On the use of symmetry-adapted crystalline orbitals in SCF-LCAO periodic calculations. I. The construction of the symmetrized orbitals

A computational procedure for generating space-symmetry-adapted Bloch functions (BF) is presented. The case is discussed when BF are built from a basis of local functions (atomic orbitals [AOs]). The method, which is completely general in the sense that it applies to any space group and AOs of any quantum number, is based on the diagonalization of Dirac characters. For its implementation, it does not require as an input character tables or related data, since this information is automatically generated starting from the space group symbol and the AO basis set. Formal aspects of the method, not available in textbooks, are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 299–309, 1998     

Group representations and matrix spectral problems

We study a connectionvia group representation theory, between the problem of describing the invariant factors of a product of two matrices over a principal ideal domain and the problem of describing the spectrum of a sum of two Hermitian matrices.     

Permutation Representations Defined by G-Clusters with Application to Quasicrystals

Starting from each finite union of orbits (called G- cluster) of an R-irreducible orthogonal representation of a finite group G, we define a representation of G in a higher-dimensional space (called permutation representation), and we prove that it can be decomposed into an orthogonal sum of two representations such that one of them is equivalent to the initial representation. This decomposition allows us to use the strip projection method and to obtain some patterns useful in quasicrystal physics. We show that certain self-similarities of such a pattern can be obtained by using the decomposition into R-irreducible components of the corresponding permutation representation, and we present two examples.     

Unitarity of Induced Representations from Coisotropic Quantum Subgroups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced by Ciccoli in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group where we have to use in all generality our definition of quasi-invariant functional.     

Cycles on Arithmetic Surfaces

We develop a localized intersection theory for arithmetic schemes on the model of Fulton's intersection theory. We prove a Lefschetz fixed point formula for arithmetic surfaces, and give an application to a conjecture of Serre on the existence of Artin's representations for regular local rings of dimension 2 and unequal characteristic.     

A character formula for a family of simple modular representations of $ GL_n $

Let K be an algebraically closed field of finite characteristic p, and let be an integer. In the paper, we give a character formula for all simple rational representations of with highest weight any multiple of any fundamental weight. Our formula is slightly more general: say that a dominant weight λ is special if there are integers such that and . Indeed, we compute the character of any simple module whose highest weight λ can be written as with all are special. By stabilization, we get a character formula for a family of irreducible rational -modules. Received: June 30, 1997.     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.     

A Comparison of Programming Languages and Algebraic Notation as Expressive Languages for Physics

The purpose of the present work is to consider some of the implications of replacing, for the purposes of physics instruction, algebraic notation with a programming language. Whatis novel is that, more than previous work, I take seriously the possibility that a programming language can function as the principle representational system for physics instruction. This means treating programming as potentially having a similar status and performing a similar function to algebraic notation in physics learning. In order to address the implications of replacing the usual notational system with programming, I begin with two informal conjectures: (1) Programming-based representations might be easier for students to understand than equation-based representations, and (2) programming-based representations might privilege a somewhat different ``intuitive vocabulary.' If the second conjecture is correct, it means that the nature of the understanding associated with programming-physics might be fundamentally different than the understanding associated with algebra-physics.In order to refine and address these conjectures, I introduce a framework based around two theoretical constructs, what I callinterpretive devices and symbolic forms. A conclusion of this work is that algebra-physics can be characterized as a physics of balance and equilibrium, and programming-physics as a physics of processes and causation. More generally, this work provides a theoretical and empirical basis for understanding how the use of particular symbol systems affects students' conceptualization.This revised version was published online in September 2005 with corrections to the Cover Date.     

The Impact of Teacher Privileging on Learning Differentiation with Technology

This study examines how two teachers taught differentiation using a hand held computer algebra system, which made numerical,graphical and symbolic representations of the derivative readily available. The teachers planned the lessons together but taught their Year 11 classes in very different ways. They had fundamentally different conceptions of mathematics with associated teaching practices,innate ‘privileging’ of representations, and of technology use. This study links these instructional differences to the different differentiation competencies that the classes acquired. Students of the teacher who privileged conceptual understanding and student construction of meaning were more able to interpret derivatives. Students of the teacher who privileged performance of routines made better use of the CAS for solving routine problems. Comparison of the results with an earlier study showed that although each teacher's teaching approach was stable over two years, each used technology differently with further experience of CAS. The teacher who stressed understanding moved away from using CAS, whilst the teacher who stressed rules,adopted it more. The study highlights that within similar overall attainment on student tests, there can be substantial variations of what students know. New technologies provide more approaches to teaching and so greater variations between teaching and the consequent learning may become evident. This revised version was published online in July 2006 with corrections to the Cover Date.