WAIS seminar:Mathematics for Web Science Part 3-2

ABSTRACT In the first two seminars we looked at the evolution of Ontologies from the current OWL level towards more powerful/expressive models and the corresponding hierarchy of Logics that underpin every stage of this evolution. We examined this in the more general context of the general evolution of the Web as a mathematical (directed and weighed) graph and the archetypical “living network” In the third seminar we will analyze further some of the startling properties that the Web has as a graph/network and which it shares with an array of “real-life” networks as well as some key elements of the mathematics (probability, statistics and graph theory) that underpin all this. No mathematical prerequisites are assumed or required. We will outline some directions that current (2005-now) research is taking and conclude with some illustrations/examples from ongoing research and applications that show great promise.

WAIS seminar: Mathematics for Web Science: Part 3-3

ABSTRACT In the first two seminars we looked at the evolution of Ontologies from the current OWL level towards more powerful/expressive models and the corresponding hierarchy of Logics that underpin every stage of this evolution. We examined this in the more general context of the general evolution of the Web as a mathematical (directed and weighed) graph and the archetypical “living network” In the third seminar we will analyze further some of the startling properties that the Web has as a graph/network and which it shares with an array of “real-life” networks as well as some key elements of the mathematics (probability, statistics and graph theory) that underpin all this. No mathematical prerequisites are assumed or required. We will outline some directions that current (2005-now) research is taking and conclude with some illustrations/examples from ongoing research and applications that show great promise.

A new all-round density functional based on spin states and SN2 barriers

We report here a new empirical density functional that is constructed based on the performance of OPBE and PBE for spin states and SN 2 reaction barriers and how these are affected by different regions of the reduced gradient expansion. In a previous study [Swart, Sol̀, and Bickelhaupt, J. Comput. Methods Sci. Eng. 9, 69 (2009)] we already reported how, by switching between OPBE and PBE, one could obtain both the good performance of OPBE for spin states and reaction barriers and that of PBE for weak interactions within one and the same (SSB-sw) functional. Here we fine tuned this functional and include a portion of the KT functional and Grimme's dispersion correction to account for π- π stacking. Our new SSB-D functional is found to be a clear improvement and functions very well for biological applications (hydrogen bonding, π -π stacking, spin-state splittings, accuracy of geometries, reaction barriers)

Inclusive approaches to teaching and learning mathematics

This chapter explores the role of mentors in supporting pre-service teachers to include all children in mathematics teaching, no matter what their individual needs.

Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

An all-substrings common subsequence algorithm

Given two strings A and B of lengths n(a) and n(b), n(a) <= n(b), respectively, the all-substrings longest common subsequence (ALCS) problem obtains, for every substring B` of B, the length of the longest string that is a subsequence of both A and B. The ALCS problem has many applications, such as finding approximate tandem repeats in strings, solving the circular alignment of two strings and finding the alignment of one string with several others that have a common substring. We present an algorithm to prepare the basic data structure for ALCS queries that takes O(n(a)n(b)) time and O(n(a) + n(b)) space. After this preparation, it is possible to build that allows any LCS length to be retrieved in constant time. Some trade-offs between the space required and a matrix of size O(n(b)(2)) the querying time are discussed. To our knowledge, this is the first algorithm in the literature for the ALCS problem. (C) 2007 Elsevier B.V. All rights reserved.

Solvatochromic Shift of the pi -> pi* Transition in All-Trans, Cis-13, Cis-11, Cis-9, and Cis-7 Retinal Isomers Induced by Water and Methanol

The solvatochromic shift of the lowest singlet it pi -> pi* electronic transition in the all-trans, cis-13, cis-11, cis-9, and cis-7 retinal isomers were computed under the influence of water, methanol, and benzene solvents. Excitation energies were calculated in gas phase and in solution. The calculations in solution were performed considering the sequential Monte Carlo (MC) /Quantum Mechanical approach. The MC simulations were performed considering the full retinal isomer molecules and 900 water molecules, 900 methanol, or 400 benzene ones. The OPLS/AA parametrization was chosen for retinal, methanol, and benzene molecules and the SPC model was used for water one. From the MC calculations 100 independent configurations were selected, with 100 solvent molecules in thermodynamical equilibrium at T = 298.15 K. Average point-charges were obtained from those independent configurations for water, methanol, and benzene solvent. TDDFT and CASSCF//CASPT2 methodologies were used to compute the vertical excitation energy of the retinal isomers in different environment. (C) 2010 Wiley Periodicals, Inc. Int J Quantum Chem 110: 2076-2087, 2010

