El tema d'aquesta tesi és l'enumeració de permutacions amb subseqüències prohibides respecte a certs estadístics, i l'enumeració de permutacions que eviten subseqüències generalitzades.Després d'introduir algunes definicions sobre subseqüències i estadístics en permutacions i camins de Dyck, comencem estudiant la distribució dels estadístics -nombre de punts fixos' i -nombre d'excedències' en permutacions que eviten una subseqüència de longitud 3. Un dels resultats principals és que la distribució conjunta d'aquest parell de paràmetres és la mateixa en permutacions que eviten 321 que en permutacions que eviten 132. Això generalitza un teorema recent de Robertson, Saracino i Zeilberger. Demostrem aquest resultat donant una bijecció que preserva els dos estadístics en qüestió i un altre paràmetre. La idea clau consisteix en introduir una nova classe d'estadístics en camins de Dyck, basada en el que anomenem túnel.A continuació considerem el mateix parell d'estadístics en permutacions que eviten simultàniament dues o més subseqüències de longitud 3. Resolem tots els casos donant les funcions generadores corresponents. Alguns casos són generalitzats a subseqüències de longitud arbitrària. També descrivim la distribució d'aquests paràmetres en involucions que eviten qualsevol subconjunt de subseqüències de longitud 3. La tècnica principal consisteix en fer servir bijeccions entre permutacions amb subseqüències prohibides i certs tipus de camins de Dyck, de manera que els estadístics en permutacions que considerem corresponen a estadístics en camins de Dyck que són més fàcils d'enumerar.Tot seguit presentem una nova família de bijeccions del conjunt de camins de Dyck a sí mateix, que envien estadístics que apareixen en l'estudi de permutacions amb subseqüències prohibides a estadístics clàssics en camins de Dyck, la distribució dels quals s'obté fàcilment. En particular, això ens dóna una prova bijectiva senzilla de l'equidistribució de punts fixos en les permutacions que eviten 321 i en les que eviten 132. A continuació donem noves interpretacions dels nombres de Catalan i dels nombres de Fine. Considerem una classe de permutacions definida en termes d'aparellaments de 2n punts en una circumferència sense creuaments. N'estudiem l'estructura i algunes propietats, i donem la distribució de diversos estadístics en aquests permutacions.En la següent part de la tesi introduïm una noció diferent de subseqüències prohibides, amb el requeriment que els elements que formen la subseqüència han d'aparèixer en posicions consecutives a la permutació. Més en general, estudiem la distribució del nombre d'ocurrències de subparaules (subseqüències consecutives) en permutacions. Resolem el problema en diversos casos segons la forma de la subparaula, obtenint-ne les funcions generadores exponencials bivariades corresponents com a solucions de certes equacions diferencials lineals. El mètode està basat en la representació de permutacions com a arbres binaris creixents i en mètodes simbòlics.La part final tracta de subseqüències generalitzades, que extenen tant la noció de subseqüències clàssiques com la de subparaules. Per algunes subseqüències obtenim nous resultats enumeratius. Finalment estudiem el comportament assimptòtic del nombre de permutacions de mida n que eviten una subseqüència generalitzada fixa quan n tendeix a infinit. També donem fites inferiors i superiors en el nombre de permutacions que eviten certes subseqüències.

No description available.

