\quad</math> \t(<math>c = AB</math>).\n\nTaking the simplest interpretation of vocabulary richness <math>w</math> as\n\n<math>w=\\frac{v}{n}</math>,\t \n\nwe obtain\n\n(2) <math>r=cs\\frac{v}{n}</math> .\n\nUsing Herdan\u00b4s (1966: 76) type-token ratio (<math>\\rightarrow</math>) to express the vocabulary of the text as a power function of its length,\n\n(3) <math> v=n^a,\\quad 0 < a < 1</math>,\n\nand inserting this in (2), one obtains\n\n(4) <math> r= csn^{a-1}=csn^b, \\quad  a-1 = b \\quad     -1 < b < 0</math>,\n\nwhich meets assumptions (i) and (ii).\n\t\n'''Example''': The course of references in a Turkish text\n\nH\u0159eb\u00ed\u010dek (1992) examined the course of references in several Turkish texts. One of these cases is shown in Table 1.\n\n<div align=\"center\">[[Image:Tabelle1_R.jpg]]</div>\n\n\n'''4. Authors''': U. Strauss, G. Altmann\n\n'''5. References'''\n\n'''Altmann, G.''' (1988a). ''Wiederholungen in Texten''. Bochum, Brockmeyer.\n\n'''H\u0159eb\u00ed\u010dek, L'''. (1985). Text as a unit and co-references. In: Ballmer, Th.T. (ed.), ''Linguistic dynamics'': 190-198. New York, de Gruyter.\n\n'''H\u0159eb\u00ed\u010dek, L'''.  (1986). Cohesion in Ottoman poetic texts. ''Archiv orient\u00e1ln\u00ed 54,252-256''.\n\n'''H\u0159eb\u00ed\u010dek, L'''. (1989). A syntactic variable on the text level. Glottometrika ''10, 204-218''.\n\n'''H\u0159eb\u00ed\u010dek, L.''' (1992). ''Text in communication: Supra-sentence structure''. Bochum, Brockmeyer.\n\n'''H\u0159eb\u00ed\u010dek, L'''. (2000). ''Variation in sequences''. Prague: Oriental Institute.\nH\u0159eb\u00ed\u010dek 1985, 1986, 1989, 1992, 2000; Altmann 1988.\n\n'''H\u0159eb\u00ed\u010dek, L'''. (2006). Text laws. In: K\u00f6hler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 348-361.'' Berlin: de Gruyter.\n\n'''Mehler, A. ''' (2006). Eigenschaften der textuellen Einheiten und Systeme. In: K\u00f6hler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 325-348.'' Berlin: de Gruyter.\n\n[[[Das zeichnen geht schwer, da zwei unabh\u00e4ngige Variablen drin sind. Mit Harvard Graphics wirds]]]\n\n[[Category:Unfertig]]"
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            "1376": {
                "pageid": 1376,
                "ns": 0,
                "title": "Repeat rate and inventory of phonemes",
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                        "*": "'''1. Problem and history'''\n\nThe repeat rate is a measure expressing the diversity of relative frequencies of the elements of a closed system. For computing it, the data must be normalized, i.e. the functions used in \u201dPhoneme frequencies\u201d (<math>rightarrow</math>) must be considered probability functions. Its expression is\n\n(1)<math> R = \\sum_{x=1}^K p_x^2</math>\t \n\nweher <math>p_x = f_x/N</math>, N being the number of phonemes in the sample, fx is the absolute frequency of phoneme x, K is the number of phonemes in the inventory (inventory size). The problem is to find whether R is associated with the inventory size K.\n\n\tLehfeldt and Altmann (1983) have shown that R is lawlikely joined with K if the frequencies are distributed according to the right truncated geometric distribution. Z\u00f6rnig and Altmann (1983) have shown that the relation holds if the frequencies follow the Zipf-Mandelbrot distribution (see Ranking).\n\n'''2. Hypothesis'''\n\n''The repeat rate of phonemes R is a monotone decreasing function of the phoneme invetory K.''\n\n'''3. Derivation'''\n\n'''3.1. From the geometric distribution'''. \nLet the rank-ordered phoneme frequencies follow the 1-displaced right truncated geometric distribution, defined as\n\n(2)<math> P_x = aq^{x-1},\\quad x = 1, 2, 3, ..., K</math> .