Spontaneous Focusing on Quantitative Relations and the Development of Rational Number Conceptual Knowledge

The aim of the present set of studies was to explore primary school children’s Spontaneous Focusing On quantitative Relations (SFOR) and its role in the development of rational number conceptual knowledge. The specific goals were to determine if it was possible to identify a spontaneous quantitative focusing tendency that indexes children’s tendency to recognize and utilize quantitative relations in non-explicitly mathematical situations and to determine if this tendency has an impact on the development of rational number conceptual knowledge in late primary school. To this end, we report on six original empirical studies that measure SFOR in children ages five to thirteen years and the development of rational number conceptual knowledge in ten- to thirteen-year-olds. SFOR measures were developed to determine if there are substantial differences in SFOR that are not explained by the ability to use quantitative relations. A measure of children’s conceptual knowledge of the magnitude representations of rational numbers and the density of rational numbers is utilized to capture the process of conceptual change with rational numbers in late primary school students. Finally, SFOR tendency was examined in relation to the development of rational number conceptual knowledge in these students. Study I concerned the first attempts to measure individual differences in children’s spontaneous recognition and use of quantitative relations in 86 Finnish children from the ages of five to seven years. Results revealed that there were substantial inter-individual differences in the spontaneous recognition and use of quantitative relations in these tasks. This was particularly true for the oldest group of participants, who were in grade one (roughly seven years old). However, the study did not control for ability to solve the tasks using quantitative relations, so it was not clear if these differences were due to ability or SFOR. Study II more deeply investigated the nature of the two tasks reported in Study I, through the use of a stimulated-recall procedure examining children’s verbalizations of how they interpreted the tasks. Results reveal that participants were able to verbalize reasoning about their quantitative relational responses, but not their responses based on exact number. Furthermore, participants’ non-mathematical responses revealed a variety of other aspects, beyond quantitative relations and exact number, which participants focused on in completing the tasks. These results suggest that exact number may be more easily perceived than quantitative relations. As well, these tasks were revealed to contain both mathematical and non-mathematical aspects which were interpreted by the participants as relevant. Study III investigated individual differences in SFOR 84 children, ages five to nine, from the US and is the first to report on the connection between SFOR and other mathematical abilities. The cross-sectional data revealed that there were individual differences in SFOR. Importantly, these differences were not entirely explained by the ability to solve the tasks using quantitative relations, suggesting that SFOR is partially independent from the ability to use quantitative relations. In other words, the lack of use of quantitative relations on the SFOR tasks was not solely due to participants being unable to solve the tasks using quantitative relations, but due to a lack of the spontaneous attention to the quantitative relations in the tasks. Furthermore, SFOR tendency was found to be related to arithmetic fluency among these participants. This is the first evidence to suggest that SFOR may be a partially distinct aspect of children’s existing mathematical competences. Study IV presented a follow-up study of the first graders who participated in Studies I and II, examining SFOR tendency as a predictor of their conceptual knowledge of fraction magnitudes in fourth grade. Results revealed that first graders’ SFOR tendency was a unique predictor of fraction conceptual knowledge in fourth grade, even after controlling for general mathematical skills. These results are the first to suggest that SFOR tendency may play a role in the development of rational number conceptual knowledge. Study V presents a longitudinal study of the development of 263 Finnish students’ rational number conceptual knowledge over a one year period. During this time participants completed a measure of conceptual knowledge of the magnitude representations and the density of rational numbers at three time points. First, a Latent Profile Analysis indicated that a four-class model, differentiating between those participants with high magnitude comparison and density knowledge, was the most appropriate. A Latent Transition Analysis reveal that few students display sustained conceptual change with density concepts, though conceptual change with magnitude representations is present in this group. Overall, this study indicated that there were severe deficiencies in conceptual knowledge of rational numbers, especially concepts of density. The longitudinal Study VI presented a synthesis of the previous studies in order to specifically detail the role of SFOR tendency in the development of rational number conceptual knowledge. Thus, the same participants from Study V completed a measure of SFOR, along with the rational number test, including a fourth time point. Results reveal that SFOR tendency was a predictor of rational number conceptual knowledge after two school years, even after taking into consideration prior rational number knowledge (through the use of residualized SFOR scores), arithmetic fluency, and non-verbal intelligence. Furthermore, those participants with higher-than-expected SFOR scores improved significantly more on magnitude representation and density concepts over the four time points. These results indicate that SFOR tendency is a strong predictor of rational number conceptual development in late primary school children. The results of the six studies reveal that within children’s existing mathematical competences there can be identified a spontaneous quantitative focusing tendency named spontaneous focusing on quantitative relations. Furthermore, this tendency is found to play a role in the development of rational number conceptual knowledge in primary school children. Results suggest that conceptual change with the magnitude representations and density of rational numbers is rare among this group of students. However, those children who are more likely to notice and use quantitative relations in situations that are not explicitly mathematical seem to have an advantage in the development of rational number conceptual knowledge. It may be that these students gain quantitative more and qualitatively better self-initiated deliberate practice with quantitative relations in everyday situations due to an increased SFOR tendency. This suggests that it may be important to promote this type of mathematical activity in teaching rational numbers. Furthermore, these results suggest that there may be a series of spontaneous quantitative focusing tendencies that have an impact on mathematical development throughout the learning trajectory.

