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This paper discusses a two-year study at an Australian university in which 154 undergraduate and 51 postgraduate students reflected on their experiences with flexible and personalised assessment where they could choose assessment tasks, submission dates and weightings of their assignments. Through pre- and post-course surveys and a focus group, feelings of empowerment and attitudes towards flexibility were investigated. Students were positive about all aspects of the flexible approach and felt it was beneficial for their learning. Only the postgraduate group showed significant improvement in feelings of empowerment and grades suggesting that more experienced learners may adapt more easily to this model. We argue that empowering learners is not about better grades but about students gaining more input, voice and control in their learning and assessment. Despite the disadvantages of increased workload for teachers and a decision-making burden on students, the potential benefits for students justify consideration of more personalised and flexible assessment. [ABSTRACT FROM AUTHOR]

This paper discusses Ethiopia's planned climate adaptation interventions and the barriers that impede implementation of adaptation policies at the local level by using the case study of Raya Azebo district. Data was collected through reviews of policy documents, focus group discussions with farmers and interviews with relevant government actors. Results indicate that climate change is addressed in various policy documents but there is limited progress in implementation of these policies. The study identified various barriers to climate adaptation policy implementation which included a lack of financial resources, poor coordination among institutional actors and local actors' low technical capacities for addressing climate change. The study contributes to the literature of climate change policy planning and implementation in low-income and lower-middle-income countries and suggests measures to overcome the existing barriers to climate change adaptation policies. [ABSTRACT FROM AUTHOR]

Combinatorial vector fields on simplicial complexes introduced by Robin Forman constitute a combinatorial analogue of classical flows. They have found numerous and varied applications in recent years. Yet, their formal relationship to classical dynamical systems has been less clear. In this paper we prove that for every combinatorial vector field on a finite simplicial complex X one can construct a semiflow on the underlying polytope X which exhibits the same dynamics. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information. [ABSTRACT FROM AUTHOR]

Point cloud data arises naturally from 3-dimensional scanners, LiDAR sensors, and industrial computed tomography (i.e. CT scans) among other sources. Most point clouds obtained through experimental means exhibit some level of noise, inhibiting mesh reconstruction algorithms and topological data analysis techniques. To alleviate the problems caused by noise, smoothing algorithms are often employed as a preprocessing step. Moving least squares is one such technique, however, many of these techniques are designed to work on surfaces in R3. As interesting point clouds can naturally live in higher dimensions, we seek a method for smoothing higher dimensional point clouds. To this end, we turn to the distance to measure function. In this paper, we provide a theoretical foundation for studying the gradient flow induced by the squared distance to measure function, as introduced by Chazal, Cohen-Steiner, and Mérigot. In particular, we frame the gradient flow as a Filippov system and find a method for solving the squared distance to measure gradient flow, induced by the uniform empirical measure, using higher order Voronoi diagrams. In contrast to some existing techniques, this gradient flow provides a smoothing algorithm which computationally scales with dimensionality. [ABSTRACT FROM AUTHOR]

Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer‐assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches. The use of computer‐assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates. [ABSTRACT FROM AUTHOR]

Sobolev spaces and their embedding properties have long been of central importance in the study of partial differential equations, particularly for the classical theoretical analysis of both linear and nonlinear problems. In recent years, computer‐assisted proof techniques have been developed for obtaining existence and uniqueness proofs of solutions to a variety of nonlinear partial differential equations that arise in applications, and they have led to a number of new results that currently lie beyond the reach of classical analytical approaches. The use of computer‐assisted methods, however, frequently relies on the explicit knowledge of a variety of embedding constants for Sobolev spaces. In the present paper, we show that in the context of certain Sobolev space Banach algebras, these constants themselves can be bounded rigorously and precisely using validated computations based on interval arithmetic, combined with analytical error estimates. [ABSTRACT FROM AUTHOR]

