Diophantine equations as a context for technology-enhanced training in conjecturing and proving

Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.

Discrete phase-locked loop systems and spreadsheets

This paper demonstrates the use of a spreadsheet in exploring non-linear difference equations that describe digital control systems used in radio engineering, communication and computer architecture. These systems, being the focus of intensive studies of mathematicians and engineers over the last 40 years, may exhibit extremely complicated behaviour interpreted in contemporary terms as transition from global asymptotic stability to chaos through period-doubling bifurcations. The authors argue that embedding advanced mathematical ideas in the technological tool enables one to introduce fundamentals of discrete control systems in tertiary curricula without learners having to deal with complex machinery that rigorous mathematical methods of investigation require. In particular, in the appropriately designed spreadsheet environment, one can effectively visualize a qualitative difference in the behviour of systems with different types of non-linear characteristic.

Crossword puzzle attack on NLS

NLS is one of the stream ciphers submitted to the eSTREAM project. We present a distinguishing attack on NLS by Crossword Puzzle (CP) attack method which is introduced in this paper. We build the distinguisher by using linear approximations of both the non-linear feedback shift register (NFSR) and the nonlinear filter function (NLF). Since the bias of the distinguisher depends on the Konst value, which is a key-dependent word, we present the graph showing how the bias of distinguisher vary with Konst. In result, we estimate the bias of the distinguisher to be around O(2^−30). Therefore, we claim that NLS is distinguishable from truly random cipher after observing O(2^60) keystream words. The experiments also show that our distinguishing attack is successful on 90.3% of Konst among 2^32 possible values. We extend the CP attack to NLSv2 which is a tweaked version of NLS. In result, we build a distinguisher which has the bias of around 2− 48. Even though this attack is below the eSTREAM criteria (2^−40), the security margin of NLSv2 seems to be too low.

Amplitude equations and asymptotic expansions for multi-scale problems

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases. © 2010 - IOS Press and the authors. All rights reserved.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems. © 2008 The American Physical Society.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution. © 2008 Optical Society of America.

Self-interaction of surface polaritons at the narrow-gap n-type semiconductor metal interface

The non-linear self-interaction of the potential surface polaritons (SP) which is due to the free carriers dispersion law where nonparabolicity is studied. The SP propagate at the interface between n-type semiconductor and a metal. The self interaction of the SP is shown to be different in semiconductors with normal and inverse zone structures. The results of the SP field envelope evolution are given. The obtained nonlinear frequency shift has been compared with shifts which are due to another self-interaction mechanisms. This comparison shows that the nonlinear self-interaction mechanism, which is due to free carriers spectrum nonparabolicity, is especially significant in narrow-gap semiconductor materials.

Effect of soil properties on the response of pile to underground explosion

This paper develops and presents a fully coupled non-linear finite element procedure to treat the response of piles to ground shocks induced by underground explosions. The Arbitrary Lagrange Euler coupling formulation with proper state material parameters and equations are used in the study. Pile responses in four different soil types, viz, saturated soil, partially saturated soil and loose and dense dry soils are investigated and the results compared. Numerical results are validated by comparing with those from a standard design manual. Blast wave propagation in soils, horizontal pile deformations and damages in the pile are presented. The pile damage presented through plastic strain diagrams will enable the vulnerability assessment of the piles under the blast scenarios considered. The numerical results indicate that the blast performance of the piles embedded in saturated soil and loose dry soil are more severe than those in piles embedded in partially saturated soil and dense dry soil. Present findings should serve as a benchmark reference for future analysis and design.

Firm-customer mobile digital connectedness : conceptualization, measurement and implications

Firm-customer digital connectedness for effective sensing and responding is a strategic imperative for contemporary competitive firms. This research-in-progress paper conceptualizes and operationalizes the firm-customer mobile digital connectedness of a smart-mobile customer. The empirical investigation focuses on mobile app users and the impact of mobile apps on customer expectations. Based on pilot data collected from 127 customers, we tested hypotheses pertaining to firm-customer mobile digital connectedness and customer expectations. Our test analysis using linear and non-linear postulations reveals those customers raise their expectations as they increase their digital interactions with a firm.

Stress relaxation analysis of single chondrocytes using porohyperelastic model based on AFM experiments

The aim of this study is to investigate the stress relaxation behavior of single chondrocytes using the Porohyperelastic (PHE) model and inverse Finite Element Analysis (FEA). Firstly, based on Atomic Force Microscopy (AFM) technique, we have found that the chondrocytes exhibited stress relaxation behavior. We explored the mechanism of this stress relaxation behavior and concluded that the intracellular fluid exuding out from the cells during deformation plays the most important role in the stress relaxation. Next, we have applied the inverse FEA technique to determine necessary material parameters for PHE model to simulate this stress relaxation behavior as this model is proven capable of capturing the non-linear behavior and the fluid-solid interaction during the stress relaxation of the single chondrocytes. It is observed that this PHE model can precisely capture the stress relaxation behavior of single chondrocytes and would be a suitable model for cell biomechanics.

