Precise description of the different far fields encountered in the problem of diffraction of acoustic waves by a quarter-plane

This paper provides a review of important results concerning the Geometrical Theory of Diffraction and Geometrical Optics. It also reviews the properties of the existing solution for the problem of diffraction of a time harmonic plane wave by a half-plane. New mathematical expressions are derived for the wave fields involved in the problem of diffraction of a time harmonic plane wave by a quarter-plane, including the secondary radiated waves. This leads to a precise representation of the diffraction coefficient describing the diffraction occurring at the corner of the quarter-plane. Our results for the secondary radiated waves are an important step towards finding a formula giving the corner diffraction coefficient everywhere. © 2012 The authors.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

The rolling ball problem on the plane revisited

By a sequence of rollings without slipping or twisting along segments of a straight line of the plane, a spherical ball of unit radius has to be transferred from an initial state to an arbitrary final state taking into account the orientation of the ball. We provide a new proof that with at most 3 moves, we can go from a given initial state to an arbitrary final state. The first proof of this result is due to Hammersley ( 1983). His proof is more algebraic than ours which is more geometric. We also showed that generically no one of the three moves, in any elimination of the spin discrepancy, may have length equal to an integral multiple of 2 pi.

A smooth evolutionary structural optimization procedure applied to plane stress problem

Topological optimization problems based on stress criteria are solved using two techniques in this paper. The first technique is the conventional Evolutionary Structural Optimization (ESO), which is known as hard kill, because the material is discretely removed; that is, the elements under low stress that are being inefficiently utilized have their constitutive matrix has suddenly reduced. The second technique, proposed in a previous paper, is a variant of the ESO procedure and is called Smooth ESO (SESO), which is based on the philosophy that if an element is not really necessary for the structure, its contribution to the structural stiffness will gradually diminish until it no longer influences the structure; its removal is thus performed smoothly. This procedure is known as "soft-kill"; that is, not all of the elements removed from the structure using the ESO criterion are discarded. Thus, the elements returned to the structure must provide a good conditioning system that will be resolved in the next iteration, and they are considered important to the optimization process. To evaluate elasticity problems numerically, finite element analysis is applied, but instead of using conventional quadrilateral finite elements, a plane-stress triangular finite element was implemented with high-order modes for solving complex geometric problems. A number of typical examples demonstrate that the proposed approach is effective for solving problems of bi-dimensional elasticity. (C) 2014 Elsevier Ltd. All rights reserved.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

On the numerical solution of a Cauchy problem in an elastostatic half-plane with a bounded inclusion

We propose an iterative procedure for the inverse problem of determining the displacement vector on the boundary of a bounded planar inclusion given the displacement and stress fields on an infinite (planar) line-segment. At each iteration step mixed boundary value problems in an elastostatic half-plane containing the bounded inclusion are solved. For efficient numerical implementation of the procedure these mixed problems are reduced to integral equations over the bounded inclusion. Well-posedness and numerical solution of these boundary integral equations are presented, and a proof of convergence of the procedure for the inverse problem to the original solution is given. Numerical investigations are presented both for the direct and inverse problems, and these results show in particular that the displacement vector on the boundary of the inclusion can be found in an accurate and stable way with small computational cost.

Anti-Periodic Boundary Value Problem for Impulsive Fractional Integro Differential Equations

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37

Enhancing road safety for young drivers : how Graduate Driver Licensing initiatives can complement "anti-hooning" legislation

Street racing and associated (hooning) behaviours have attracted increasing concern in recent years. While New Zealand and all Australian jurisdictions have introduced “antihooning” legislation and allocated significant police resources to managing the problem, there is limited evidence of the road safety implications of hooning. However, international and Australian data suggests that drivers charged with a hooning offence tend to be young males who are accompanied by one or more peers, and hooning-related crashes tend to occur at night. In this regard, there is considerable evidence that drivers under the age of 25 are over-represented in crash statistics, and are particularly vulnerable soon after obtaining a Provisional licence, when driving at night, and when carrying peer-aged passengers. The similarity between the nature of hooning offenders, offences and crashes, and road safety risks for young drivers in general, suggests that hooning is an issue that may be viewed as part of the broader young driver problem. Many jurisdictions have recently implemented a range of evidence-based strategies to address young driver road safety, and this paper will present Queensland crash and offence data to highlight the potential benefit of Graduated Driver Licensing initiatives, such as night driving restrictions and peer-aged passenger restrictions, to related road safety issues, including hooning. An understanding of potential flow-on effects is important for evaluations of anti-hooning legislation and Graduated Driver Licensing programs, and may have implications for future law enforcement resource allocation and policy development.

Problem-solving courts, therapeutic jurisprudence and the constitution : if two is company, is three a crowd?

Court costs, resource-intensive trials, booming prison populations and the obduracy of recidivism rates all present as ugly excesses of the criminal law adversarial paradigm. To combat these excesses, problem-solving courts have evolved with an edict to address the underlying issues that have caused an individual to commit a crime. When a judge seeks to help a problem-solving court participant deal with issues like addiction, mental health or poverty, they are performing a very different role to that of a judicial officer in the traditional court hierarchy. They are no longer the removed, independent arbiter — a problem-solving court judge steps into the ‘arena’ with the participant and makes active use of their judicial authority to assist in rehabilitation and positive behavioural change. Problem-solving court judges employing the principles of therapeutic jurisprudence appreciate that their interaction with participants can have therapeutic and anti-therapeutic consequences. This article will consider how the deployment of therapeutic measures (albeit with good intention) can lead to the behavioural manifestation of partiality and bias on the part of problem-solving court judges. Chapter III of the Commonwealth Constitution will then be analysed to highlight why the operation and functioning of problem solving courts may be deemed unconstitutional. Part IV of this article will explain how a problem-solving court judge who is not acting impartially or independently will potentially contravene the requirements of the Constitution. It will finally be suggested that judges who possess a high level of emotional intelligence will be the most successful in administering an independent and impartial problem solving court.

