This paper deals with the applications of symbolic computation of Mathematica 7.0 (Wolfram, 2008) in distribution theory. The purpose of this study is twofold. Firstly, we will implement some functions to extend Mathematica capabilities to handle symbolic computations of the characteristic function for linear combination of independent univariate random variables. These functions utilizes pattern-matching codes that enhance Mathematica's ability to simplify expressions involving the product and summation of algebraic terms. Secondly, characteristic function can be classified into commonly used distributions, including six discrete distributions and seven continuous distributions, via the pattern-matching feature of Mathematica. Finally, several examples will be presented. The examples include calculating limit of characteristic function of linear combinations of independent random variables, and applications of coded functions and illustrate the central limit theorem, the law of large numbers and properties of some distributions.

Mathematica is an extremely powerful and flexible symbolic computer algebra system that enables the user to deal with complicated algebraic tasks. It can also easily handle the numerical and graphical sides. One such task is the derivation of moment generating functions (MGF) and characteristic functions (CF), demonstrably effective tools to characterize a distribution. In this paper, we define some rules in Mathematica to help in computing the MGF and CF for linear combination of independent random variables. These commands utilizes pattern-matching code that enhances Mathematica's ability to simplify expressions involving the product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for MGF and CF. Applications of these rules to determine mean, variance and distribution are illustrated for various independent random variables.

There are discussed implementational aspects of the special-purpose computer algebra system FELIX designed for computations in constructive algebra. In particular, data types developed for the representation of and computation with commutative and non-commutative polynomials are described. Furthermore, comparisons of time and memory requirements of different polynomial representations are reported.

The special-purpose computer algebra system FELIX is designed for computations in constructive commutative and non-commutative algebra. In this paper we discuss some features of the system supporting the computation of rather complex problems, especially standard basis computations, using standard hardware. There is a frst aspect concerning the definition and implementation of the basic data types which should be a good compromise between space and time efficient representations of the algebraic objects. Usually, rather complex computations are very time consuming (up to weeks) and often require several attempts. So, there are included special session saving methods in FELIX which allows to backup the attained intermediate results in form of memory images into special session files and to restart later on. Finally, we describe our efforts crunching complex problems by parallelization. The implemented interface is based on stream sockets and includes a special protocol for the data exchange. It supports the distributed computation on heterogeneous, loosely coupled systems.

<p>SammanfattningSkolverket beslutade 2006 att tillåta symbolhanterande räknare på nationella prov i matematik för alla gymnasieelever från höstterminen 2007. Beslutet debatterades flitigt t.ex. i tidskriften <em>Nämnaren</em>.</p><p>Prototypen till en ny symbolhanterande räknare testades i sju ’pilotklasser’ vid fyra svenska gymnasieskolor från december 2006. Jag har följt två av dessa klasser t.o.m. höstterminen 2007 då de använde den färdiga räknaren TI-Nspire CAS.</p><p>Syftet med denna pilotstudie är att ta reda på hur den nya symbolhanterande räknaren fungerar i matematikundervisningen i de två pilotklasserna på gymnasiets NV-program, främst ur elevernas perspektiv, men också ur lärarnas.</p><p>Tanken är att pröva olika metoder, som kan användas i en större undersökning.</p><p>Bakgrunden ger bl.a. en översikt av de handburna räknehjälpmedlens historia och en kortfattad historik över elektroniska hjälpmedel i den svenska skolan. Tidigare forskning tar upp miniräknaren i skolan och då främst grafräknaren.</p><p>Observationsmetoderna är miniräknardagböcker, diagnostiska test, elev- och lärarenkät. Jag jämför också klassernas kursbetyg och klassificerar ett antal matematikuppgifter m.a.p. möjlig användning av CAS.</p><p>Resultatet gav en mångfacetterad bild med både positiva och negativa sidor. Skolorna behöver ta ställning till vilka klasser som ska använda symbolhanterande räknare och erbjuda lärarna fortbildning. Det behövs metodutveckling, nya läromedel och en reformerad lärarutbildning för att fullt kunna dra nytta av den nya teknikens möjligheter.</p><p>Jag lämnar flera förslag på fortsatt forskning.</p> / kontakt: david_u_strom@hotmail.com