On Commutation and Conjugacy of Rational Languages and the Fixed Point Method

The research on language equations has been active during last decades. Compared to the equations on words the equations on languages are much more difficult to solve. Even very simple equations that are easy to solve for words can be very hard for languages. In this thesis we study two of such equations, namely commutation and conjugacy equations. We study these equations on some limited special cases and compare some of these results to the solutions of corresponding equations on words. For both equations we study the maximal solutions, the centralizer and the conjugator. We present a fixed point method that we can use to search these maximal solutions and analyze the reasons why this method is not successful for all languages. We give also several examples to illustrate the behaviour of this method.

Characterizing Propionibacterium freudenreichii ssp. shermanii JS and Lactobacillus rhamnosus LC705 as a new probiotic combination: basic properties of JS and pilot in vivo assessment of the combination

Characterizing Propionibacterium freudenreichii ssp. shermanii JS and Lactobacillus rhamnosus LC705 as a new probiotic combination: basic properties of JS and pilot in vivo assessment of the combination Each candidate probiotic strain has to have the documentation for the proper identification with current molecular tools, for the biological properties, for the safety aspects and for the health benefits in human trials if the intention is to apply the strain as health promoting culture in the commercial applications. No generalization based on species properties of an existing probiotic are valid for any novel strain, as strain specific differences appear e.g. in the resistance to GI tract conditions and in health promoting benefits (Madsen, 2006). The strain evaluation based on individual strain specific probiotic characteristics is therefore the first key action for the selection of the new probiotic candidate. The ultimate goal in the selection of the probiotic strain is to provide adequate amounts of active, living cells for the application and to guarantee that the cells are physiologically strong enough to survive and be biologically active in the adverse environmental conditions in the product and in GI tract of the host. The in vivo intervention studies are expensive and time consuming; therefore it is not rational to test all the possible candidates in vivo. Thus, the proper in vitro studies are helping to eliminate strains which are unlikely to perform well in vivo. The aims of this study were to characterize the strains of Propionibacterium freudenreichii ssp. shermanii JS and Lactobacillus rhamnosus LC705, both used for decades as cheese starter cultures, for their technological and possible probiotic functionality applied in a combined culture. The in vitro studies of Propionibacterium freudenreichii ssp. shermanii JS focused on the monitoring of the viability rates during the acid and bile treatments and on the safety aspects such as antibiotic susceptibility and adhesion. The studies with the combination of the strains JS and LC705 administered in fruit juices monitored the survival of the strains JS and LC705 during the GI transit and their effect on gut wellbeing properties measured as relief of constipation. In addition, safety parameters such as side effects and some peripheral immune parameters were assessed. Separately, the combination of P. freudenreichii ssp. shermanii JS and Lactobacillus rhamnosus LC705 was evaluated from the technological point of view as a bioprotective culture in fermented foods and wheat bread applications. In this study, the role ofP. freudenreichii ssp. shermanii JS as a candidate probiotic culture alone and in a combination with L. rhamnosus LC705 was demonstrated. Both strains were transiently recovered in high numbers in fecal samples of healthy adults during the consumption period. The good survival through the GI transit was proven for both strains with a recovery rate from 70 to 80% for the JS strain and from 40 to 60% for the LC705 strain from the daily dose of 10 log10 CFU. The good survival was shown from the consumption of fruit juices which do not provide similar matrix protection for the cells as milk based products. The strain JS did not pose