Self and peer-assessment are becoming central aspects of student-centred assessment processes in higher education. Despite increasing evidence that both forms of assessment are helpful for developing key capabilities in students, such as taking more responsibility for their own learning, developing a better understanding of the subject matter, assessment criteria and their own values and judgements, and developing critical reflection skills, both forms of assessment are still not the norm at universities. This paper provides the findings of a two-year study of formative self and peerassessment at an Australian university. The study supports other research showing that students tend to regard formative self and peer-assessment as beneficial for gaining more insights about the assessment process and for improving their own work. We argue that self and peer-assessment requires careful design and implementation for it to be an effective tool for formative assessment processes; and that the development of students'capacities for giving feedback, and the continuous and timely involvement of the teacher, are central aspects for successful self and peer-assessment. The move to self and peer-assessment is not simple for teachers and students but is worthwhile and necessary for twenty-first century higher education. [ABSTRACT FROM AUTHOR]

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function. [ABSTRACT FROM AUTHOR]

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state and local and global energy minimizers. The proof of the above statement is conceptually simple and combines several techniques from some of the authors' and Zgliczynski's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic connections at various nontrivial parameter values. [ABSTRACT FROM AUTHOR]

In September 2015, Australia, along with 193 member states of the United Nations, signed the United Nations Sustainable Development Goals. The facilitation of international trade and increasing foreign aid for developing countries were emphasised as crucial means for achieving the Sustainable Development Goals. An important role was assigned to the international Aid for Trade initiative, which is about stimulating economic growth in developing countries through removing constraints to trade. Australia has been a strong supporter of the Aid for Trade initiative since it was launched at the World Trade Organization's Ministerial Conference in Hong Kong in 2005. Aid for trade has become a central plank of the 'new development paradigm' of Australia's international aid program. This article analyses the conceptualisation and practice of aid for trade in Australia's aid, with a focus on how it links to ecologically sustainable development. It argues that Australia's aid for trade is reinforcing the neo-liberal development paradigm in which environmental dimensions are overall neglected and private sector development and free trade are prioritised. In order to achieve international and national development goals of poverty reduction and sustainable development, environmental sustainability needs to be fully integrated into the growing aid for trade portfolio of Australia's international aid. [ABSTRACT FROM AUTHOR]

Human rights-based approaches (HRBAs) can challenge the underlying structures and power relations that perpetuate poverty. They have thus emerged in the development field as a prominent instrument for addressing development issues. Access to clean water and sanitation are now internationally acknowledged as human rights, and have become a stand-alone Sustainable Development Goal of the international community’s commitment to international development. This paper analyses the potential use of HRBAs by local Non-Governmental Organisations (NGOs) working on sanitation issues in slums in Mumbai. It is argued that it is more productive for local NGOs to build (i) partnerships with duty-bearers (in this case the state and the Municipal Corporation of Greater Mumbai) and (ii) the capacity of rights-holders, in particular women, than to rely on litigation strategies to create momentum for change. HRBAs are more useful as a political tool for NGOs for establishing good working relationships with government agencies rather than as a legal instrument, which can be counter-productive to the poverty reduction objectives of NGOs. [ABSTRACT FROM PUBLISHER]

Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn–Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations. [ABSTRACT FROM AUTHOR]

The use of rigorous verification methods is a powerful tool which permits progress in the analysis of dynamical processes that is not possible using purely analytical techniques. In this paper we develop a set of tools for branch validation, which allows for the rigorous verification of branch behavior, bifurcation, and solution index on branches generated through a saddle-node bifurcation. While the presented methodology can be applied in a variety of settings, we illustrate the use of these tools in the context of materials science. In particular, lattice models have been proposed as a more realistic reflection of the behavior of materials than traditional continuum models. For example, unlike their continuum counterparts, lattice models can account for phenomena such as pinning, and a significant body of work has been developed to study traveling waves. However, in a variety of other contexts such as bifurcation theory, questions about lattice dynamical systems are significantly harder to answer than those for continuum models. In t he present paper, we show that computer-assisted proof techniques can be used to answer some of these questions. We apply these tools to the discrete Allen-Cahn equation, giving us results on the existence of branches of mosaic solutions and their robustness as it relates to grain size. We also demonstrate that there are situations in which classical continuation methods can fail to identify the correct branching behavior. [ABSTRACT FROM AUTHOR]