In the shadow of the enlightenment : Le Corbusier, Le Faisceau and Georges Valois

On 9 January 1927 Le Corbusier materialised on the front cover of the Faisceau journal edited by Georges Valois Le Nouveau Siècle which printed the single-point perspective of Le Corbusier’s Plan Voisin and an extract from the architect’s discourse in Urbanisme. In May Le Corbusier presented slides of his urban designs at a fascist rally. These facts have been known ever since the late 1980s when studies emerged in art history that situated Le Corbusier’s philosophy in relation to the birth of twentieth-century fascism in France—an elision in the dominant reading of Le Corbusier’s philosophy, as a project of social utopianism, whose received genealogy is Saint-Simon and Charles Fourier. Le Corbusier participated with the first group in France to call itself fascist, Valois’s militant Faisceau des Combattants et Producteurs, the “Blue Shirts,” inspired by the Italian “Fasci” of Mussolini. Thanks to Mark Antliff, we know the Faisceau did not misappropriate Le Corbusier’s plans, in some remote quasi-symbolic sense, rather Valois’s organisation was premised on the redesign of Paris based on Le Corbusier’s schematic designs. Le Corbusier’s Urbanisme was considered the “prodigious” model for the fascist state Valois called La Cité Française – after his mentor the anarcho-syndicalist Georges Sorel. Valois stated that Le Corbusier’s architectural concepts were “an expression of our profoundest thoughts,” the Faisceau, who “saw their own thought materialized” on the pages of Le Corbusier’s plans. The question I pose is, In what sense is Le Corbusier’s plan a complete representation of La Cité? For Valois, the fascist city “represents the collective will of La Cité” invoking Enlightenment philosophy, operative in Sorel, namely Rousseau, for whom the notion of “collective will” is linked to the idea of political representation: to ‘stand in’ for someone or a group of subjects i.e. the majority vote. The figures in Voisin are not empty abstractions but the result of “the will” of the “combatant-producers” who build the town. Yet, the paradox in anarcho-syndicalist anti-enlightenment thought – and one that became a problem for Le Corbusier – is precisely that of authority and representation. In Le Corbusier’s plan, the “morality of the producers” and “the master” (the transcendent authority that hovers above La Cité) is lattened into a single picture plane, thereby abolishing representation. I argue that La Cité pushed to the limits of formal abstraction by Le Corbusier thereby reverts to the Enlightenment myth it first opposed, what Theodor Adorno would call the dialectic of enlightenment.

Examining gender differences in drivers' thoughts and feelings about different outcomes of speeding: Implications for future anti-speeding messages

Speeding remains a pervasive road safety problem, increasing both crash frequency and severity. Advertising countermeasures which aim to change individuals’ attitudes and behaviours are a key component in the array of countermeasures aimed at reducing this risky behaviour. Enhancing individuals’ perceptions of the personal relevance of such messages is important for increasing persuasiveness. This study examined what males and females reported as the most concerning aspects associated with (i) receiving a speeding fine, (ii) losing one’s license, and (iii) being involved in a crash. For each of these outcomes, a range of specific and appropriate aspects were assessed. For instance, in relation to receiving a fine, individuals reported the extent to which they would, for example, feel concerned about losing demerit points and paying more in insurance premiums. An online survey of 751 drivers (579 males; 16-79 years) was administered. When controlling for age, overall significant gender differences were found in relation to two of the three outcomes; receiving a fine and being in a crash. Follow-up tests of univariate effects revealed that females consistently reported being significantly more concerned than males on all aspects. Thus, for being fined, females were significantly more concerned with, for example, being caught and receiving a ticket in the mail; while, for being in a crash, specific aspects included, for example, injuring/killing oneself and seeing oneself as not a good/safe driver. The findings are discussed in terms of their implications for developing well-targeted messages aimed at discouraging drivers from speeding.

Anti-wind-up designs for dynamic positioning of marine vehicles with control allocation

This paper presents a framework for the design of a joint motion controller and a control allocation strategy for dynamic positioning of marine vehicles. The key aspects of the proposed designs are a systematic approach to deal with actuator saturation and to inform the motion controller about saturation. The proposed system uses a mapping that translates the actuator constraint sets into constraint sets at the motion controller level. Hence, while the motion controller addresses the constraints, the control allocation algorithm can solve an unconstrained optimisation problem. The constrained control design is approached using a multivariable anti-wind-up strategy for strictly proper controllers. This is applicable to the implementation of PI and PID type of motion controllers.

Broadcast anti-jamming systems

In a traditional anti-jamming system a transmitter who wants to send a signal to a single receiver spreads the signal power over a wide frequency spectrum with the aim of stopping a jammer from blocking the transmission. In this paper, we consider the case that there are multiple receivers and the transmitter wants to broadcast a message to all receivers such that colluding groups of receivers cannot jam the reception of any other receiver. We propose efficient coding methods that achieve this goal and link this problem to well-known problems in combinatorics. We also link a generalisation of this problem to the Key Distribution Pattern problem studied in combinatorial cryptography.