Search

The relativistic formulation of abstract evolution equations is introduced. The corresponding logarithmic representation is shown to exist without assuming the invertible property of evolution operators. Consequently, by means of the logarithmic representation of operators, nonlinear and discrete properties are shown to be valid to the infinitesimal generators of relativistic abstract evolution equations. [ABSTRACT FROM AUTHOR]

A few years of experience of teaching mathematics make us notice that the same types of mistakes are done repeatedly by students. This paper presents an insight into categories of mistakes, how male and female students differ in terms of mistakes that are commonly done and the ability of the students to identify the mistakes. Sample of mistakes were taken from Calculus 1 final exam answer scripts, then it was listed and analyzed. Data analysis revealed that students' misconceptions fall into four categories. The first category is misunderstanding the meaning of brackets, followed by misconception of basic mathematics rules, misconception in notation and misconception in properties of trigonometry. A mistake identification test which consists of ten false mathematical statements was designed based on the mistake done by the previous batch of students that covered topics algebra, trigonometry, index, limit, differentiation and integration. Then, the test was given to students who enrolled in Calculus I course. Respondents of this study were randomly selected among two hundreds engineering students. Data obtained were analyzed using basic descriptive analysis and Chi Square test to capture gender differences in the mistake done for each category. Findings indicate that thirty five percent of the students have the ability to identify the mistakes and make a proper correction for at most two statements. Thirty one percent of the students are able to identify the mistakes but unable to make proper correction. Twenty five percent of the students failed to identify the mistakes in six out of ten false statements. Female students' misconception is more likely in basic mathematics rules compared to male. The findings of this study could serve as baseline information to be stressed in improving teaching and learning mathematics. [ABSTRACT FROM AUTHOR]

A Cartesian grid-based unified gas kinetic scheme is developed. In this approach, any oriented boundary in a Cartesian grid is represented by many directional boundary points. The numerical flux is evaluated on each boundary point. Then, a boundary flux interpolation method (BFIM) is constructed to distribute the boundary effect to the flow evolution on regular Cartesian grid points. The BFIM provides a general strategy to implement any kind of boundary condition on Cartesian grid. The newly developed technique is implemented in the unified gas kinetic scheme, where the scheme is reformulated into a finite difference format. Several typical test cases are simulated with different geometries. For example, the thermophoresis phenomenon for a plate with infinitesimal thickness immersed in a rarefied flow environment is calculated under different orientations on the same Cartesian grid. These computational results validate the BFIM in the unified scheme for the capturing of different thermal boundary conditions. The BFIM can be extended to the moving boundary problems as well. [ABSTRACT FROM AUTHOR]

The modern geometrical approach to nonlinear PDEs is the out-come of a nontrivial synthesis of differential calculus over commutative al-gebras and cohomological algebra in the context of infinite jet spaces. In this paper we propose a very natural generalization of the notion of a jet space, which allows to treat the space of initial data of a nonlinear PDE on the same footing as the space of its solutions. [ABSTRACT FROM AUTHOR]

The calculation of the cross section in atomic collision theory when the collision is inelastic and without exchange effect requires the solution of a system of coupled differential equations. The Numerov technique is applied to solve a system of two coupled differential equations (schematic model), which simulate the solution of real problems but are themselves exactly solvable in the resonance cases. [ABSTRACT FROM AUTHOR]

The view that imagery can hinder mathematical thinking might surprise some mathematics educators who contend that understanding is enriched whenever visual imagery is used. In two contrasting cases, we present the view that dynamic visual images, without support from analytic thinking, can be a hindrance in constructing mathematical meaning - contrary to the "panacea" view of imagery which is sometimes expressed. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

It can be shown that a process modeled by a strongly continuous real-valued semigroup (that has a space convolution operator as infinitesimal generator) cannot satisfy causality By causality we mean that a characteristic feature of a process like an interface or a front must propagate with a finite speed. We present and discuss a causal model of diffusion that satisfies the semigroup property at a discrete set of time instants M := {mτ| m ∈ N0} and that in contrast to the classical diffusion model is not smooth. More precisely, if v denotes the concentration of a substance diffusing with constant speed, then v is continuous but its time derivative is discontinuous at the discrete set M of time instants. It is this property of (causal) diffusion that forbids the classical limit procedure τ → 0 that leads to the noncausal diffusion model in Stochastics. Finally, we give two explanations why in some cases the discretization of the noncausal diffusion model can be considered as an approximation of the causal diffusion model. In particular, we present an inhomogeneous wave equation with a time dependent coefficient that is satisfied by causal diffusion. [ABSTRACT FROM AUTHOR]

Studies on students' understanding of functions have reported that many students have a limited understanding of functions. Students' understanding of function transformation, function inverse, function composition, and the relationship among these three concepts is thus far understudied. This study investigates calculus students' understandings of the three concepts. Data obtained from two phases, a questionnaire and task-based interviews, will be presented. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]