Mathematics Commons^™

Unknotting Tunnels Of Hyperbolic Tunnel Number N Manifolds, Stephan Daniel Burton

Theses and Dissertations

Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the …

Decision Making Under Interval And Fuzzy Uncertainty: Towards An Operational Approach, Rafik Aliev, Oleg H. Huseynov, Vladik Kreinovich

Departmental Technical Reports (CS)

Traditional decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is either completely unable to select one of these alternatives, or selects one of the alternatives only "to some extent". How can we extend the traditional decision theory to such realistic interval and fuzzy cases? In their previous papers, the first two authors proposed a natural generalization of the usual decision theory axioms to interval and fuzzy cases, and described decision coming from this generalization. …

Membership Functions Or Alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies An Interval Approach, Vladik Kreinovich

Departmental Technical Reports (CS)

In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number -- by a membership function and by alpha-cuts -- are not algorithmically equivalent, and only the alpha-cut representation enables us to efficiently process fuzzy data.

Inquiry- Without Posing Questions?, Helle Alrø, Marit Johnsen-Høines

The Mathematics Enthusiast

This article discusses what inquiry conversations could mean when learning mathematics.³ Referring to Gadamar’s distinction of true and apparent questions it is discussed what it takes to be inquiring and if this attitude necessarily includes posing questions. Which qualities are expressed in inquiring questions, and what other ways of communicating may have an inquiring function in learning conversations? The intention is to develop and frame the concept of ’inquiry’ in learning conversations, and this is the focus of analysis of an authentic classroom situation, where teacher and pupils are exploring the concept of ’volume’. Further, this analysis informs a …

Investigating Engineers’ Needs As A Part Of Designing A Professional Development Program For Engineers Who Are To Become Mathematics Teachers, Ragnhild Johanne Rensaa

The Mathematics Enthusiast

The average age for teachers in mathematics and physics in Norway is about 60 (Sørensen, 2003) and few students taking a masters degree in a science subject choose a career as a teacher (KD, 2006). This implies that there may be a critical lack of science teachers in schools in the near future (Næss, 2002). At the same time, rather many graduated engineers – who have worked as engineers sometimes several years – want to contribute to the educational setting by taking part time jobs as teachers; lector-II (Vassnes, 2008). To combine this supply and demand for qualified teachers in …

The Role Of Theory When Studying Epistemological Characterizations Of Mathematics Lecture(R)S, Magnus Österholm

The Mathematics Enthusiast

The study presented in this paper is a contribution to the scientific discussion about the role and use of theory in mathematics education research. In particular, focus is here on the use of and comparison between different types of theories and frameworks, which is discussed primarily through the example of an empirical study examining what types of messages about mathematics are conveyed in lectures. The main purpose of this paper is to examine how different types of theories and frameworks can affect different parts of the research process.

Numerical Analysis And Design Optimization Of Organic Bulk Hetero-Junction Solar Cells Using The Mixed Finite Element And Arc-Length Methods, James L. Dean

Mechanical & Aerospace Engineering Theses & Dissertations

Due to the high coupling and non-linear nature of the semiconductor equations, numerical stability becomes a major problem when using a purely Newtonian numerical method in solving these equations. Presented is a robust mixed finite element formulation with application for organic solar cell devices that incorporates the arc-length predictor-corrector method to bypass such stability issues. The simulated device represents a multilayered solar cell using organic bulk hetero-junction materials including donor materials Poly[2,1,3-benzothiadiazole-4,7-diyl[4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b:3,4- b']dithiophene-2,6-diyl]] (PCPDTBT) and poly 3-hexylthiophene (P3HT) with acceptor fullerene materials [6,6]-phenyl C61 butyric acid methyl ester (PC61BM) or [6,6]-phenyl C71 butyric acid methyl ester (PC71BM) as the photoactive …

University Mathematics Teachers' Views On The Required Reasoning In Calculus Exams, Ewa Bergqvist

The Mathematics Enthusiast

Students often use imitative reasoning, i.e. copy algorithms or recall facts, when solving mathematical tasks. Research shows that this type of imitative reasoning might weaken the students' understanding of the underlying mathematical concepts. In a previous study, the author classified tasks from 16 final exams from introductory calculus courses at Swedish universities. The results showed that it was possible to pass 15 of the exams, and solve most of the tasks, using imitative reasoning. This study examines the teachers' views on the reasoning that students are expected to perform during their own and others mathematics exams. The results indicate that …

Tme Volume 9, Number 3

The Mathematics Enthusiast

No abstract provided.

Generalized Diamond-Alpha Dynamic Opial Inequalities, Nuriye Atasever, Billûr Kaymakçalan, Goran Lesaja, Kenan Taş

Mathematical Sciences: Faculty Publications

We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result.

2012 (Summer), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2012 Summer Colloquium.