Solving Leontief Input-Output Model with Trapezoidal Fuzzy Numbers and Gauss-Seidel Algorithm

In this work a fuzzy linear system is used to solve Leontief input-output model with fuzzy entries. For solving this model, we assume that the consumption matrix from di erent sectors of the economy and demand are known. These assumptions heavily depend on the information obtained from the industries. Hence uncertainties are involved in this information. The aim of this work is to model these uncertainties and to address them by fuzzy entries such as fuzzy numbers and LR-type fuzzy numbers (triangular and trapezoidal). Fuzzy linear system has been developed using fuzzy data and it is solved using Gauss-Seidel algorithm. Numerical examples show the e ciency of this algorithm. The famous example from Prof. Leontief, where he solved the production levels for U.S. economy in 1958, is also further analyzed.

A quantitative view on fuzzy numbers

Since its introduction, fuzzy set theory has become a useful tool in the mathematical modelling of problems in Operations Research and many other fields. The number of applications is growing continuously. In this thesis we investigate a special type of fuzzy set, namely fuzzy numbers. Fuzzy numbers (which will be considered in the thesis as possibility distributions) have been widely used in quantitative analysis in recent decades. In this work two measures of interactivity are defined for fuzzy numbers, the possibilistic correlation and correlation ratio. We focus on both the theoretical and practical applications of these new indices. The approach is based on the level-sets of the fuzzy numbers and on the concept of the joint distribution of marginal possibility distributions. The measures possess similar properties to the corresponding probabilistic correlation and correlation ratio. The connections to real life decision making problems are emphasized focusing on the financial applications. We extend the definitions of possibilistic mean value, variance, covariance and correlation to quasi fuzzy numbers and prove necessary and sufficient conditions for the finiteness of possibilistic mean value and variance. The connection between the concepts of probabilistic and possibilistic correlation is investigated using an exponential distribution. The use of fuzzy numbers in practical applications is demonstrated by the Fuzzy Pay-Off method. This model for real option valuation is based on findings from earlier real option valuation models. We illustrate the use of number of different types of fuzzy numbers and mean value concepts with the method and provide a real life application.

Numbers and Languages

The thesis presents results obtained during the authors PhD-studies. First systems of language equations of a simple form consisting of just two equations are proved to be computationally universal. These are systems over unary alphabet, that are seen as systems of equations over natural numbers. The systems contain only an equation X+A=B and an equation X+X+C=X+X+D, where A, B, C and D are eventually periodic constants. It is proved that for every recursive set S there exists natural numbers p and d, and eventually periodic sets A, B, C and D such that a number n is in S if and only if np+d is in the unique solution of the abovementioned system of two equations, so all recursive sets can be represented in an encoded form. It is also proved that all recursive sets cannot be represented as they are, so the encoding is really needed. Furthermore, it is proved that the family of languages generated by Boolean grammars is closed under injective gsm-mappings and inverse gsm-mappings. The arguments apply also for the families of unambiguous Boolean languages, conjunctive languages and unambiguous languages. Finally, characterizations for morphisims preserving subfamilies of context-free languages are presented. It is shown that the families of deterministic and LL context-free languages are closed under codes if and only if they are of bounded deciphering delay. These families are also closed under non-codes, if they map every letter into a submonoid generated by a single word. The family of unambiguous context-free languages is closed under all codes and under the same non-codes as the families of deterministic and LL context-free languages.

