Translation between external representation systems in mathematics: All-or-none or skill conglomerate?

The study described herein represents an initial, exploratory attempt to understand what it means to translate between external representation systems. Researchers have traditionally considered translation as an all-or-none activity. We hypothesize that translation is comprised of both knowledge and skill components, and accordingly construe translation as an activity that our framework allows us to define as partial or complete. We examined the effects of part-whole knowledge in change unknown subtraction situations and the structure of three different external representations on first grade students’ ability to translate in a partitioning task. Based on three interviews conducted with each student, results from our mixed methods analysis showed that translation skill was affected by what students knew about subtraction and the structure of the ER. Further, whereas high knowledge students demonstrated greater difficulty with translating to the ERs, low knowledge students had difficulty translating from the problem statement as well as translating to the ERs.     

一维光子晶体的带隙分析

本文用转移矩阵法推导出一维光子晶体的禁带存在的判据;讨论了在光线正入射时,光子禁带存在的条件及特性.     

Abelian Subgroups of Garside Groups

In this article, we show that for every abelian subgroup H of a Garside group, some conjugate g ^?1 Hg consists of ultra summit elements and the centralizer of H is a finite index subgroup of the normalizer of H. Combining with the results on translation numbers in Garside groups, we obtain an easy proof of the algebraic flat torus theorem for Garside groups and solve several algorithmic problems concerning abelian subgroups of Garside groups.     

Auslander–Reiten Components with Sectional Bypasses

In this work, we will prove that the modules lying in a sectional bypass of an arrow in the Auslander–Reiten quiver of an artin algebra, are either all left stable or all right stable, but not τ-periodic. Moreover, if such a bypass exists, then the Auslander–Reiten quiver has an infinite left or right stable component which contains a section with a bypass.     

A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement

The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 19² admitting SL(2,5), one of which is obtained by 12-nest replacement.      

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

We study the structure of a finite groupG admitting a Kantor family (F, F ^*) of type (s, t) and a nontrivial normal subgroupX which isfactorized byF F ^*. The most interesting cases, giving necessary conditions on the structure ofG and the parameterss andt, are those where a further Kantor family is induced inX, or where a partial congruence partition is induced in the factor groupG/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Tight Multiwavelet Frames with Different Matrix Dilations and Matrix Translations

The characterization of tight multiwavelet frames with different matrix dilations and matrix translations for L ²(R ^d ) is established. The result contains and further extends the generalizations that have appeared in the literature. Two sufficient conditions for affine frames are also presented.     

Translation groups of Steiner loops

If the order of any product of two different translations of a finite Steiner quasigroup of size n>3 is odd, then the group G generated by the translations of the corresponding Steiner loop of order n+1 contains the alternating group of degree n+1.     

Shift-invariant spaces on LCA groups

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd.