Modelling the landing of a plane in a calculus lab

We exhibit a simple model of a plane landing that involves only basic concepts of differential calculus, so it is suitable for a first-year calculus lab. We use the computer algebra system Maxima and the interactive geometry software GeoGebra to do the computations and graphics.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

Book review

Probability in Social Science by Samuel Goldberg     

Area Integral Functions and H Functional Calculus for Sectorial Operators on Hilbert Spaces

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi's results on H^∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.     

Gradient type noises II - Systems of stochastic partial differential equations

The present paper is the second and main part of a study of partial differential equations under the influence of noisy perturbations. Existence and uniqueness of function solutions in the mild sense are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems. If the noise data are random, the results may be seen as a pathwise approach to SPDE's. For typical examples, such as spatially one-dimensional stochastic heat equations with additive or multiplicative perturbations of fractional Brownian type, we recover and extend known results. In addition, we propose to consider partial noises of low order.     

Newton Methods for Quasidifferentiable Equations and Their Convergence

The Newton method and the inexact Newton method for solving quasidifferentiable equations via the quasidifferential are investigated. The notion of Q-semismoothness for a quasidifferentiable function is proposed. The superlinear convergence of the Newton method proposed by Zhang and Xia is proved under the Q-semismooth assumption. An inexact Newton method is developed and its linear convergence is shown.Project sponsored by Shanghai Education Committee Grant 04EA01 and by Shanghai Government Grant T0502.     

LITTLEWOOD-PALEY THEOREM FOR SCHR(O)DINGER OPERATORS

Let H be a Schr(o)dinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Difference discrete connection and curvature on cubic lattice Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.     

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schrodinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Noisy image segmentation based on nonlinear diffusion equation model

This paper addresses the segmentation problem in noisy image based on nonlinear diffusion equation model and proposes a new adaptive segmentation model based on gray-level image segmentation model. This model also can be extended to the vector value image segmentation. By virtue of the prior information of regions and boundary of image, a framework is established to construct different segmentation models using different probability density functions. A segmentation model exploiting Gauss probability density function is given in this paper. An efficient and unconditional stable algorithm based on locally one-dimensional (LOD) scheme is developed and it is used to segment the gray image and the vector values image. Comparing with existing classical models, the proposed approach gives the best performance.