Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

Primitive partial permutation representations of the polycyclic monoids and branching function systems

We characterise the maximal proper closed inverse submonoids of the polycyclic inverse monoids, also known as Cuntz inverse semigroups, and so determine all their primitive partial permutation representations. We relate our results to the work of Kawamura on certain kinds of representations of the Cuntz C*-algebras and to the branching function systems of Bratteli and Jorgensen.      

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Filter products of C0-semigroups and ultraproduct representations for Lie groups

We provide a new approach to filter products of C₀-semigroups and prove a spectral theorem for the generator and its filter product. In a similar fashion, we construct ultraproducts of strongly continuous unitary representations of locally compact groups and study spectral theoretic connections between the representations and their ultraproducts. In the case of Lie groups, our investigations are extended to the infinitesimal representation.     

On the Existence of Auslander–Reiten Sequences of Group Representations. I

This is the first part of our study of the existence ofAuslander–Reiten sequences of group representations. In this part weconsider representations of group schemes in characteristic 0; in Part II weconsider representations of group schemes in characteristic p; andin Part III we give applications to representations of groups and Liealgebras.     

On the Existence of Auslander–Reiten Sequences of Group Representations II

This is the second part of our study of the existence ofAuslander–Reiten sequences of group representations. In Part I weconsidered representations of group schemes in characteristic 0; in thispart we consider representations of group schemes in characteristicp; and in Part III we give applications to representations ofabstract groups and Lie algebras.     

Linear algebra students’ modes of reasoning: Geometric representations

Main goal of our research was to document differences on the types of modes linear algebra students displayed in their responses to the questions of linear independence from two different assignments. In this paper, modes from the second assignment are discussed in detail. Second assignment was administered with the support of graphical representations through an interactive web-module. Additionally, for comparison purposes, we briefly talk about the modes from the first assignment. First assignment was administered with the support of computational devices such as calculators providing the row reduced echelon form (rref) of matrices. Sierpinska’s framework on thinking modes (2000) was considered while qualitatively documenting the aspects of 45 matrix algebra students’ modes of reasoning. Our analysis revealed 17 categories of the modes of reasoning for the second assignment, and 15 categories for the first assignment. In conclusion, the findings of our analysis support the view of the geometric representations not replacing one’s arithmetic or algebraic modes but encouraging students to utilize multiple modes in their reasoning. Specifically, geometric representations in the presence of algebraic and arithmetic modes appear to help learners begin to consider the diverse representational aspects of a concept flexibly.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

On the Existence of Auslander–Reiten Sequences of Group Representations. III

This is the third and final part of our study of the existence of Auslander–Reiten sequences ofgroup representations. In Part I we considered representations of group schemes in characteristic 0.In Part II we considered representations of group schemes in characteristic p. In this partwe give applications to representations of abstract groups and Lie algebras.     

Linearization relations for the generalized Bedient polynomials of the first and second kinds via their integral representations

The main object of this paper is to investigate several general families of hyper-geometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials of the first and second kinds. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

Grammar compressed sequences with rank/select support

Sequence representations supporting not only direct access to their symbols, but also rank/select operations, are a fundamental building block in many compressed data structures. Several recent applications need to represent highly repetitive sequences, and classical statistical compression proves ineffective. We introduce, instead, grammar-based representations for repetitive sequences, which use up to 6% of the space needed by statistically compressed representations, and support direct access and rank/select operations within tens of microseconds. We demonstrate the impact of our structures in text indexing applications.     

Plain Representations of Lie Algebras

In this paper we study representations of finite dimensionalLie algebras. In this case representations are not necessarilycompletely reducible. As the general problem is known to beof enormous complexity, we restrict ourselves to representationsthat behave particularly well on Levi subalgebras. We call suchrepresentations plain (Definition 1.1). Informally, we showthat the theory of plain representations of a given Lie algebraL is equivalent to representation theory of finitely many finitedimensional associative algebras, also non-semisimple. The senseof this is to distinguish representations of Lie algebras thatare of complexity comparable with that of representations ofassociative algebras. Non-plain representations are intrinsicallymuch more complex than plain ones. We view our work as a steptoward understanding this complexity phenomenon. We restrict ourselves also to perfect Lie algebras L, that is,such that L = [L, L]. In our main results we assume that L isperfect and sl₂-free (which means that L has no quotient isomorphicto sl₂). The ground field F is always assumed to be algebraicallyclosed and of characteristic 0.     

On calibrated representations and the Plancherel Theorem for affine Hecke algebras

This paper has two main purposes. Firstly, we generalise Ram’s combinatorial construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced BC _n case). We then derive the Plancherel formulae for all rank 1 and rank 2 affine Hecke algebras, using our calibrated representations to construct all representations involved.     

ℒ-invariants and logarithm derivatives of eigenvalues of Frobenius

Let K be a p-adic local field. We study a special kind of p-adic Galois representations of it. These representations are similar to the Galois representations occurred in the exceptional zero conjecture for modular forms. In particular, we verify that a formula of Colmez can be generalized to our case. We also include a degenerated version of Colmez’s formula.     

Smooth group representations on bornological vector spaces

We develop the basic theory of smooth representations of locally compact groups on bornological vector spaces. In this setup, we are able to formulate better general theorems than in the topological case. Nonetheless, smooth representations of totally disconnected groups on vector spaces and of Lie groups on Fréchet spaces remain special cases of our theory. We identify smooth representations with essential modules over an appropriate convolution algebra. We examine smoothening functors on representations and modules and show that they agree if they are both defined. We establish the basic properties of induction and compact induction functors using adjoint functor techniques. We describe the center of the category of smooth representations.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Universal similarity factorization equalities for commutative quaternions and their matrices

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.     

Exploration and Visualization: Making Critical Connections About Linear Systems of Equations

Linear systems of equations are used to model relationships in both science and mathematics. Studies have shown that gaps exist in student understanding of important ideas about such systems. Students show weaknesses in understanding the connections between algebraic and geometric representations, the impact of scaling and approximation methods, and the validity of methods like Gaussian elimination. As demonstrated with the activities presented in this article, current technology allows readily accessible representations in algebraic, numeric, and geometric forms. Integrating these different representations into our curriculum helps students to think more critically about systems, to foster new perspectives, to feel more confident in their results, and to understand better the relations between different representations.     

Controlled Nonlinear Stochastic Delay Equations: Part II: Approximations and Pipe-Flow Representations

This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.     

THE INFLUENCE OF TEACHERS’ PRESENTATIONS ON PUPILS’ MENTAL REPRESENTATIONS

Teachers use a variety of external representations to communicate mathematical ideas to their pupils. This paper reports a preliminary study of the internal mental representations that 6- and 7- year-old pupils form as a result of their interactions with the teacher's verbal, written, pictorial and concrete material representations, involving two-digit numbers and operations on them. The results presented here concern the picture-like mental representations that pupils use in performing two-digit calculations mentally. The evidence suggests that pupils seldom spontaneously visualise teachers’ representations or attempt mental manipulation of visual images to help with calculation. Pupils can, however, have mental representations which reproduce some aspects of the teachers’ representations.