Abstract The aim of this study was to examine and develop statistics education by implementing computer supported collaborative learning (CSCL). This study has been influenced by design-based research, and it focuses on describing the statistical learning of upper secondary school students (N=138) in a CSCL environment, both quantitative and qualitative methods have been utilised. The present study is filling a gaping void in classroom study and disseminates new knowledge with a novel approach in combining CSCL, mathematics education at secondary level and statistical literacy. First, the students’ starting level in statistical literacy was assessed in the pre-test in which students’ perceptions and knowledge of statistics was evaluated. The results showed that students had a severe lack of understanding of basic statistical concepts. Second, CSCL supports students in collaborating asynchronously in different small-groups by using technology. Results suggest that studying in a group fostered their learning and the electronic and interactive material clarified learned topics which was designed by integrating the principles of cognitive theory of multimedia learning. Third, the shift from traditional didactic instruction towards student-centred CSCL learning was challenging for students. According to the results, students had only a few earlier experiences in learning CSCL environments. The quality of the students’ conversational acts varied considerably. It seems that learning how to collaborate productively needs practice. According to the results, the articulation and quality of mathematical discussion increased as students’ acquaintance with their teammates improved. Students’ collaboration in small groups was examined by using video analyses and content analyses. Contact summary sheet -instrument used in analyses facilitated observation of the magnitude and quality in student’s inter-subjective phenomena in collaborative learning. Fourth, students in the treatment group had better learning outcomes than students in the control group. The results suggest a statistically significant difference between treatment and control groups only in the delayed post-test and the effect size indicates a medium effect. The interactive material and CSCL seemed to foster and facilitate the development of statistical literacy. Nevertheless, students were critical of studying in the CSCL environment. / Tiivistelmä Tutkimuksen tarkoituksena oli tutkia ja kehittää tilastojen opetusta hyödyntäen tietokoneavusteista yhteisöllistä oppimista (CSCL). Tutkimus on saanut vaikutteita design-perustaisesta tutkimuksesta ja se keskittyy kuvaamaan lukio-opiskelijoiden (N=138) tilastojen oppimista CSCL-ympäristössä ja tutkimuksessa on hyödynnetty kvantitatiivisia ja kvalitatiivisia menetelmiä. Tämä tutkimus lisää tietämystä luokkahuonetutkimuksesta ja yhdistää CSCL:n ja tilastollisen lukutaidon opetuksen toisella asteella. Ensimmäiseksi oppilaiden tilastollisen lukutaidon lähtötaso mitattiin alkutestissä, missä arvioitiin heidän ennakkokäsityksiä ja tietoa tilastoista. Tutkimustulokset osoittivat, että oppilailla oli suuria vaikeuksia ymmärtää tilastollisia peruskäsitteitä. Toiseksi CSCL-teknologia tukee asynkronisesti pienryhmätyöskentelyä. Tulosten mukaan opiskelu ryhmissä tuki opiskelijoiden oppimista ja sähköinen ja interaktiivinen oppimateriaali selkiytti opeteltavia asioita, joka oli suunniteltu kognitiivisen multimedia oppimisteorian periaatteiden mukaisesti. Kolmanneksi opiskelijat kokivat haasteellisuutta siirryttäessä perinteisestä opettajajohtoisesta opetusmenetelmästä oppijakeskeiseen CSCL-oppimismenetelmään. Tulosten mukaan opiskelijoilla on ollut vain vähän aikaisempia kokemuksia oppimisesta CSCL-ympäristöissä. Opiskelijoiden funktionaalisten roolien laadut vaihtelivat huomattavasti. Näyttää ilmeiseltä, että produktiivisen yhteistoiminnallisuuden oppimiseen tarvitaan harjoittelua. Tutkimustulosten mukaan artikulaatio ja laatu matemaattisissa keskusteluissa lisääntyivät oppimistilanteissa, kun opiskelijoiden ryhmätyöskentelytaidot kehittyivät. Opiskelijoiden pienryhmätyöskentelyä tutkittiin video- ja sisällönanalyysin avulla. Analysoinnissa käytetty contact summary sheet -instrumentti auttoi havainnoimaan opiskelijoiden intersubjektiivista yhteistoiminnallisen oppimisen laatua ja määrää. Neljänneksi opetuskokeiluun osallistuneilla oli parempia oppimistuloksia verrattaessa kontrolliryhmän oppilaisiin. Tutkimustulosten mukaan tilastollisesti merkittävä ero oli havaittavissa opetuskokeilu- ja kontrolliryhmän välillä ainoastaan viivästetyssä lopputestissä ja vaikutuksen suuruus on keskivoimakasta. Interaktiivinen opetusmateriaali ja CSCL näyttäisivät edistävän tilastollisen lukutaidon kehittymistä. Tästä huolimatta, opiskelijat suhtautuivat kriittisesti opiskeluun CSCL-ympäristössä.

Caloric materials show prospect in replacing the function of vaporcompression systems in todays cooling devices, resulting in more energy efficient cooling and eliminating the need for refrigerents which contribute to climate change. This project has focused on magnetocaloric materials, which experience changes in temperature when exposed to magnetic fields. A step to finding viable materials is developing realistic simulations. To this end, this project has investigated if the calculated magnetocaloric effect is impacted by the choice of statistic. Three systems have been studied, bcc Fe, FeRh and Fe2P, using Monte Carlo simulations. The results have shown differences in the calculated entropy change depending on the statistic of choice. The quantum statistics have shown a ∆S = 0 below the phase transition, unlike the classical statistics. At the phase tranisitions quantum statistics resulted in either similar or smaller values for the calculated change in entropy. / Kaloriska material har potential att i framtiden ersätta funktionen hos ångkomprimeringssystem i dagens kylapparater, vilket i sin tur kan leda till mer energieffektiv kylning samt eliminerar behovet av kylmedier som bidrar till klimatförändringen. I detta projekt ligger fokus på magnetokaloriska material, vilka erfar temperaturförändringar då de utsätts för magnetfält. Ett steg mot att hitta gångbara material är att utveckla realistiska simulationer. För detta ändamål undersöktes huruvida den beräknade magnetokaloriska effekten påverkas av valet av statistik. Tre system studerades, bcc Fe, FeRh samt Fe2P, med hjälp av Monte Carlo simulationer. Resultaten visade skillnader i den beräknade entropiförändringen beroende på valet av statistik. För kvantstatistiken var  ∆S = 0 för temperaturer under fasövergångerna, vilket skiljde sig från de klassiska resultaten. Vid fasövergångarna gav kvantstatistiken liknande eller mindre värden för den beräknade entropiförändringen.

