Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

Legal issues in the design and implementation of environmental offset and environmental trading frameworks

The Queensland Government released its new Environmental Offset Policy in July 2008. This policy creates a set of overarching principles which are to be incorporated into existing environmental offset policy. This article is the final article in a set of three interrelated articles discussing the operation and implementation of environmental offsets in Queensland. The first article discusses the Environmental Offsets Discussion Paper and the existing environmental offset requirements. No significant changes have been made to these existing offset requirements under the new Environmental Offset Policy. This article also touches briefly on the legal issues associated with design and implementation of environmental offset and trading frameworks. The second article considered the compatibility of different land tenure arrangements in Queensland against the requirements for the creation and trade of environmental offsets. The third article being the present article, discusses the application of the new Environmental Offset Policy while also analysing the legal issues associated with environmental offsets in further detail.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

The International regulation of persons displaced by climate change

It is certain that there will be changes in environmental conditions across the globe as a result of climate change. Such changes will require the building of biological, human and infrastructure resilience. In some instances the building of such resilience will be insufficient to deal with extreme changes in environmental conditions and legal frameworks will be required to provide recognition and support for people dislocated because of environmental change. Such dislocation may occur internally within the country of original origin or externally into another State’s territory. International and national legal frameworks do not currently recognise or assist people displaced as a result of environmental factors including displacement occurring as a result of climate change. Legal frameworks developed to deal with this issue will need to consider the legal rights of those people displaced and the legal responsibilities of those countries required to respond to such displacement. The objective of this article is to identify the most suitable international institution to host a program addressing climate displacement. There are a number of areas of international law that are relevant to climate displacement, including refugee law, human rights law and international environmental law. These regimes, however, were not designed to protect people relocating as a result of environmental change. As such, while they indirectly may be of relevance to climate displacement, they currently do nothing to directly address this complex issue. In order to determine the most appropriate institution to address and regulate climate displacement, it is imperative to consider issues of governance. This paper seeks to examine this issue and determine whether it is preferable to place climate displacement programs into existing international legal frameworks or whether it is necessary to regulate this area in an entirely new institution specifically designed to deal with the complex and cross-cutting issues surrounding the topic. Commentators in this area have proposed three different regulatory models for addressing climate displacement. These models include: (a) Expand the definition of refugee under the Refugee Convention to encompass persons displaced by climate change; (b) Implement a new stand alone Climate Displacement Convention; and (c) Implement a Climate Displacement Protocol to the UNFCCC. This article will examine each of these proposed models against a number of criteria to determine the model that is most likely to address the needs and requirements of people displaced by climate change. It will also identify the model that is likely to be most politically acceptable and realistic for those countries likely to attract responsibilities by its implementation. In order to assess whether the rights and needs of the people to be displaced are to be met, theories of procedural, distributive and remedial justice will be used to consider the equity of the proposed schemes. In order to consider the most politically palatable and realistic scheme, reference will be made to previous state practice and compliance with existing obligations in the area. It is suggested that the criteria identified by this article should underpin any future climate displacement instrument.

The legal nature of justice in REDD+

Formation of Reduced Emissions from Deforestation and Degradation (REDD+) policy within the international climate regime has raised a number of discussions about ‘justice’. REDD+ aims to provide an incentive for developing countries to preserve or increase the amount of carbon stored in their forested areas. Governance of REDD+ is multi-layered: at the international level, a guiding framework must be determined; at the national level, strong legal frameworks are a pre-requisite to ensure both public and private investor confidence and at the sub-national level, forest-dependent peoples need to agree to participate as stewards of forest carbon project areas. At the international level the overall objective of REDD+ is yet to be determined, with competing mitigation, biological and justice agendas. Existing international law pertaining to the environment (international environmental principles and law, IEL) and human rights (international human rights law, IHRL) should inform the development of international and national REDD+ policy especially in relation to ensuring the environmental integrity of projects and participation and benefit-sharing rights for forest dependent communities. National laws applicable to REDD+ must accommodate the needs of all stakeholders and articulate boundaries which define their interactions, paying particular attention to ensuring that vulnerable groups are protected. This paper i) examines justice theories and IEL and IHRL to inform our understanding of what ‘justice’ means in the context of REDD+, and ii) applies international law to create a reference tool for policy-makers dealing with the complex sub-debates within this emerging climate policy. We achieve this by: 1) Briefly outlining theories of justice (for example – perspectives offered by anthropogenic and ecocentric approaches, and views from ‘green economics’). 2) Commenting on what ‘climate justice’ means in the context of REDD+. 3) Outlining a selection of IEL and IHRL principles and laws to inform our understanding of ‘justice’ in this policy realm (for example – common but differentiated responsibilities, the precautionary principle, sovereignty and prevention drawn from the principles of IEL, the UNFCCC and CBD as relevant conventions of international environmental law; and UNDRIP and the Declaration on the Right to Development as applicable international human rights instruments) 4) Noting how this informs what ‘justice’ is for different REDD+ stakeholders 5) Considering how current law-making (at both the international and national levels) reflects these principles and rules drawn from international law 6) Presenting how international law can inform policy-making by providing a reference tool of applicable international law and how it could be applied to different issues linked to REDD+. As such, this paper will help scholars and policy-makers to understand how international law can assist us to both conceptualise and embody ‘justice’ within frameworks for REDD+ at both the international and national levels.