Flexible teaching and learning and the ‘flipped classroom’ are current buzzwords in higher education in Australia and elsewhere in the world. They are reflections of the progressive change in higher education over the last few decades towards more student-and learning centred pedagogies and practices, which are made possible through new technologies and more delivery of online and blended (combination of face-to-face and online components) courses. The increasing personalising and flexibility of learning in higher education requires equal attention spent to assessment practices to ensure a cohesive learning experience. This paper provides the findings and conclusions of a study about a flipped classroom, which also included flexible assessment components. The study showed that students enjoy and are more engaged in a flipped classroom, prefer a blended learning to a fully online learning approach, want and require clear structure and guidelines, and strongly value flexible assessment through more choices and control. The main concern of higher education teachers is the time commitment and lack of institutional support for flipping classrooms and providing flexible assessment. It is argued that personalising learning requires more personalising of assessment, and that it is mainly the responsibility of teachers and institutions to develop ‘flexible students’. [ABSTRACT FROM AUTHOR]

This paper analyses the rapidly emerging discourse of a green economy based on green growth. It highlights inherent conflicts and contradictions of this discourse such as the myth of decoupling growth from the environment, pollution generations and resource consumption. Using key theoretical constructs of both Gramsci and Polanyi, the paper argues that the green economy/growth discourse can be seen as a Gramscian ‘passive revolution’ whereby the dominant sustainable development discourse, subsumed by capitalist hegemony, is protected in the context of global environmental, economic and development crises. The ‘neoliberalising of nature’, or in other words, the privatisation, marketisation and commodification of nature, akin to Polanyi's fictitious commodities, continues and intensifies with green economy/growth strategies. Greening the economy and associated strategies of green growth divert attention from the social and political dimensions of sustainability and issues of social and international justice. In this way, the inexorable march or ‘sustainable development’ of neoliberal capitalism is maintained. This paper goes on to argue that contesting the claims of green economy/growth discourse through political struggles by civil society against the neoliberalisation of nature is a sign of a slowly emerging counter-hegemonic ‘double movement’ which challenges capitalist hegemony and the commodification of society–nature relations. [ABSTRACT FROM PUBLISHER]

The inclusion of men and masculinities in gender and development policies and practices has emerged in the last decade as a critical component for achieving gender equality and women's empowerment. This article analyses the contemporary character and progress towards 'men-streaming' of gender and development, and argues that implicit in any action towards gender equality and working with men is the requirement to base action on a clear understanding of the politics of masculinities and the relational aspects of gender equity and equality. [ABSTRACT FROM AUTHOR]

Isolated invariant sets and their associated Conley indices are valuable tools for studying dynamical systems and their global invariant structures. Through their design, they aim to capture invariant behavior which is robust under small perturbations, and this in turn makes them amenable to a computational treatment. Over the years, a number of algorithms have been proposed tofind index pairs for an isolated invariant set and then to use an index pair to compute the associated Conley index. Nevertheless, most of these methods are restricted to discrete, albeit possibly multivalued, dynamical systems. Only relatively few general methods exist for dynamical systems generated by differential equations. In this paper, we present a new method for finding and rigorously verifying a special type of index pairs, namely isolating blocks and their exit sets. Our method makes use of a recently developed adaptive algorithm for rigorously determining the location of nodal sets of smooth functions, which combines an adaptive subdivision technique with interval arithmetic. By characterizing an exit set as a nodal domain, we are able to determine a valid index pair and proceed to compute its Conley index. Our method is illustrated using several examples for three-dimensional flows. [ABSTRACT FROM AUTHOR]

