Single scale factor for the universe from the creation of radiation and matter till the present

A scheme for incorporating the creation of radiation and matter into the cosmological evolution is introduced so that it becomes possible to merge the times before and after the creation of radiation and matter in a single scale factor in Robertson-Walker metric. This scheme is illustrated through a toy model that has the prospect of constituting a basis for a realistic model.Comment: Minor typos are corrected, an acknowledgment is added, to be published in The European Physical Journal

A symmetry for vanishing cosmological constant in an extra dimensional toy model

We introduce a symmetry principle that forbids a bulk cosmological constant in six and ten dimensions. Then the symmetry is extended so that it insures absence of 4-dimensional cosmological constant induced by the six dimensional curvature scalar, at least, for a class of metrics. A small cosmological constant may be induced by breaking of the symmetry by a small amount.Comment: Typos are corrected. Two paragraphs and two references are added. To appear in PL

Is it possible to obtain cosmic accelerated expansion through energy transfer between different energy densities?

The equation of state of an energy density may be significantly modified by coupling it to another energy density. In the light of this observation we check the possibility of producing cosmic accelerated expansion in this way. In particular we consider the case where matter is converted to radiation (or vice versa by particle physics processes). We find that cosmic accelerated expansion can be obtained in this way only if an intermediate state with negative equation of state forms during the conversion.Comment: To be published in Physics of the Dark Univers

Fermion families, and chirality through extra dimensions

We give a simple model to explain the origin of fermion families, and chirality through the use of a domain wall placed in a five dimensional space-time.Comment: 12 page

Finite number of Kaluza-Klein modes, all with zero masses

Kaluza-Klein modes of fermions in a 5-dimensional toy model are considered. The number of Kaluza-Klein modes that survive after integration over extra dimensions is finite in this space. Moreover the extra dimensional piece of the kinetic term induces no mass for the higher Kaluza-Klein modes on contrary to the standard lore.Comment: Presentation is improved and typos are corrected, two appendices and some references are added. No change in the essential content of the paper. 11 page

Reconsidering extra time-like dimensions

In this study we reconsider the phenomenological problems related to tachyonic modes in the context of extra time-like dimensions. First we reconsider a lower bound on the size of extra time-like dimensions. Next we discuss the issues of spontaneous decay of stable fermions through tachyonic decays and disappearance of fermions due to tachyonic contributions to their self-energies. We find that the tachyonic modes due to extra time-like dimensions are less problematic than the tachyonic modes in the usual 4-dimensional setting because the most troublesome Feynman diagrams are forbidden once the conservation of momentum in the extra time-like dimensions is imposed.Comment: The version to appear in EPJ

Higgs field as the gauge field corresponding to parity in the usual space-time

We find that the local character of field theory requires the parity degree of freedom of the fields to be considered as an additional dicrete fifth dimension which is an artifact emerging due to the local description of space-time. Higgs field arises as the gauge field corresponding to this discrete dimension. Hence the noncommutative geometric derivation of the standard model follows as a manifestation of the local description of the usual space-time.Comment: 14 pages, latex, no figure

Investigating the changes in teachers\u27 pedagogical practices through the use of the Mathematics Reasoning Heuristic (MRH) approach

Our changing world needs many more mathematically literate individuals. Mathematical literacy can be defined, parallel to reading and writing literacy, as not only being able to understand the fundamental notions of mathematics, develop sophisticated mathematical models and evaluate someone else\u27s use of numbers and mathematical models but also being able to represent quantitative relations using algebraic reasoning and interpret and reflect on mathematical language patterns. In order to help students become mathematically literate, the National Council of Teachers of Mathematics (NCTM) has focused attention on students\u27 conceptual understanding of mathematics suggesting students need to be actively involved in the learning process using their experiences and prior knowledge. Along with this view on learning, understanding of teaching has also been revised in mathematics classrooms. Teachers now need to provide students with a challenging and supportive classroom environment in which they can build new knowledge by engaging in exploration of mathematical ideas by themselves. Since the publication of Curriculum and Evaluation Standards for School Mathematics in 1989, the National Council of Teachers of Mathematics (NCTM) has paid special attention on teacher change, problem solving, and, more recently, using writing in mathematics classrooms for helping students develop thorough mathematical understanding and to becoming more mathematically literate.;This change in the views of learning and teaching has placed students in the center of learning occurring in the classroom by altering students\u27 roles and requiring them to be actively involved in talking and writing in mathematics classrooms. The NCTM mandated that students at all levels should be able to use mathematical ideas in a variety of situations. For this purpose, students must have the opportunity to discuss their ideas publicly, to reflect on their thoughts and problem solving processes, and to communicate their ideas using various modes of representation (graphical, pictorial, oral, written, etc.). Writing in mathematics was emphasized in The Principles and Standards for School Mathematics (NCTM, 2000, p. 61), which said, Writing in mathematics can...help students consolidate their thinking... because it requires an active involvement of learners such that they use writing as a vehicle for learning and become the center of their own learning processes by engaging in reflection on mathematical experiences.;This study focused on examining the changes in pedagogical practices when three high school algebra teachers shift from their traditional teaching to more student-centered practices through the use of the Mathematics Reasoning Heuristic (MRH) approach. The study also looked at the performance differences on the Iowa Test of Educational Development (ITED) between the students in the control classes where the teachers engaged in their traditional instructional routines and the students in the treatment classes where the teachers used the MRH approach. The goal of the MRH approach is to help teachers improve their pedagogical practices to scaffold students\u27 understanding of mathematical concepts and their problem solving skills.;The major findings of this study are that teachers\u27 adoption of the required pedagogical practices varied as they attempted to move away from their traditional practices and that implementing a student-oriented approach such as the MRH approach which includes embedded writing-to-learn strategies does have an impact on student performance. The student performance on the standardized test was significantly enhanced for those students in the MRH classrooms compared to students who engaged in the more traditional approaches. The results from the analysis of the teachers\u27 pedagogical practices in their treatment and control classes indicate to us the importance of pedagogical skills to promote dialogical interaction during problem solving. In examining the results the researcher would suggest that there are two critical elements of the MRH approach. The first is the pedagogical approach needed and the second is the consistent use of the heuristic concept through the scaffolded writing component of the MRH approach