Interval Arithmetic and Standardization

Interval arithmetic is arithmetic for continuous sets. Floating-point intervals are intervals of real numbers with floating-point bounds. Operations for intervals can be efficiently implemented. There is an unanimous agreement, how to define the basic operations, if we exclude division by an interval containing zero. Hence, it should be standardized. For division by zero, two options are possible, the clean exception free interval arithmetic or the containment arithmetic. They can be standardized as options. Elementary functions for intervals can be defined. In some application areas loose evaluation of functions, i.e. evaluation over an interval which is not completely contained in the function domain, is recommended, In this case, however, a discontinuity flag has to be set to inform that Brouwer\u27s fixed point theorem is no longer applicable in that case

Latest Developments on the IEEE 1788 Effort for the Standardization of Interval Arithmetic

(Standardization effort supported by the INRIA D2T.)International audienceInterval arithmetic undergoes a standardization effort started in 2008 by the IEEE P1788 working group. The structure of the proposed standard is presented: the mathematical level is distinguished from both the implementation and representation levels. The main definitions are introduced: interval, mathematical functions, either arithmetic operations or trigonometric functions, comparison relations, set operations. While developing this standard, some topics led to hot debate. Such a hot topic is the handling of exceptions. Eventually, the system of decorations has been adopted. A decoration is a piece of information that is attached to each interval. Rules for the propagation of decorations have also been defined. Another hot topic is the mathematical model used for interval arithmetic. Historically, the model introduced by R. Moore in the 60s covered only non-empty and bounded intervals. The set-based model includes the empty set and unbounded intervals as well. Tenants of Kaucher arithmetic also insisted on offering "reverse" intervals. It has eventually been decided that an implementation must provide at least one of these flavors of interval arithmetic. The standard provides hooks for these different flavors. As the preparation of the draft should end in December 2013, no chapter is missing. However, a reference implementation would be welcome

Überidentifikation von Lernstörungen bei Kindern mit Deutsch als Zweitsprache. Implikationen für die Normierung von standardisierten Schulleistungstests

This German prevalence study examined disproportionate representation of language- minority students among children identified with learning disorder (LD) according to ICD-10 (WHO, 1992). Most German school achievement tests used in LD diagnostics do not provide separate norms for language-minority students, and thus do not take these children’s second language status into account when evaluating their academic performance. Although this is likely to result in an LD over identification of language-minority students, little is known about the magnitude of this effect. Therefore, we compared the estimation of LD prevalence between native German speaking students (n = 566) and language-minority students (n = 478) when pooled versus group-specific achievement norms were used for LD classification. Three important findings emerged from our study: Firstly, and as expected, significant disproportionality effects occurred under pooled norms. Specifically, the likelihood of being diagnosed with LD amounted to 14–18 % among native German speakers and nearly doubled to 25–30 % among language-minority students. Secondly, disproportionality varied as a function of LD subtype: Whereas no disproportionate representation was revealed for arithmetic LD (F81.2), overidentification of language-minority students was found for verbal LD subtypes (namely, reading disorder [F81.0], spelling disorder [F81.1], and mixed disorder of scholastic skills [F81.3]). Thirdly, disproportionality effects were absent when group-specific norms were used for LD classification that controlled for second-language issues. Challenges that have to be met when testing language-minority students for LD are discussed. (DIPF/Orig.)Die Prävalenzstudie untersucht bei Kindern, die Deutsch als Muttersprache (DaM) bzw. als Zweitsprache (DaZ) sprechen, die Häufigkeit von Lernstörungen nach ICD-10 (WHO, 1992). Die meisten deutschen Schulleistungstests, die zur Lernstörungsdiagnose herangezogen werden, stellen keine gesonderten Normen für Kinder mit DaZ bereit. Es ist anzunehmen, dass dies zu einer Überidentifikation von Lernstörungen bei Kindern mit DaZ führt, da die besondere Spracherwerbssituation dieser Kinder nicht berücksichtigt wird. Dennoch ist bislang wenig über das Ausmaß dieses Effektes bekannt. Die vorliegende Studie vergleicht daher die Lernstörungsprävalenz zwischen Drittklässlern mit DaM (n = 566) bzw. mit DaZ (n = 478) wenn gemeinsame versus getrennte Schulleistungsnormen zur Leistungsbeurteilung herangezogen werden. Die Studie erbrachte drei wesentliche Ergebnisse: (1) Wie erwartet kam es bei Verwendung gemeinsamer Schulleistungsnormen zu einer deutlichen Erhöhung der Lernstörungsprävalenz bei Kindern mit DaZ. Die Wahrscheinlichkeit einer Lernstörungsdiagnose belief sich für diese Teilstichprobe auf 25–30 % und war damit annähernd doppelt so groß wie bei Kindern mit DaM, für die sich eine Gesamtprävalenz von 14–18 % ergab. (2) Die Gruppenunterschiede variierten dabei in Abhängigkeit des Lernstörungstypus: Während keine signifikant unterschiedlichen Prävalenzraten für die isolierte Rechenstörung (F81.2) nachweisbar waren, zeigten sich für die verbalen Lernstörungstypen (d. h. Lese-Rechtschreibstörung [F81.0], isolierte Rechtschreibstörung [F81.1] und kombinierte Störung schulischer Fertigkeiten [F81.3]) signifikant erhöhte Prävalenzraten für Kinder mit DaZ. (3) Werden hingegen getrennte Schulleistungsnormen zur Lernstörungsdiagnose herangezogen um für die besondere Spracherwerbssituation von Kindern mit DaZ zu kontrollieren, nähern sich die Prävalenzraten beider Gruppen wie erwartet auf ein vergleichbares Niveau an. Es wird diskutiert, welche Herausforderungen sich bei der Lernstörungsdiagnostik von Kindern mit DaZ ergeben. (DIPF/Orig.

The theory of index-based futures and options markets

This paper reviews the development of index-based futures and options markets. The thesis is that, while the growth of these markets to date has been dramatic, their development could be extended much further, if some problems of measurement can be solved.