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.

The slumbarumba jumbuck problem

The potential for simple linear relationships arising from a computer game to build student modelling and "world problem" skills is explored. The fundamental capability of the spreadsheet to tabulate and graph possible solutions is used to lay bare the problem structure for the students.

Cryptanalysis of block ciphers with overdefined systems of equations

Several recently proposed ciphers, for example Rijndael and Serpent, are built with layers of small S-boxes interconnected by linear key-dependent layers. Their security relies on the fact, that the classical methods of cryptanalysis (e.g. linear or differential attacks) are based on probabilistic characteristics, which makes their security grow exponentially with the number of rounds N r r. In this paper we study the security of such ciphers under an additional hypothesis: the S-box can be described by an overdefined system of algebraic equations (true with probability 1). We show that this is true for both Serpent (due to a small size of S-boxes) and Rijndael (due to unexpected algebraic properties). We study general methods known for solving overdefined systems of equations, such as XL from Eurocrypt’00, and show their inefficiency. Then we introduce a new method called XSL that uses the sparsity of the equations and their specific structure. The XSL attack uses only relations true with probability 1, and thus the security does not have to grow exponentially in the number of rounds. XSL has a parameter P, and from our estimations is seems that P should be a constant or grow very slowly with the number of rounds. The XSL attack would then be polynomial (or subexponential) in N r> , with a huge constant that is double-exponential in the size of the S-box. The exact complexity of such attacks is not known due to the redundant equations. Though the presented version of the XSL attack always gives always more than the exhaustive search for Rijndael, it seems to (marginally) break 256-bit Serpent. We suggest a new criterion for design of S-boxes in block ciphers: they should not be describable by a system of polynomial equations that is too small or too overdefined.

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.

Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

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.

Rapid, simultaneous activation of thin nanowire growth in low-temperature, low-pressure chemically active plasmas

Multiscale numerical modeling of the species balance and transport in the ionized gas phase and on the nanostructured solid surface complemented by the heat exchange model is used to demonstrate the possibility of minimizing the Gibbs-Thompson effect in low-temperature, low-pressure chemically active plasma-assisted growth of uniform arrays of very thin Si nanowires, impossible otherwise. It is shown that plasma-specific effects drastically shorten and decrease the dispersion of the incubation times for the nucleation of nanowires on non-uniform Au catalyst nanoparticle arrays. The fast nucleation makes it possible to avoid a common problem of small catalyst nanoparticle burying by amorphous silicon. These results explain a multitude of experimental observations on chemically active plasma-assisted Si nanowire growth and can be used for the synthesis of a range of inorganic nanowires for environmental, biomedical, energy conversion, and optoelectronic applications.

Plasma/ion-controlled metal catalyst saturation : enabling simultaneous growth of carbon nanotube/nanocone arrays

It is shown that the simultaneous saturation of Ni nanoparticles used as catalyst for vertically aligned carbon nanotube and nanocone arrays can be improved in low-temperature plasma- or ion-assisted processes compared with neutral gas-based routes. The results of hybrid multiscale numerical simulations of the catalyst nanoarrays (particle sizes of 2 and 10 nm) saturation with carbon show the possibility of reducing the difference in catalyst incubation times for smallest and largest catalyst particles by up to a factor of 2. This approach is generic and provides process conditions for simultaneous nucleation and growth of uniform arrays of vertically aligned nanostructures. © 2008 American Institute of Physics.

Simultaneous gene silencing and supressing gene silencing in the same cell

The present invention relates to genetically modified cells that are capable of optimal transgene expression by co-expressing a silencing suppressor whilst at the same time are also capable of silencing a gene, such as a naturally occurring gene of the cell. The present invention also relates to methods of producing the modified cells, as well as relates to processes for obtaining a genetically modified cell with a desired property.

