The theory of images in the representation of functions.

Thesis (Ph.D.)--Cornell University, 1888.

State goals for learning and sample learning objectives.

"Grades 3,6, 8, 10, 12."

Four lectures on mathematics, delivered at Columbia University in 1911,

I. The defintion of solutions of linear partial differnetial equations by boundary conditions.--II. Contemporary researches in differential equations, integral equations, and integro-differential equations.--III. Analysis situs in connection with correspondences and differential equations.--IV. Elementary solutions of partial differential equations and Green's functions.

The Princeton Colloquium; lectures on mathematics, delivered September 15 to 17, 1909, before members of the American Mathematical Society in connection with the summer meeting held at Princeton University, Princeton, N.J.

Fundamental existence theorems, by G. A. Bliss.--Differential-geometric aspects of dynamics, by E. Kasner.

Pre-Service Teachers’ Mathematics Language and Reflection in the Context of an Early Childhood Mathematics Methods Course

Thesis (Ph.D.)--University of Washington, 2016-06

Utility functions on locally connected spaces

We investigate the role of local connectedness in utility theory and prove that any continuous total preorder on a locally connected separable space is continuously representable. This is a new simple criterion for the representability of continuous preferences, and is not a consequence of the standard theorems in utility theory that use conditions such as connectedness and separability, second countability, or path-connectedness. Finally we give applications to problems involving the existence of value functions in population ethics and to the problem of proving the existence of continuous utility functions in general equilibrium models with land as one of the commodities. (C) 2003 Elsevier B.V. All rights reserved.

Extremal circular permutations with concave functions

Let {a(1), a(2), ..., a(n)} be a set of n distinct real numbers and let alpha(1), alpha(2), ..., alpha(n) an be a permutation of the numbers. We construct the permutation to maximise L-f = Sigma(i=1)(n) f(\alpha(i+1) - alpha(i)\), for any increasing concave function f, where we denote alpha(n+1) equivalent to alpha(1). The optimal permutation depends on the particular numbers {a(1), a(2), ..., a(n)} and the function f, contrary to a postulate by Chao and Liang (European J. Combin. 13 (1992) 325). (C) 2004 Elsevier Ltd. All rights reserved.

Learning difficulties in numeracy in Australia

In this article, we provide an understanding of the term numeracy as it is used in Australia and a description of numeracy education in this country. In particular, we discuss the role of outcomes-based curriculum frameworks and outline the dominant teaching approaches. The focus is on students with learning difficulties and how they are identified and supported in schools. We create two vignettes based on real students with difficulties in numeracy, which highlight two of the most common problems. We report on the prevalence of learning difficulties in numeracy in Australia and describe some of the initiatives related to the assessment and enhancement of learning in numeracy that are being undertaken in various states in Australia. Finally, we identify some of the future challenges facing the Australian education system in this area. These challenges relate to teacher knowledge, mandated assessment, and the role of parents. We conclude the article by calling for the evaluation of current assessment and intervention initiatives as well as the development of a national program to support the goal of numeracy for all students.

A Bayesian approach to imposing curvature on distance functions

The estimated parameters of output distance functions frequently violate the monotonicity, quasi-convexity and convexity constraints implied by economic theory, leading to estimated elasticities and shadow prices that are incorrectly signed, and ultimately to perverse conclusions concerning the effects of input and output changes on productivity growth and relative efficiency levels. We show how a Bayesian approach can be used to impose these constraints on the parameters of a translog output distance function. Implementing the approach involves the use of a Gibbs sampler with data augmentation. A Metropolis-Hastings algorithm is also used within the Gibbs to simulate observations from truncated pdfs. Our methods are developed for the case where panel data is available and technical inefficiency effects are assumed to be time-invariant. Two models-a fixed effects model and a random effects model-are developed and applied to panel data on 17 European railways. We observe significant changes in estimated elasticities and shadow price ratios when regularity restrictions are imposed. (c) 2004 Elsevier B.V. All rights reserved.

Improving learning in undergraduate control engineering courses using context-based learning models

Control Engineering is an essential part of university electrical engineering education. Normally, a control course requires considerable mathematical as well as engineering knowledge and is consequently regarded as a difficult course by many undergraduate students. From the academic point of view, how to help the students to improve their learning of the control engineering knowledge is therefore an important task which requires careful planning and innovative teaching methods. Traditionally, the didactic teaching approach has been used to teach the students the concepts needed to solve control problems. This approach is commonly adopted in many mathematics intensive courses; however it generally lacks reflection from the students to improve their learning. This paper addresses the practice of action learning and context-based learning models in teaching university control courses. This context-based approach has been practised in teaching several control engineering courses in a university with promising results, particularly in view of student learning performances.

Identification of MIMO Hammerstein systems using cardinal spline functions

A new approach to identify multivariable Hammerstein systems is proposed in this paper. By using cardinal cubic spline functions to model the static nonlinearities, the proposed method is effective in modelling processes with hard and/or coupled nonlinearities. With an appropriate transformation, the nonlinear models are parameterized such that the nonlinear identification problem is converted into a linear one. The persistently exciting condition for the transformed input is derived to ensure the estimates are consistent with the true system. A simulation study is performed to demonstrate the effectiveness of the proposed method compared with the existing approaches based on polynomials. (C) 2006 Elsevier Ltd. All rights reserved.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Determinant representations of correlation functions for the supersymmetric t-J model

Working in the F-basis provided by the factorizing F-matrix, the scalar products of Bethe states for the supersymmetric t-J model are represented by determinants. By means of these results, we obtain determinant representations of correlation functions for the model.

On the construction of correlation functions for the integrable supersymmetric fermion models

We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing F-matrices (or the so-called F-basis) play an important role in the construction. In the F-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the U-q(gl(2 vertical bar 1)) (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analyzing physical properties of the integrable models in the thermodynamical limit.