Creating 'saviour siblings' : the notion of harming by conceiving in the context of healthy children

Over the past decade there have been a number of families who have utilised assisted reproductive technologies (ARTs) to create a tissue-matched child, with the purpose of using the child’s tissue to cure an existing sick child. This inevitably brings such families a sense of hope as the ultimate aim is to overcome a family health crisis. However, this specific use of reproductive technologies has been the subject of significant criticism, most of which is levelled against the potential harm to the ‘saviour’ child. In Australia, families seeking to access reproductive technologies in this context are therefore required to justify their motives to an ethics committee in order to establish, amongst other things, whether the child will suffer harm once born. This paper explores the concept of harm in the context of conception, focusing on whether it is possible to ‘harm’ a healthy child who has been conceived to save another. To achieve this, the paper will evaluate the impact of the ‘non-identity’ principle in the ‘saviour sibling’ context, and assess the existing body of literature which addresses ‘harm’ in the context of conception. As will be established, the majority of such literature has focused on ‘wrongful life’ cases which seek to address whether an existing child who has been born with a disability, has been harmed. Finally, this paper will distinguish the harm arguments in the ‘saviour sibling’ context based on the fact that the harm evaluation concerns the ‘future-life’ assessment of a healthy child.

Peer assisted learning, skills development and Generation Y : a case study of a first year undergraduate law unit

The majority of current first year university students belong to Generation Y. Consequently, research suggests that, in order to more effectively engage them, their particular learning preferences should be acknowledged in the organisation of their learning environments and in the support provided. These preferences are reflected in the Torts Student Peer Mentor Program, which, as part of the undergraduate law degree at the Queensland University of Technology, utilises active learning, structured sessions and teamwork to supplement student understanding of the substantive law of Torts with the development of life-long skills. This article outlines the Program, and its relevance to the learning styles and experiences of Generation Y first year law students transitioning to university, in order to investigate student perceptions of its effectiveness – both generally and, more specifically, in terms of the Program’s capacity to assist students to develop academic and work-related skills.

Legal practitioners’ negligence liability for claim investigation and advice

Nigam v Harm (No 2) [2011] WASCA 221, Western Australia Court of Appeal, 18 October 2011

Collective sampling and analysis of high order tensors for chatroom communications

This work investigates the accuracy and efficiency tradeoffs between centralized and collective (distributed) algorithms for (i) sampling, and (ii) n-way data analysis techniques in multidimensional stream data, such as Internet chatroom communications. Its contributions are threefold. First, we use the Kolmogorov-Smirnov goodness-of-fit test to show that statistical differences between real data obtained by collective sampling in time dimension from multiple servers and that of obtained from a single server are insignificant. Second, we show using the real data that collective data analysis of 3-way data arrays (users x keywords x time) known as high order tensors is more efficient than centralized algorithms with respect to both space and computational cost. Furthermore, we show that this gain is obtained without loss of accuracy. Third, we examine the sensitivity of collective constructions and analysis of high order data tensors to the choice of server selection and sampling window size. We construct 4-way tensors (users x keywords x time x servers) and analyze them to show the impact of server and window size selections on the results.

The doctrine of command responsibility in the midst of war : legal fallacy or reflection of reality

The history of war is blighted with astonishing reminders of man’s ability to cast his sense of humanity aside and inflict unspeakable harm upon one another. The ruthless bombing of Dresden, the callousness of the Nazi concentration camps and the massacre of the Tutsis are but a few of the atrocities that have haunted our past. In response to these atrocities, society has imposed an ever-increasing number of laws and rules to regulate warfare. Amongst these is the doctrine of command responsibility. The doctrine of command responsibility states that a commander is criminally liable for the crimes of his subordinates if he knew or should have known of their crimes. This paper will examine whether the doctrine is an appropriate and realistic legal standard to hold commanders accountable to or whether the doctrine is more a reflection of social sentiment and legal rhetoric. If the doctrine, and indeed the law of war in general, is unrealistic then the law cannot fulfil its purpose - that is, the prevention of atrocities. Instead of being solely a reflection of moral authority and social sentiments the law must also be a tool that guides and shapes the decisions and actions of the military through the chaotic and brutal nature of war...

A framework for a dense matching algorithm : Results using matching metrics and order statistic filters

An algorithm for computing dense correspondences between images of a stereo pair or image sequence is presented. The algorithm can make use of both standard matching metrics and the rank and census filters, two filters based on order statistics which have been applied to the image matching problem. Their advantages include robustness to radiometric distortion and amenability to hardware implementation. Results obtained using both real stereo pairs and a synthetic stereo pair with ground truth were compared. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.

Order statistic filters for image matching

The rank and census are two filters based on order statistics which have been applied to the image matching problem for stereo pairs. Advantages of these filters include their robustness to radiometric distortion and small amounts of random noise, and their amenability to hardware implementation. In this paper, a new matching algorithm is presented, which provides an overall framework for matching, and is used to compare the rank and census techniques with standard matching metrics. The algorithm was tested using both real stereo pairs and a synthetic pair with ground truth. The rank and census filters were shown to significantly improve performance in the case of radiometric distortion. In all cases, the results obtained were comparable to, if not better than, those obtained using standard matching metrics. Furthermore, the rank and census have the additional advantage that their computational overhead is less than these metrics. For all techniques tested, the difference between the results obtained for the synthetic stereo pair, and the ground truth results was small.

Legal Profession Act 2007 : entitlement to itemised bill – meaning of “client” – meaning of “third party payer”

In Legal Services Commissioner and Wright [2010] QSC 168 and Amos v Ian K Fry & Company, the Supreme Court of Queensland considered the scope of some of the provisions of the Legal Profession Act 2007 (Qld), including the definition of “third party payer” in s 301 of the Act.