Carbon saturation of arrays of Ni catalyst nanoparticles of different size and pattern uniformity on a silicon substrate

The kinetics of saturation of Ni catalyst nanoparticle patterns of the three different degrees of order, used as a model for the growth of carbon nanotips on Si, is investigated numerically using a complex model that involves surface diffusion and ion motion equations. It is revealed that Ni catalyst patterns of different degrees of order, with Ni nanoparticle sizes up to 12.5 nm, exhibit different kinetics of saturation with carbon on the Si surface. It is shown that in the cases examined (surface coverage in the range of 1-50%, highly disordered Ni patterns) the relative pattern saturation factor calculated as the ratio of average incubation times for the processes conducted in the neutral and ionized gas environments reaches 14 and 3.4 for Ni nanoparticles of 2.5 and 12.5 nm, respectively. In the highly ordered Ni patterns, the relative pattern saturation factor reaches 3 for nanoparticles of 2.5 nm and 2.1 for nanoparticles of 12.5 nm. Thus, more simultaneous saturation of Ni catalyst nanoparticles of sizes in the range up to 12.5 nm, deposited on the Si substrate, can be achieved in the low-temperature plasma environment than with the neutral gas-based process.

Simultaneous localization and planning on multiple map hypotheses

This paper presents a novel method to rank map hypotheses by the quality of localization they afford. The highest ranked hypothesis at any moment becomes the active representation that is used to guide the robot to its goal location. A single static representation is insufficient for navigation in dynamic environments where paths can be blocked periodically, a common scenario which poses significant challenges for typical planners. In our approach we simultaneously rank multiple map hypotheses by the influence that localization in each of them has on locally accurate odometry. This is done online for the current locally accurate window by formulating a factor graph of odometry relaxed by localization constraints. Comparison of the resulting perturbed odometry of each hypothesis with the original odometry yields a score that can be used to rank map hypotheses by their utility. We deploy the proposed approach on a real robot navigating a structurally noisy office environment. The configuration of the environment is physically altered outside the robots sensory horizon during navigation tasks to demonstrate the proposed approach of hypothesis selection.

Vision-based simultaneous localization and mapping in changing outdoor environments

For robots operating in outdoor environments, a number of factors, including weather, time of day, rough terrain, high speeds, and hardware limitations, make performing vision-based simultaneous localization and mapping with current techniques infeasible due to factors such as image blur and/or underexposure, especially on smaller platforms and low-cost hardware. In this paper, we present novel visual place-recognition and odometry techniques that address the challenges posed by low lighting, perceptual change, and low-cost cameras. Our primary contribution is a novel two-step algorithm that combines fast low-resolution whole image matching with a higher-resolution patch-verification step, as well as image saliency methods that simultaneously improve performance and decrease computing time. The algorithms are demonstrated using consumer cameras mounted on a small vehicle in a mixed urban and vegetated environment and a car traversing highway and suburban streets, at different times of day and night and in various weather conditions. The algorithms achieve reliable mapping over the course of a day, both when incrementally incorporating new visual scenes from different times of day into an existing map, and when using a static map comprising visual scenes captured at only one point in time. Using the two-step place-recognition process, we demonstrate for the first time single-image, error-free place recognition at recall rates above 50% across a day-night dataset without prior training or utilization of image sequences. This place-recognition performance enables topologically correct mapping across day-night cycles.

Experimental studies on web crippling behaviour of hollow flange channel beams under two flange load cases

This paper presents the details of an experimental study of a cold-formed steel hollow flange channel beam known as LiteSteel beam (LSB) subject to web crippling under End Two Flange (ETF) and Interior Two Flange (ITF) load cases. The LSB sections with two rectangular hollow flanges are made using a simultaneous cold-forming and electric resistance welding process. Due to the geometry of the LSB, and its unique residual stress characteristics and initial geometric imperfections, much of the existing research for common cold-formed steel sections is not directly applicable to LSB. Experimental and numerical studies have been carried out to evaluate the behaviour and design of LSBs subject to pure bending, predominant shear and combined actions. To date, however, no investigation has been conducted on the web crippling behaviour and strength of LSB sections. Hence an experimental study was conducted to investigate the web crippling behaviour and capacities of LSBs. Twenty-eight web crippling tests were conducted under ETF and ITF load cases, and the ultimate web crippling capacities were compared with the predictions from the design equations in AS/NZS 4600 and AISI S100. This comparison showed that AS/NZS 4600 and AISI S100 web crippling design equations are unconservative for LSB sections under ETF and ITF load cases. Hence new equations were proposed to determine the web crippling capacities of LSBs based on experimental results. Suitable design rules were also developed under the direct strength method (DSM) format.