Topology is a natural mathematical tool for quantifying complex structures. In many applications, such as, for example, in the context of phase-field models in materials science, the structures of interest arise as sub- or superlevel sets of continuous functions, i.e., as nodal domains. From a computational point of view, any attempt at constructing a truthful representation of the topology of nodal domains has to involve a discretization step, and it is natural to wonder whether this step introduces topological artifacts. In this paper, we present a randomized subdivision algorithm which, given a smooth function, constructs an adaptive rectangular grid containing the essential information necessary for approximating nodal domains. Furthermore, under mild regularity assumptions the algorithm will also provide a computer-assisted proof for the correctness of the approximation by showing that the rectangular grid can be used to construct rectangular complexes which are homotopy equivalent to the nodal domains of the function. Our method extends the results of [S. Day, W. D. Kalies, and T. Wanner, Multiscale Model. Simul., 7 (2009), pp. 1695-1726], by employing a more accurate and efficient interval arithmetic range enclosure algorithm, as well as developing a randomized subdivision technique to virtually eliminate grid alignment effects. [ABSTRACT FROM AUTHOR]

This article analyses emerging approaches by donors to risk management, focusing on the case of the Australian Agency for International Development (AusAID). It suggests that AusAID's understanding and management of risk are expressions of particular political and social interests and, subsequently, that its approach to risk management is inherently political in nature. While AusAID's policies and guidelines on risk management identify a wide range of risks as being relevant to the agency's work – including risks to poverty reduction and sustainable development – in practice it has focused on managing risks to the building of neo-liberal markets, both within Australia and in developing countries, and the Australian government's foreign policy objectives. To illustrate these points, we present an analysis of AusAID's approach to risk management in a major economic governance project, the Philippines-Australia Partnership for Economic Governance Reforms. We conclude by indicating ways in which donor approaches to risk management can be revised to give greater attention to the needs of the poor and civil society, while acknowledging that such revisions may be contingent upon prior changes in the structure of power and interest and/or institutional characteristics of aid policy-making processes in donor countries. [ABSTRACT FROM PUBLISHER]

Abstract: We present an algorithm for computing the homology of inclusion maps which is based on the idea of coreductions and leads to significant speed improvements over current algorithms. It is shown that this algorithm can be extended to compute both persistent homology and an extension of the persistence concept to two-sided filtrations. In addition to describing the theoretical background, we present results of numerical experiments, as well as several applications to concrete problems in materials science. [ABSTRACT FROM AUTHOR]

Abstract: Quantitative homology metrics are proposed for characterizing the thermal–elastic response of polycrystalline materials. Simulations for a calcite-based polycrystal, marble, are used as an illustrative example. The homology metrics are based on topological measurements, such as the number of components and the number of handles of the thermal–elastic response fields for a complex microstructure. These homology metrics are applied to characterize not only the elastic energy density and maximum principal stress response fields in a polycrystal but also the correlated grain-boundary misorientation distributions that influenced the formation of these response fields. It is demonstrated that the topological analysis can quantitatively distinguish between different types of grain-boundary misorientations, as well as between differences in the resulting response fields. [Copyright &y& Elsevier]

Abstract: We present a spectral method for the computation of homoclinic orbits in ordinary differential equations. The method is based on Hermite–Fourier expansions of the complete homoclinic solution and exhibits exponential convergence. In addition, our method can be used to approximate nonlinear functionals which depend on the complete homoclinic solution. This is demonstrated using examples from phase separation dynamics and metastability. [Copyright &y& Elsevier]

Several models have been proposed for describing the formation of animal coat patterns. We consider reaction–diffusion models due to Murray, which rely on a Turing instability for the pattern selection. In this paper, we describe the early stages of the pattern formation process for large domain sizes. This includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one-, two-, and three-dimensional base domains. These results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the Cahn–Hilliard equation. We use techniques of dynamical systems, viewing solutions of the reaction–diffusion model in terms of nonlinear semiflows. Our results are applicable to any parabolic system exhibiting a Turing instability. [Copyright &y& Elsevier]

