Non-linear principal components analysis : an alternative method for finding patterns in environmental data

The main purpose of this article is to gain an insight into the relationships between variables describing the environmental conditions of the Far Northern section of the Great Barrier Reef, Australia. Several of the variables describing these conditions had different measurement levels and often they had non-linear relationships. Using non-linear principal component analysis, it was possible to acquire an insight into these relationships. Furthermore, three geographical areas with unique environmental characteristics could be identified.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

Colour by numbers in Excel 2007 : solving algebraic equations without algebra

This is an update of an earlier paper, and is written for Excel 2007. A series of Excel 2007 models is described. The more advanced versions allow solution of f(x)=0 by examining change of sign of function values. The function is graphed and change of sign easily detected by a change of colour. Relevant features of Excel 2007 used are Names, Scatter Chart and Conditional Formatting. Several sample Excel 2007 models are available for download, and the paper is intended to be used as a lesson plan for students having some familiarity with derivatives. For comparison and reference purposes, the paper also presents a brief outline of several common equation-solving strategies as an Appendix.

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.

Authenticating Multicast Streams in Lossy Channels Using Threshold Techniques

We first classify the state-of-the-art stream authentication problem in the multicast environment and group them into Signing and MAC approaches. A new approach for authenticating digital streams using Threshold Techniques is introduced. The new approach main advantages are in tolerating packet loss, up to a threshold number, and having a minimum space overhead. It is most suitable for multicast applications running over lossy, unreliable communication channels while, in same time, are pertain the security requirements. We use linear equations based on Lagrange polynomial interpolation and Combinatorial Design methods.

Algebraic attacks on SOBER-t32 and SOBER-t16 without stuttering

This paper presents algebraic attacks on SOBER-t32 and SOBER-t16 without stuttering. For unstuttered SOBER-t32, two different attacks are implemented. In the first attack, we obtain multivariate equations of degree 10. Then, an algebraic attack is developed using a collection of output bits whose relation to the initial state of the LFSR can be described by low-degree equations. The resulting system of equations contains 2^69 equations and monomials, which can be solved using the Gaussian elimination with the complexity of 2^196.5. For the second attack, we build a multivariate equation of degree 14. We focus on the property of the equation that the monomials which are combined with output bit are linear. By applying the Berlekamp-Massey algorithm, we can obtain a system of linear equations and the initial states of the LFSR can be recovered. The complexity of attack is around O(2^100) with 2^92 keystream observations. The second algebraic attack is applicable to SOBER-t16 without stuttering. The attack takes around O(2^85) CPU clocks with 2^78 keystream observations.

Cheating prevention in linear secret sharing

Cheating detection in linear secret sharing is considered. The model of cheating extends the Tompa-Woll attack and includes cheating during multiple (unsuccessful) recovery of the secret. It is shown that shares in most linear schemes can be split into subshares. Subshares can be used by participants to trade perfectness of the scheme with cheating prevention. Evaluation of cheating prevention is given in the context of different strategies applied by cheaters.

Distinguishing attack on SOBER-128 with linear masking

We present a distinguishing attack against SOBER-128 with linear masking. We found a linear approximation which has a bias of 2^− − 8.8 for the non-linear filter. The attack applies the observation made by Ekdahl and Johansson that there is a sequence of clocks for which the linear combination of some states vanishes. This linear dependency allows that the linear masking method can be applied. We also show that the bias of the distinguisher can be improved (or estimated more precisely) by considering quadratic terms of the approximation. The probability bias of the quadratic approximation used in the distinguisher is estimated to be equal to O(2^− − 51.8), so that we claim that SOBER-128 is distinguishable from truly random cipher by observing O(2^103.6) keystream words.

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.

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.

Generalized azimuthal shear deformations in compressible isotropic elastic materials

In this article we study the azimuthal shear deformations in a compressible Isotropic elastic material. This class of deformations involves an azimuthal displacement as a function of the radial and axial coordinates. The equilibrium equations are formulated in terms of the Cauchy-Green strain tensors, which form an overdetermined system of partial differential equations for which solutions do not exist in general. By means of a Legendre transformation, necessary and sufficient conditions for the material to support this deformation are obtained explicitly, in the sense that every solution to the azimuthal equilibrium equation will satisfy the remaining two equations. Additionally, we show how these conditions are sufficient to support all currently known deformations that locally reduce to simple shear. These conditions are then expressed both in terms of the invariants of the Cauchy-Green strain and stretch tensors. Several classes of strain energy functions for which this deformation can be supported are studied. For certain boundary conditions, exact solutions to the equilibrium equations are obtained. © 2005 Society for Industrial and Applied Mathematics.

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.

Nonlinear Model Predictive Control for a multi-rotor with heavy slung load

In this paper a novel controller for stable and precise operation of multi-rotors with heavy slung loads is introduced. First, simplified equations of motions for the multi-rotor and slung load are derived. The model is then used to design a Nonlinear Model Predictive Controller (NMPC) that can manage the highly nonlinear dynamics whilst accounting for system constraints. The controller is shown to simultaneously track specified waypoints whilst actively damping large slung load oscillations. A Linear-quadratic regulator (LQR) controller is also derived, and control performance is compared in simulation. Results show the improved performance of the Nonlinear Model Predictive Control (NMPC) controller over a larger flight envelope, including aggressive maneuvers and large slung load displacements. Computational cost remains relatively small, amenable to practical implementation. Such systems for small Unmanned Aerial Vehicles (UAVs) may provide significant benefit to several applications in agriculture, law enforcement and construction.

Computational strategies for surface fitting using thin plate spline finite element methods

Thin plate spline finite element methods are used to fit a surface to an irregularly scattered dataset [S. Roberts, M. Hegland, and I. Altas. Approximation of a Thin Plate Spline Smoother using Continuous Piecewise Polynomial Functions. SIAM, 1:208--234, 2003]. The computational bottleneck for this algorithm is the solution of large, ill-conditioned systems of linear equations at each step of a generalised cross validation algorithm. Preconditioning techniques are investigated to accelerate the convergence of the solution of these systems using Krylov subspace methods. The preconditioners under consideration are block diagonal, block triangular and constraint preconditioners [M. Benzi, G. H. Golub, and J. Liesen. Numerical solution of saddle point problems. Acta Numer., 14:1--137, 2005]. The effectiveness of each of these preconditioners is examined on a sample dataset taken from a known surface. From our numerical investigation, constraint preconditioners appear to provide improved convergence for this surface fitting problem compared to block preconditioners.

Surface reconstruction of wheat leaf morphology from three-dimensional scanned data

Realistic virtual models of leaf surfaces are important for a number of applications in the plant sciences, such as modelling agrichemical spray droplet movement and spreading on the surface. In this context, the virtual surfaces are required to be sufficiently smooth to facilitate the use of the mathematical equations that govern the motion of the droplet. While an effective approach is to apply discrete smoothing D2-spline algorithms to reconstruct the leaf surfaces from three-dimensional scanned data, difficulties arise when dealing with wheat leaves that tend to twist and bend. To overcome this topological difficulty, we develop a parameterisation technique that rotates and translates the original data, allowing the surface to be fitted using the discrete smoothing D2-spline methods in the new parameter space. Our algorithm uses finite element methods to represent the surface as a linear combination of compactly supported shape functions. Numerical results confirm that the parameterisation, along with the use of discrete smoothing D2-spline techniques, produces realistic virtual representations of wheat leaves.