Contactless and replaceable modularized battery energy storage system for electric vehicles

A novel replaceable, modularized energy storage system with wireless interface is proposed for a battery operated electric vehicle (EV). The operation of the proposed system is explained and analyzed with an equivalent circuit and an averaged state-space model. A non-linear feedback linearization based controller is developed and implemented to regulate the DC link voltage by modulating the phase shift ratio. The working and control of the proposed system is verified through simulation and some preliminary results are presented.

The impact of diabetic ketoacidosis and age on behavior six months post-diagnosis in children with type 1 diabetes

Objectives: Children with type 1 diabetes mellitus (DM1) may be at increased risk of psychosocial and adjustment difficulties. We examined behavioral outcomes six months post-diagnosis in a group of children with newly diagnosed DM1. Methods: This study formed part of a larger longitudinal project examining pathophysiology and neuropsychological outcomes in diabetic patients with or without diabetic ketoacidosis (DKA). Participants were 61 children (mean age 11.8 years, SD 2.7 years) who presented with a new diagnosis of DM1 at the Royal Children’s Hospital, Melbourne. Twenty-three (11 female) presented in DKA and 38 (14 female) without DKA. Parents completed the behavior assessment system for children, second edition six months post-diagnosis. Results: There was a non-linear relationship between age and behavior. Internalising problems (i.e. anxiety depression, withdrawal) peaked in the transition from childhood to adolescence; children aged 10–13 years had elevated rates relative to the normal population (t = 2.55, P = 0.018). There was a non-significant trend for children under 10 to display internalising problems (P = 0.052), but rates were not elevated in children over 13 (P = 0.538). Externalising problems were not significantly elevated in any age group. Interestingly, children who presented in DKA were at lower risk of internalising problems than children without DKA (t = 3.83, P < 0.001). There was no effect of DKA on externalising behaviors. Conclusions: Children transitioning from childhood to adolescence are at significant risk for developing internalising problems such as anxiety and lowered mood after diagnosis of DM1. Somewhat counter-intuitively, parents of children presenting in DKA reported fewer internalising symptoms than parents of children without DKA. These results highlight the importance of monitoring and supporting psychosocial adjustment in newly diagnosed children even when they seem physically well.

Sequential monitoring of hospital adverse events when control charts fail : the example of fall injuries in hospitals

Objective To evaluate methods for monitoring monthly aggregated hospital adverse event data that display clustering, non-linear trends and possible autocorrelation. Design Retrospective audit. Setting The Northern Hospital, Melbourne, Australia. Participants 171,059 patients admitted between January 2001 and December 2006. Measurements The analysis is illustrated with 72 months of patient fall injury data using a modified Shewhart U control chart, and charts derived from a quasi-Poisson generalised linear model (GLM) and a generalised additive mixed model (GAMM) that included an approximate upper control limit. Results The data were overdispersed and displayed a downward trend and possible autocorrelation. The downward trend was followed by a predictable period after December 2003. The GLM-estimated incidence rate ratio was 0.98 (95% CI 0.98 to 0.99) per month. The GAMM-fitted count fell from 12.67 (95% CI 10.05 to 15.97) in January 2001 to 5.23 (95% CI 3.82 to 7.15) in December 2006 (p<0.001). The corresponding values for the GLM were 11.9 and 3.94. Residual plots suggested that the GLM underestimated the rate at the beginning and end of the series and overestimated it in the middle. The data suggested a more rapid rate fall before 2004 and a steady state thereafter, a pattern reflected in the GAMM chart. The approximate upper two-sigma equivalent control limit in the GLM and GAMM charts identified 2 months that showed possible special-cause variation. Conclusion Charts based on GAMM analysis are a suitable alternative to Shewhart U control charts with these data.

Student interactivity and teacher participation: An application of legitimate peripheral participation in higher education online learning environments

Lave and Wenger’s legitimate peripheral participation is an important aspect of online learning environments. It is common for teachers to scaffold varying levels of online participation in Web 2.0 contexts, such as online discussion forums and blogs. This study argues that legitimate peripheral participation needs to be redefined in response to students’ decentralised multiple interactions and non-linear engagement in hyperlinked learning environments. The study examines students’ levels of participation in online learning through theories of interactivity, distinguishing between five levels of student participation in the context of a first-year university course delivered via a learning management system. The data collection was implemented through two instruments: i) a questionnaire about students’ interactivity perception in the online reflective learning (n = 238) and then ii) an open discussion on the reason for the diverse perceptions of interactivity (n = 34). The study findings indicate that student participants, other than those who were active, need high levels of teacher or moderator intervention, which better enables legitimate peripheral participation to occur in online learning contexts.

Assessment of the mortality displacement in temperature-related deaths in Brisbane, Australia

This thesis is a population-based ecological study designed to investigate the issue of mortality displacement (or "harvesting" effect) in the assessment of temperature-related deaths in Brisbane, Australia. It examines the temperature impacts on mortality, and assesses the harvesting effects on the temperature–related deaths. This study contributes to the knowledge base of understanding the temperature-mortality relationship and assists in formulating and evaluating public health intervention strategies within the context of climate change.