In this paper, we continue a previous study of the geometric properties of a certain prototype system of nonautonomous difference equations. In fact, we show that by means of the fundamental theorem on the existence of invariant fiber bundles two foliations of the extended state space can be found. This detailed information about the geometric structure of the underlying system is then used to construct transformations establishing the topological equivalence between a given coupled and a completely uncoupled system. [ABSTRACT FROM AUTHOR]

This paper gives theoretical results on spinodal decomposition for the Cahn-Hillard equation. We prove a mechanism which explains why most solutions for the Cahn-Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected. The Cahn-Hilliard equation depends on a small parameter ε, modeling the (atomic scale) interaction length; we quantify the behavior of solutions as ε → 0. Specifically, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance O(ε[sup 2-n/2]) up to a ball of radius R in the H²(Ω)norm, where R is proportional to ε[sup -1+g+n/4] as ε → 0. Here, n ∈ {1, 2, 3} denotes the dimension of the considered domain, and g > 0 can be chosen arbitrarily small. Not only does this approach significantly increase the radius of explanation for spinodal decomposition, but it also gives a clear picture of how the phenomenon occurs. While these results hold for the standard cubic nonlinearity, we also show that considerably better results can be obtained for similar higher order nonlinearities. In particular, we obtain R ∼ ε[sup -2+g+n/2] for every g > 0 by choosing a suitable nonlinearity. [ABSTRACT FROM AUTHOR]

This short contribution comments on parts of a series of two papers [2] and [3] on spinodal decomposition for the Cahn-Hilliard equation In a small neighborhood of a homogeneous equilibrium u = μ of (1), with μ ϵ ℛ. in the spinodal region, we find that a strongly unstable manifold of u has dominating effects. Although this is demonstrated here for the linearized equation only, it will be essential for the nonlinear situation as well (cf. [3]). Furthermore we comment on the notion of a characteristic wavelength for the strongly unstable directions. These patterns indeed correspond to typically observed spinodally decomposed states which arise for instance in materials sciences where u corresponds to the concentration of one of the components of a binary alloy (cf. [1] or the references in [2]).where Ω C ℛ, n ∈ {1,2,3}, is a bounded domain with sufficiently smooth boundary, 0 < ϵ <<1 is a small parameter, and f is cubic-like, for example f (u) = u - u. [ABSTRACT FROM AUTHOR]

This short contribution comments on parts of a series of two papers [2] and [3] on spinodal decomposition for the Cahn-Hilliard equation In a small neighborhood of a homogeneous equilibrium uo = μ of (1), with μ ϵ ℛ. in the spinodal region, we find that a strongly unstable manifold of uo has dominating effects. Although this is demonstrated here for the linearized equation only, it will be essential for the nonlinear situation as well (cf. [3]). Furthermore we comment on the notion of a characteristic wavelength for the strongly unstable directions. These patterns indeed correspond to typically observed spinodally decomposed states which arise for instance in materials sciences where u corresponds to the concentration of one of the components of a binary alloy (cf. [1] or the references in [2]). [ABSTRACT FROM AUTHOR]

Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes or with a welldefined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial noise frequency, and we can explain via numerical continuation methods that if this spatial frequency is too high, then pattern formation is significantly impaired by the underlying structure of the system. In addition, we provide rigorous bounds on the nucleation time frame in the degenerate stochastic setting. [ABSTRACT FROM AUTHOR]

This special issue is devoted to showcasing recent uses of topological methods in the study of dynamical behavior and the analysis of both numerical and experimental data. The twelve original research papers span a wide spectrum of results from abstract index theories, over homology- and persistence-based data analysis techniques, to computer-assisted proof techniques based on topological fixed point arguments. [ABSTRACT FROM AUTHOR]