M.S.

This dissertation is a study of certain aspects of restricted ranking, a method intended for use by a panel of m judges evaluating the relative merits of N subjects, candidates tor scholarships, awards, etc. Each judge divides the N subjects into R classes, so that n₁ individuals receive a grade i (i = 1, 2, R; Σnᵢ = N) where the R numbers nᵢ are close to N/R (nᵢ = N/R when N is divisible by R) and are preassigned and the same for all judges. When this method is used, all subjects are treated alike, the grading system is the same for all judges and the grades of each judge are given equal weight. Equally important, the meaning of a particular grade is clear to each judge and the same for each judge. Under the null hypothesis that all nR = N subjects are of equal merit, tests of significance are developed to determine whether (1) a particular individual is superior or inferior to the rest of the subjects; (2) two particular subjects are of equal merit; (3) the individuals with the highest and lowest scores are respectively superior and interior to the rest of the subjects and (4) the nR subjects form a homogeneous group. The critical values of the test statistics for (1), (2) and (3) are tabled for small to moderate values of m, an approximation based on the asymptotic normality of the appropriate test statistic proving suitable for large m. The test of homogeneity (4) employs a sum of squares of subjects’ scores which is shown to be asymptotically distributed for m→∞ as chi-square with nR-1 degrees of freedom. For the special case of complete ranking (R=N), this statistic is identical to one proposed by Friedman (1937) form rankings. The behavior of two of these tests is theoretically investigated for the non-null case of nR-1 subjects having equal merit and one "outlying" subject whose merit exceeds the others. The assumption is made that each judge j assigns a grade to every subject i on the basis of a "subjective random variable" xᵢⱼ with mean equal to the "true" merit of subject i and that the distribution of xᵢⱼ is the same for all j. The probability, P(δ), that subject #1 with true mean differing from the others by an amount δ would receive a significantly high score according to the test for outliers is obtained and presented graphically as a function of for xᵢⱼ distributed as (1/2) sech² (x-δ) and also as N ( δ, 1). Using a result due to Hannan (1956), an expression for the asymptotic relative efficiency of the chi-squared homogeneity test for restricted vs. complete ranking for the aforementioned non-null case is obtained and values of this A.R.E. for 2 ≤n≤10 and. 2≤R≤8 are tabled. This A.R.E. is found to be at least 0.9 for all cases where n≤10 and R≥4. A further comparison of the performances of restricted (R) and complete (C) ranking is made by way of some simulation studies performed on a high speed digital computer tor the non- null ease where xᵢⱼ is normally distributed with unit variance and a mean δ₁ having as many as three different possible values. The complete and restricted ranks assigned by the jth judge to the ith subject are assigned on the basis of the value of xᵢⱼ obtained by experimental sampling using a random normal number generator in the computer program. A group of Nₛ subjects with the highest rank sums for (R) and for (C) are then selected in each study. The observed difference in true means between selected and remaining groups is then used as a measure of goodness of the two selection procedures. The results of these studies are presented graphically, displaying a very close agreement between (R) and (C) in all instances. / Ph. D.