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might …

Extended Pcr Rules For Dynamic Frames, Florentin Smarandache, Jean Dezert

Branch Mathematics and Statistics Faculty and Staff Publications

In most of classical fusion problems modeled from belief functions, the frame of discernment is considered as static. This means that the set of elements in the frame and the underlying integrity constraints of the frame are fixed forever and they do not change with time. In some applications, like in target tracking for example, the use of such invariant frame is not very appropriate because it can truly change with time. So it is necessary to adapt the Proportional Conflict Redistribution fusion rules (PCR5 and PCR6) for working with dynamical frames. In this paper, we propose an extension of …

Bayesian Analysis Of Hypothesis Testing Problems For General Population: A Kullback–Leibler Alternative, Naveen K. Bansal, Gholamhossein Hamedani, Ru Sheng

Mathematics, Statistics and Computer Science Faculty Research and Publications

We consider a hypothesis problem with directional alternatives. We approach the problem from a Bayesian decision theoretic point of view and consider a situation when one side of the alternatives is more important or more probable than the other. We develop a general Bayesian framework by specifying a mixture prior structure and a loss function related to the Kullback–Leibler divergence. This Bayesian decision method is applied to Normal and Poisson populations. Simulations are performed to compare the performance of the proposed method with that of a method based on a classical z-test and a Bayesian method based on the …

General Allee Effect In Two-Species Population Biology, G Livadiotis, Saber Elaydi

Mathematics Faculty Research

The main objective of this work is to present a general framework for the notion of the strong Allee effect in population models, including competition, mutualistic, and predator–prey models. The study is restricted to the strong Allee effect caused by an inter-specific interaction. The main feature of the strong Allee effect is that the extinction equilibrium is an attractor. We show how a ‘phase space core’ of three or four equilibria is sufficient to describe the essential dynamics of the interaction between two species that are prone to the Allee effect. We will introduce the notion of semistability in planar …

Some Unified Results On Comparing Linear Combinations Of Independent Gamma Random Variables, Subhash C. Kochar, Maochao Xu

Mathematics and Statistics Faculty Publications and Presentations

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].

Beyond Post-Modernity In Mathematics Education, Ole Skovsmose

The Mathematics Enthusiast

A radical form of postmodernity is presented with reference to Nietzsche’s ideas with respect to truth, knowledge, sciences, progress, democracy, and ethical values in general. Thereafter is presented Foucault’s archaeology of knowledge. This brings us forward to the notion of genealogy, which is a defining idea for the postmodern conception of critique. However, it is emphasised that a critique can address the generativity of mathematical rationality by considering mathematics-based fabrications. Finally, imagination is presented as yet another feature of a critical enterprise. It is illustrated how such a three dimensional critical enterprise is relevant for both mathematics and mathematics education. …

Loss Of Dimension In The History Of Calculus And In Student Reasoning, Robert Ely

The Mathematics Enthusiast

Research indicates that calculus students often imagine objects losing a dimension entirely when a limit is taken, and that this image serves as an obstacle to their understanding of the fundamental theorem of calculus. Similar imagery, in the form of “indivisibles”, was similarly unsupportive of the development of the fundamental theorem in the mid-1600s, unlike the more powerful subsequent imagery of infinitesimals. This parallel between student issues and historical issues suggests several implications for how to provide students with imagery that is more productive for understanding the fundamental theorem, such as the imagery of infinitesimals or the more modern quantitative …

Demographics Of Self-Selected Participants In Mathematical Tracks In Swedish Upper Secondary School, Linda Mattsson

The Mathematics Enthusiast

This study investigates the three demographic factors gender, geographical origin and parents educational level among self-selected students in mathematical tracks in Swedish Upper Secondary School (n=147). Comparisons are made, through statistical tests of significance, with corresponding student distributions on the closely related Natural Science program (n = 1528). Data is obtained from Statistics Sweden. Results confirm that females are under-represented in mathematical tracks (p<0,01) and that parents of students in mathematical tracks have higher educational level than parents of students in the comparison group (p<0,05). Results do not show that students with foreign background are underrepresented in mathematical tracks, but raise questions about under-representation of Swedish students. Possible implications of these findings for these and similar programs are discussed.

Foundations For Success In Mathematical Competitions: A Study Of Best Praxis In Lower Secondary Schools In Norway, Steinar Thorvaldsen, Lars Vavik

The Mathematics Enthusiast

The purpose of this study is to investigate the following questions: What factors leads to success in mathematics, and how can these success factors and qualities be described? Will the teacher’s education and pedagogical praxis have an impact on good learning results? We report results from a case-control association study on among high achievement classes in mathematics in Norway. The data were collected from matched pairs of schools, paired on the basis of location and socioeconomic status. The questionnaire was first distributed to teachers in 38 Norwegian secondary schools at grade 9, which have had repeated success in the annual …