Homology has long been accepted as an important computational tool for quantifying complex structures. In many applications these structures arise as nodal domains of real-valued functions and are therefore amenable only to a numerical study based on suitable discretizations. Such an approach immediately raises the question of how accurately the resulting homology can be computed. In this paper we present an algorithm for correctly computing the homology of one- and two-dimensional nodal domains. The approach relies on constructing an appropriate cubical approximation for the nodal domain based on the behavior of the defining function at the vertices of a fixed grid. Betti numbers for these cubical sets are readily computable using [T. Kaczynski, K. Mischaikow, and M. Mrozek, Computational Homology, Springer-Verlag, New York, 2004; W. Kalies, M. Mrozek, and P. Pilarczyk, Computational Homology Project, http://chomp.rutgers.edu/ (2006)]. Here, we present a technique to verify that the cubical representation is homeomorphic to the nodal domain and therefore preserves homology. To illustrate this approach we consider examples from three classes of nodal domains, including the time-dependent patterns generated by the Cahn-Hilliard model for spinodal decomposition. We use these results to examine the probability of correct homology computations given specific grid sizes as related to the analytic estimates presented in [S. Day et al., Electron. Res. Announc. Amer. Math. Soc., 13 (2007), pp. 60-73; K. Mischaikow and T. Wanner, Ann. Appl. Probab., 17 (2007), pp. 980-1018; K. Mischaikow and T. Wanner, submitted; P. Niyogi, S. Smale, and S. Weinberger, Discrete Comput. Geom., 39 (2008), pp. 419-441]. [ABSTRACT FROM AUTHOR]

We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape & Miller, 2002] with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path-following of the parameter range of Cahn–Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we, for example, find nonunique connection matrices and saddle–saddle connections of Morse sets. [ABSTRACT FROM AUTHOR]

Abstract: Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. In this paper we propose the use of computational algebraic topology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during spinodal decomposition and early coarsening in both the deterministic Cahn–Hilliard theory, as well as in the stochastic Cahn–Hilliard–Cook model. While both models produce microstructures that are qualitatively similar to the ones observed experimentally, our topological characterization points to significant differences. One particular aspect of our method is its ability to quantify boundary effects in finite size systems. [Copyright &y& Elsevier]

Enstrophy is an averaged measure of fluid vorticity. This quantity is particularly important in rotating geophysical flows. We investigate the dynamical evolution of enstrophy for large-scale quasi-geostrophic flows under random wind forcing. We obtain upper bounds on the enstrophy, as well as results establishing its Ho¨lder continuity and describing the small-time asymptotics. © 2002 American Institute of Physics. [ABSTRACT FROM AUTHOR]

This paper discusses the roughness of surfaces described by nonlinear stochastic partial differential equations on bounded domains. Roughness is an important characteristic for processes arising in molecular beam epitaxy, and is usually described by the mean interface width of the surface, i.e. the expected value of the squared Lebesgue norm. By employing results on the mean interface width for linear stochastic partial differential equations perturbed by colored noise, which have been previously obtained, we describe the evolution of the surface roughness for two classes of nonlinear equations, asymptotically both for small and large times. [ABSTRACT FROM AUTHOR]

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case. [ABSTRACT FROM AUTHOR]

In this paper we present a new algorithm for computing the homology of regular CW-complexes. This algorithm is based on the coreduction algorithm due to Mrozek and Batko and consists essentially of a geometric preprocessing algorithm for the standard chain complex generated by a CW-complex. By employing the concept of S-complexes the original chain complex can-in all known practical cases-be reduced to a significantly smaller S-complex with isomorphic homology, which can then be computed using standard methods. Furthermore, we demonstrate that in the context of non-uniform cubical grids this method significantly improves currently available algorithms based on uniform cubical grids. [ABSTRACT FROM AUTHOR]

Cahn--Morral systems serve as models for several phase separation phenomena in multicomponent alloys. In this paper we study the dynamical aspects of nucleation in a stochastic version of these models using numerical simulations, concentrating on ternary, i.e., three-component, alloys on two-dimensional square domains. We perform numerical studies and give a statistical classification for the distribution of droplet types as the component structure of the alloy is varied. We relate these statistics to the low-energy equilibria of the deterministic equation. [ABSTRACT FROM AUTHOR]