Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs

This research explored students’ views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is aimed at prospective teachers. A preponderance of the evidence suggests that the participating students, for the most part, viewed isometries operationally and viewed geometric objects (triangle, etc.) as “perceived.” Results also suggest that these two views influenced the students’ abilities to understand and to construct geometric proofs in transformational geometry.     

Lie algebroids,Lie groupoids and TFT

We construct the moduli spaces associated to the solutions of equations of motion (modulo gauge transformations) of the Poisson sigma model with target being an integrable Poisson manifold. The construction can be easily extended to a case of a generic integrable Lie algebroid. Indeed for any Lie algebroid one can associate a BF-like topological field theory which localizes on the space of algebroid morphisms, that can be seen as a generalization of flat connections to the groupoid case. We discuss the finite gauge transformations and discuss the corresponding moduli spaces. We consider the theories both without and with boundaries.     

Phase-locking of a multi-channel radial array CO2 laser

A theoretical analysis of phase-locking in a multi-channel radial array CO₂ laser is presented. The concepts are based on the theories of injection phase-locking and matrix optics. The mutual optical coupling occurring within the central region is demonstrated by theoretical deduction. Interrelated graphs of the coupling coefficient and the distance between the coupling position and the cavity mirror are calculated by numerical simulation. The results of the analysis are in accord with experimental results.     

The spectral curve theory for (k,l)-symmetric CMC surfaces

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in Hitchin (1990), Pinkall and Sterling (1989) and for genus 2 surfaces (Heller, 2014) to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting ${\mathbb{Z}}_{k+1}$ and ${\mathbb{Z}}_{l+1}$ symmetries. We determine their associated family of flat connections via certain flat line bundle connections parametrized by the spectral curve. We generalize the flow on spectral data introduced in Heller (2015) and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain countably many $1-$parameter families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.     

Thermo-elastic properties of heterogeneous materials with imperfect interfaces: Generalized Levin's formula and Hill's connections

We study the thermo-elastic properties of heterogeneous materials containing spherical particles or cylindrical fibres. The interface between the matrix and second-phase inhomogeneity is imperfect with either the displacement or the stress experiencing a jump across it. We relate the effective coefficient of thermal expansion (CTE) to the effective elastic moduli and thereby generalize Levin's formula, and reveal two connections among the effective elastic moduli, thereby generalizing Hill's connections. In contrast to the classical results, the effective CTE in the presence of an imperfect interface is strongly dependent on the size of the inhomogeneity, besides the interface elastic and thermo-elastic properties. This size dependence has been accurately captured by simple scaling laws.     

Remarks on symplectic twistor spaces

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study self-holomorphic sections of the general twistor space, with which we define a new moduli space of complex structures. We also recall the theory of flag manifolds to study the Siegel domain and other domains alike, which are the fibres of various symplectic twistor spaces. We prove that they are all Stein. In the context of a Riemann surface, with its canonical symplectic-metric connection and local structure equations, the moduli space is studied again.     

基于FPGA的固态硬盘控制器的研究与设计

随着Flash存储芯片性价比越来越高,基于Flash存储芯片的固态硬盘已成为手持电子终端设备中的主流存储设备,并已开始大举进入笔记本计算机等领域.固态硬盘控制器是固态硬盘的核心部件,固态硬盘控制器性能的好坏直接决定固态硬盘的好坏,由于FPGA具备可编程、设计灵活性、设计周期短、可在线升级等优点,本文研究基于FPGA的16通道、双核固态硬盘控制器.     

Intra-mathematical connections made by high school students in performing Calculus tasks

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection.     

Memcapacitor response under step and sinusoidal voltage excitations

Recently, mem-elements have become fundamental in the circuit theory through promising potential applications based on the built-in memory-properties of these elements. In this paper, the mathematical analysis of the memcapacitor model is derived and the effect of different voltage excitation signals is studied for the linear dopant model. General closed form expressions and analyses are presented to describe the memcapacitor behavior under DC step and sinusoidal voltage excitations. Furthermore, the step and sinusoidal responses are used to analyze the memcapacitor response under any periodic signal using Fourier series expansion where the effect of the DC component on the output response is investigated. In addition, the stored energy in the memcapacitor under step, sinusoidal and square wave excitations is discussed. Moreover, the analysis of series and parallel connection of N non-matched memcapacitors in general is introduced and special cases of matched memcapacitors are discussed. The derived equations are verified using SPICE simulations showing great matching.     

On the purely irregular fundamental group

Based on the (not yet fully understood analogy) between irregular connections and wild ramification, we define a purely irregular fundamental group for complex algebraic varieties and prove some results about this fundamental group which are analogous to the p‐adic étale fundamental group of algebraic varieties over fields of characteristic p (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)