\n\nwith \n\n(3)<math> a = \\frac{p}{1-q^K}</math>\n\t \nfrom which\n\n(4)<math> q^K = \\frac{a-1+q}{a}</math>\t .\n\nFor the Repeat rate we have\n\n(5)<math> R = \\sum_{x=1}^K (aq^{x-1})^2 = \\frac{a^2(1-q^{2K})}{1-q^2}</math>.\n\nUsing (4) from which <math>q^{2K} = (a-1+q)^2/a^2</math>  one obtains\n\n(6)<math>R = \\frac{2a+q-1}{1+q}</math>.\n\nThe first approximation to a theoretical value of R is given by setting\n\n(7)<math> q = \\frac{K-2}{K+2}</math>\t .\n\nInserting (3) and (7) in (6) and neglecting <math>q^K</math> which is very small as compared to the other values one obtains\n\n(8)<math> R =\\frac{2}{k}</math>,\n\ni.e. the Repaet rate is a very simple function of the inventory size. \n\n'''Example'''.  Repeat rate for 63 languages\n\nAltmann and Lehfeldt (1980: 154-156) collected the R-values for 63 languages from the published literature and compared them with the theoretical value (8) . The result is shown in Table 1 and Figure 1. Some of the points concern letters, some are for phoneme in texts and in dictionary, in some cases alternative counting has been performed (e.g. considering long vowels as one or two phonemes, etc.).\n\n<div align=\"center\">[[Image:Tabelle1_RRAIOP.jpg]]</div>\n\n\n'''3.2. From the Zipf-Mandelbrot distribution'''\n\nZ\u00f6rnig and Altmann (1983) used the Zipf-Mandelbrot distribution to derive R, i.e.\n\n(9)<math> P_x = \\frac{A}{(B+x)^c}, \\quad x = 1, 2, ..., K</math>.\n\nSince <math> R = A^2 \\sum_{x=1}^K (B+x)^{-2c}</math>\n\t \nwhere A is the normalizing constant. Using an approximation of the sum by an appropriate integral they obtained\n\n(10)<math> R = \\begin{cases} \\frac{(1-c)^2\\lbrack (B+K)^{1-2c} - (B+1)\\rbrack ^{1-2c}}{(1-2c) \\lbrack (B+K)^{1-c} - (B+1)^{1-c}\\rbrack ^2},\\quad c \\ne 1, c\\ne 0.5  \\\\ \\frac{(1-c)^2}{\\lbrack (B+K)^{1-c}-(B+1)^{1-c}\\rbrack ^2}\\ln \\frac{B+K}{B+1},\\quad c = 0.5 \\\\\\frac{(B+1)^{-1}-(B+K)^{-1}}{\\ln{B+K \\choose B+1}^2},\\quad c=1 \\end{cases}</math>\t \n\nBy an iterative procedure they have shown that the best fitting to the existing data is the third formula in (10) i.e.  with c = 1 and B = 0.61, thus\n\n(11)<math> R = \\frac{1.61^{-1}-(0.61 + K)^{-1}}{\\left( \\ln \\frac{0.61 + K}{1.61} \\right)^2}</math>\t \n\nwhich can be seen in Fig. 2. The results of fitting are in Table 2. The parameter B could not be interpreted as yet. It must be computed anew at any enlarging of the stock of examined languages.\n\n<div align=\"center\">[[Image:Tabelle2_RRAIOP.jpg]]</div>\n\nBoth results are satisfactory. However, other formulas in Phoneme frequency (<math>\\rightarrow</math>)  can be tested, too. Of course, the best fit follows from <math>R = aK^{-b}</math> but considering the fact that the data are very mixed (letters, phonemes, texts, dictionaries, different interpretations etc.) the result is a preliminary stimulus for further research.\n\n<div align=\"center\">[[Image:Grafik1_RRAIOP.jpg]]</div>\n<div align=\"center\">Figure 2. Fitting formulas (8) \u2013\u2013\u2013\u2013\u2013\u2013 and  (11) -------- to Repeat rates in 63 languages</div> \n\n\n'''4. Authors:''' U. Strauss, G. Altmann, V. Kromer\n\n'''5. References'''\n\n'''Altmann, G., Lehfeldt, W'''. (1980). ''Einf\u00fchrung in die quantitative Phonologie''. Buchum: Brockmeyer.\n\n'''Z\u00f6rnig, P., Altmann, G.''' (1983). The repeat rate of phoneme frequencies and the Zipf-Mandel-brot law. ''Glottometrika 5, 205-211''.\n\n'''Z\u00f6rnig, P., Altmann, G'''. (1984). The entropy of phoneme frequencies and the Zipf-Mandelbrot law. ''Glottometrika 6, 41-47''."
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