Children’s conceptions about mathematics and mathematics education

This paper deals with younger students’ (grade 2 and 5) conceptions about mathematics and mathematics education. The questionnaire consisted of three parts: (1) statements with a Likert-scale; (2) open-end questions where the students could explain further their conceptions; and, (3) a request to draw a picture of yourself doing mathematics. The results from the statements were summarised and the pictures were analysed. Most students in grade 2 had a positive attitude towards mathematics whereas a larger proportion in grade 5 gave negative answers. All students presented mathematics as an individual activity with a focus on the textbook. The elder students narrow the activity down to calculating. A post-questionnaire confirmed the results.

Mathematics teaching life histories in the study of teachers' perceptions of change

The aim of this paper is to discuss teachers' perceptions of change in their thought and/or practice over time and their perceptions of what kind of experiences or challenges might have influenced those changes. Two mathematics teaching life histories of Brazilian teachers are examined, considering a context of curriculum development in the state of São Paulo, Brazil. Reflection on teachers' thought and practice and interest in their own development, including interest in their own learning of mathematics, seemed to be the most important internal aspects influencing change and development. Close support seemed to be the most important external aspect. The retrospective analysis put a good face on personal change and development. (C) 2000 Elsevier B.V. Ltd. All rights reserved.

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx's concept of commodity and Jean Baudrillard's concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today's society and ideology. After, we engage in showing mathematics education's historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida's famous saying that there is nothing beyond the text. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy. © FIZ Karlsruhe 2009.

Bad Medicine: Homework or Headache? Responsibility and Accountability for Middle Level Mathematics Students

In this action research study of my 5th grade mathematics class, I investigated the issue of homework and its relationship with students and parents. I made some interesting observations and discovered that the majority of students and parents felt that the math homework that was given was fairly easy, yet issues of incomplete assignments and failing homework quizzes were notorious for some individuals. Comments were also made to make homework even easier and have shortened assignments despite the already indicated ease of the work. As a result of this research, I plan to look more closely at the history and development of homework, as well as the psychological implications and “hereditary” issues involving homework, which I believe are passed from one generation of learners to the next. My intent is to continue to study this phenomenon in future school years, trying to develop methods of instilling successful, intrinsic motivational skills to aid students in their homework endeavors. Finally, I will take a close inventory of my own beliefs and understandings toward homework: What is the purpose of having students do work away from the classroom, and how can homework serve as a proactive service for all who are involved?

Meaningful Independent Practice in Mathematics

In this action research of my seventh grade mathematics classroom, I investigated how students’ explanations of math homework would improve their learning in math. I discovered these explanations can be very beneficial in helping students to improve their understanding of current skills although it did not affect all students. As a result of this study, I plan to incorporate these student explanations in my instruction next year but not as a daily expectation.

The Importance of Vocabulary Instruction in Everyday Mathematics

In this action research study of my 6th grade math students I try to answer the question of how mathematical vocabulary plays an integral role in the understanding and learning of middle level mathematics. It is my belief that mathematics is a language, and to be fluent in that language one must be able to use and understand vocabulary. With the use of vocabulary quizzes and mathematically-centered vocabulary activities, student scores and understanding of math concepts can be increased. I discovered that many of the students had never been exposed to consistent mathematical terminology in their elementary education, which led many to an unfavorable impression of math. As a result of my research, I plan to incorporate vocabulary as a regular part of my mathematical teaching. As the students understood the language of math, their confidence, attitudes, and scores all began to improve.

Locally finite groups with all subgroups either subnormal or nilpotent-by-Chernikov

Let G be a locally finite group satisfying the condition given in the title and suppose that G is not nilpotent-by-Chernikov. It is shown that G has a section S that is not nilpotent-by-Chernikov, where S is either a p-group or a semi-direct product of the additive group A of a locally finite field F by a subgroup K of the multiplicative group of F, where K acts by multiplication on A and generates F as a ring. Non-(nilpotent-by-Chernikov) extensions of this latter kind exist and are described in detail.