Composite artistry in the Book of Numbers : a study in biblical narrative conventions

The present thesis discusses the coherence or lack of coherence in the book of Numbers, with special regard to its narrative features. The fragmented nature of Numbers is a well-known problem in research on the book, affecting how we approach and interpret it, but to date there has not been any thorough investigation of the narrative features of the work and how they might contribute to the coherence or the lack of coherence in the book. The discussion is pursued in light of narrative theory, and especially in connection to three parameters that are typically understood to be invoked in the interpretation of narratives: 1) a narrative paradigm, or ‘story,’ meaning events related to each other temporally, causally, and thematically, in a plot with a beginning, middle, and end; 2) discourse, being the expression plane of a narrative, or the devices that an author has at hand in constructing a narrative; 3) the situation or languagegame of the narrative, prototypical examples being factual reports, which seeks to depict a state of affairs, and storytelling narratives, driven by a demand for tellability. In view of these parameters the present thesis argues that it is reasonable to form four groups to describe the narrative material of Numbers: genuine narratives (e.g. Num 12), independent narrative sequences (e.g. Num 5:1-4), instrumental scenes and situations (e.g. Num 27:1-5), and narrative fragments (e.g. Num 18:1). These groups are mixed throughout with non-narrative materials. Seen together, however, the narrative features of these groups can be understood to create an attenuated narrative sequence from beginning to end in Numbers, where one thing happens after another. This sequence, termed the ‘larger story’ of Numbers, concerns the wandering of Israel from Sinai to Moab. Furthermore, the larger story has a fragmented plot. The end-point is fixed on the promised land, Israel prepares for the wandering towards it (Num 1-10), rebels against wandering and the promise and is sent back into the wilderness (Num 13-14), returns again after forty years (Num 21ff.), and prepares for conquering the land (Num 22-36). Finally, themes of the promised land, generational succession, and obedience-disobedience, operate in this larger story. Purity is also a significant theme in the book, albeit not connected to plot in the larger story. All in all, sequence, plot, and theme in the larger story of Numbers can be understood to bring some coherence to the book. However, neither aspect entirely subsumes the whole book, and the four groups of narrative materials can also be understood to underscore the incoherence of the work in differentiating its variegated narrative contents. Numbers should therefore be described as an anthology of different materials that are loosely connected through its narrative features in the larger story, with the aim of informing Israelite identity by depicting a certain period in the early history of the people.

Probing MST1 Kinase Sequence and Structure for Rational Drug Discovery (Targeting diabetes by blocking protein-protein interactions)

Apoptotic beta cell death is an underlying cause majorly for type I and to a lesser extent for type II diabetes. Recently, MST1 kinase was identified as a key apoptotic agent in diabetic condition. In this study, I have examined MST1 and closely related kinases namely, MST2, MST3 and MST4, aiming to tackle diabetes by exploring ways to selectively block MST1 kinase activity. The first investigation was directed towards evaluating possibilities of selectively blocking the ATP binding site of MST1 kinase that is essential for the activity of the enzymes. Structure and sequence analyses of this site however revealed a near absolute conservation between the MSTs and very few changes with other kinases. The observed residue variations also displayed similar physicochemical properties making it hard for selective inhibition of the enzyme. Second, possibilities for allosteric inhibition of the enzyme were evaluated. Analysis of the recognized allosteric site also posed the same problem as the MSTs shared almost all of the same residues. The third analysis was made on the SARAH domain, which is required for the dimerization and activation of MST1 and MST2 kinases. MST3 and MST4 lack this domain, hence selectivity against these two kinases can be achieved. Other proteins with SARAH domains such as the RASSF proteins were also examined. Their interaction with the MST1 SARAH domain were evaluated to mimic their binding pattern and design a peptide inhibitor that interferes with MST1 SARAH dimerization. In molecular simulations the RASSF5 SARAH domain was shown to strongly interact with the MST1 SARAH domain and possibly preventing MST1 SARAH dimerization. Based on this, the peptidic inhibitor was suggested to be based on the sequence of RASSF5 SARAH domain. Since the MST2 kinase also interacts with RASSF5 SARAH domain, absolute selectivity might not be achieved.