I. A Comparison of the Effectiveness of Tournaments. A paired-comparison experiment involving t treatments is analogous to a tournament with t players. A balanced experiment, in which every possible pair occurs once per replication, is the counterpart of a round robin tournament. When the objective is to pick the best treatment, the balanced design may prove to be more expensive than necessary. The knock-out tournament has been suggested as an alternative requiring fewer units of each treatment per replication. In this paper round robin, replicated knock-out, and double elimination tournaments are investigated for their effectiveness in selecting the best one or tour players. Effectiveness is gauged in terms of the two criteria (a) the probability that the best player wins and (b) the expected number of games. For general values of the parameters involved, expressions are derived for the evaluation of the criteria. Comparisons are made on the basis of series of assigned parameter values. Possibilities for the extension of the study are briefly discussed. II. Ties in Paired-Comparison Experiments. In making paired comparisons a judge frequently is unable to express a real preference in a number of the pairs he judges. In spite of this, some or the methods in current use do not permit the judge to declare a tie. In other methods tied observations are either ignored or divided equally or randomly between the tied members. It appears that there is a need, at least in the estimation of response-scale values, for a method which takes tied observations into account. In the Thurstone-Mosteller method the standardized distribution of the difference of two stimulus responses is normal with unit variance and mean equal to the difference or the two mean stimulus responses. In prohibiting ties the assumption is in effect made that all differences, however small, are perceptible to the judge. In this paper the assumption is made that a tie will occur whenever the difference between the judge's responses to the two stimuli lie below a certain threshold, i.e. if the difference lies between -t and t the judge will declare a tie. The parameter t and the mean stimulus responses are estimated by least squares. To overcome a difficulty presented by correlated data, an angular response law is postulated for the response-scale differences. In the resulting transformed data non-homogeneity of variances is encountered. In effecting a weighted solution, weights are first determined by using a preliminary unweighted analysis, and an iterative procedure is proposed. Large-sample variances and covariances of the estimates are obtained. A test of the validity of the model is described. A computational procedure is set up, and exemplified through application to experimental data. / Ph. D.

I. Small-Sample Power of the One-Sample Sign Test for Approximately Normal Distributions. The power function of the one-sided, one-sample sign test is studied for populations which deviate from exact normality, either by skewness, kurtosis, or both. The terms of the Edgeworth asymptotic expansion of order more than N^-3/2 are used to represent the population density. Three sets of hypotheses and alternatives, concerning the location of (1) the median, (2) the median as approximated by the mean and coefficient of skewness, and (3) the mean, are considered in an attempt to make valid comparisons between the power of the sign test and Student's t test under the same conditions. Numerical results are given for samples of size 10, significance level .05, and for several combinations of the coefficients of skewness and kurtosis. II. Power of Two-Sample Rank Teats on the Equality of Two Distribution Functions. A comparative study is made of the power of two-sample rank tests of the hypothesis that both samples are drawn from the same population. The general alternative is that the variables from one population are stochastically larger than the variables from the other. One of the alternatives considered is that the variables in the first sample are distributed as the smallest of k variates with distribution F, and the variables in the second sample are distributed as the largest of these k – H₁ : H = 1 - (1 - F)^k, G = F^k. These two alternative distributions are mutually symmetric if F is symmetrical. Formulae are presented, which are independent of F, for the evaluation of the probability under H₁ of any joint arrangement of the variables from the two samples. A theorem is proved concerning the equality of the probabilities of certain pairs of orderings under assumptions of mutually symmetric populations. The other alternative is that both samples are normally distributed with the same variance but different means, the standardized difference between the two extreme distributions in the first alternative corresponding to the difference between the means. Numerical results of power are tabulated for small sample sizes, k = 2, 3 and 4, significance levels .01, .05 and .10. The rank tests considered are the most powerful rank test, the one and two-sided Wilcoxon tests, Terry's c₁ test, the one and two-aided median tests, the Wald-Wolfowitz runs test, and two new tests called the Psi test and the Gamma test. The two-sample rank test which is locally most powerful against any alternative·expressing an arbitrary functional relationship between the two population distribution functions and an unspecified parameter θ is derived and its asymptotic properties studied. The method is applied to two specific functional alternatives, H₁* : H = (1-θ)F^k + θ[1 - (1-F)^k], G = F^k, and H₁**: H = 1 - (1-F)^1+θ, G = F^1+θ, where θ ≥ 0, which are similar to the alternative of two extreme distributions. The resulting test statistics are the Gamma test and the Psi test, respectively. The latter test is shown to have desirable small-sample properties. The asymptotic power functions of the Wilcoxon and WaldWolfowitz tests are compared for the alternative of two extreme distributions with k = 2, equal sample sizes and significance level .05. / Ph. D.

Ph. D.

A theorem by Shah and Khatri [A.M.S. (1961) 321883-887] is extended to give the distribution of Q/X² , where Q is a positive definite quadratic form involving non-central normal variates and X² is an independently distributed chi-square variate. Conditions are given under which the distribution of this ratio reduces to that of non-central F. Kramer’s method for analysis of variance of a two-way classification with disproportionate subclass numbers is reviewed and shown to satisfy these conditions. Various functional forms of the non-centrality parameter for evaluating the power of his method are given. Additional algebraic and numerical results are obtained to compare the power of Kramer's method and the method of fitting constants (least squares) outlined by Yates [J.A.S.A. (1934) 24151-66]. A mnemonic rule, based on 310 randomly generated two-way classifications, is given for discriminating against use of Kramer’s method in situations where his method potentially may be very deficient in power as compared to the method of fitting constants. / Ph. D.