SammanfattningSkolverket beslutade 2006 att tillåta symbolhanterande räknare på nationella prov i matematik för alla gymnasieelever från höstterminen 2007. Beslutet debatterades flitigt t.ex. i tidskriften Nämnaren. Prototypen till en ny symbolhanterande räknare testades i sju ’pilotklasser’ vid fyra svenska gymnasieskolor från december 2006. Jag har följt två av dessa klasser t.o.m. höstterminen 2007 då de använde den färdiga räknaren TI-Nspire CAS. Syftet med denna pilotstudie är att ta reda på hur den nya symbolhanterande räknaren fungerar i matematikundervisningen i de två pilotklasserna på gymnasiets NV-program, främst ur elevernas perspektiv, men också ur lärarnas. Tanken är att pröva olika metoder, som kan användas i en större undersökning. Bakgrunden ger bl.a. en översikt av de handburna räknehjälpmedlens historia och en kortfattad historik över elektroniska hjälpmedel i den svenska skolan. Tidigare forskning tar upp miniräknaren i skolan och då främst grafräknaren. Observationsmetoderna är miniräknardagböcker, diagnostiska test, elev- och lärarenkät. Jag jämför också klassernas kursbetyg och klassificerar ett antal matematikuppgifter m.a.p. möjlig användning av CAS. Resultatet gav en mångfacetterad bild med både positiva och negativa sidor. Skolorna behöver ta ställning till vilka klasser som ska använda symbolhanterande räknare och erbjuda lärarna fortbildning. Det behövs metodutveckling, nya läromedel och en reformerad lärarutbildning för att fullt kunna dra nytta av den nya teknikens möjligheter. Jag lämnar flera förslag på fortsatt forskning. / kontakt: david_u_strom@hotmail.com

This diploma thesis deals with conict economic situations based on game theory. In the beginning, basic models of conict situations and current popular software tools are dened not only for the general support of student education or for science, but also for solving economic problems in game theory. Based on this analysis, the conicting situation of two competing rms is being solved. Gradually, work goes deeper into areas of delay dierential equations that better show the behavior of two players on the market. Subsequently, these delayed dierential equations are projected into the Cournot model, for which a critical value is identied that switches the stability of two rms on the market due to the delayed realization of their outputs.

For the problem of producing pedagogical step-by-step solutions to mathematical problems in education, standard methods and algorithms used in construction of computer algebra systems are often not suitable. A method of using rules to manipulate mathematical expressions in small steps is suggested and implemented. The problem of creating a step-by-step solution by choosing which rule to apply and when to do it is redefined as a graph search problem and variations of the A* algorithm are used to solve it. It is all put together into one prototype solver that was evaluated in a study. The study was a questionnaire distributed among high school students. The results showed that while the solutions were not as good as human-made ones, they were competent. Further improvements of the method are suggested that would probably lead to better solutions. / För problemet att producera pedagogiska steg-för-steg-lösningar till matematiska problem inom utbildning, är vanliga metoder och algoritmer som används i konstruktion av datoralgebrasystem ofta inte lämpliga. En metod som använder regler för att manipulera matematiska uttryck i små steg föreslås och implementeras. Problemet att välja vilka regler som ska appliceras och när de ska göra det för att skapa en steg-för-steg-lösning omdefineras som ett grafsökningsproblem och varianter av algoritmen A* används för att lösa det. Allt sätts ihop till en prototyp av en lösare vilken utvärderas i en studie. Studien var ett frågeformulär som delades ut till gymnasiestudenter. Resultaten visade att även fast lösningar skapade av programmet inte var lika bra som lösningar skapade av människor, så var de anständiga. Fortsatta föbättringar av metoden föreslås, vilka troligtvis skulle leda till bättre lösningar.

Recently, there has been an increasing interest in high school mathematics education, especially in the teaching and learning of calculus. For example, studies conducted by Bressoud (2010); Judson and Nishimori (2005); Koh and Divaharan (2011); and St. Jarre (2008) all looked at how to improve the understanding of calculus students and what roles the educator must take to ensure that their students are successful. The purpose of this study was to determine if there was a significant difference between instruction using computer algebra system (CAS) compared to instruction using the graphing calculator in high school calculus on students’ conceptual and procedural understanding. This study explored and compared two different types of instruction based on the use of two different types of technology, CAS and graphing calculator. The total population for this study consisted of 333 students. There were 187 students classified as using the graphing calculator and 146 students classified as using CAS. The data for this study were collected from four Advanced Placement (AP) calculus AB courses from high schools in Florida. The study used observations and two sets of calculus tasks in order to gather data. The research questions for this study looked at comparing the grades of students categorized based on the type of instruction received during the learning of calculus. The statistical procedure that was used was a simple oneway analysis of variance (ANOVA). The results indicated that there was no significant difference between the two types of instruction on the students’ procedural knowledge, iii however, there was statistical significance on the students’ conceptual understanding in favor of the CAS students. The study introduces a framework on how to obtain information about the effects of different types of instruction on students’ understanding of calculus. The results of this study contribute in assisting teachers and future researchers on how to analyze student work in order to obtain information about the students’ conceptual and procedural understanding of first semester calculus.