On the Bäcklund and Darboux transformations for the Tzitzéica model

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators of rank and order two.

Backlund-Darboux Transformations in Sato's Grassmannian

We define Bäcklund–Darboux transformations in Sato’s Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on related objects: wave functions, tau-functions and spectral algebras.

Darboux Transformations and Random Point Processes

International audience

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

Darboux-Bäcklund transformations and rational solutions of the painlevé IV equation

Rational solutions of the Painlevé IV equation are constructed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy subject to the additional non-isospectral Virasoro symmetry constraint. Convenient Wronskian representations for rational solutions are obtained by successive actions of the Darboux-Bäcklund transformations. ©2010 American Institute of Physics.

Practical Transformations and Transformational Practices : Globalization, Postmodernism, and Early Childhood Education : Advances in Early Education and Day Care

Both traditional and progressive curricula are inadequate for the task of responding to the economic, political, social, and cultural changes that have occurred as a result of globalization. This book documents some of the ongoing work occurring in early childhood settings that is aimed at improving, and ultimately transforming, early childhood practice in these changed and changing times. The authors do not simply critique developmental approaches or the increasing standardization of the field. Instead, they describe how they are playing around with postmodern ideas in practice and developing unique approaches to the diverse educational circumstances that confront early childhood educators. Whether it is preparing teachers, using materials, or developing policies, each chapter provides readers with possibilities for enacting pedagogies that are responsive to the contemporary circumstances shaping the lives of young children.

Nitrogen and phosphorus transformations in fixed-film biofilters subjected to aeration/no-aeration cycles

Despite recent developments in fixed-film combined biological nutrients removal (BNR) technology; fixed-film systems (i.e., biofilters), are still at the early stages of development and their application has been limited to a few laboratory-scale experiments. Achieving enhanced biological phosphorus removal in fixed-film systems requires exposing the micro-organisms and the waste stream to alternating anaerobic/aerobic or anaerobic/anoxic conditions in cycles. The concept of cycle duration (CD) as a process control parameter is unique to fixed-film BNR systems, has not been previously investigated, and can be used to optimise the performance of such systems. The CD refers to the elapsed time before the biomass is re-exposed to the same environmental conditions in cycles. Fixed-film systems offer many advantages over suspended growth systems such as reduced operating costs, simplicity of operation, absence of sludge recycling problems, and compactness. The control of nutrient discharges to water bodies, improves water quality, fish production, and allow water reuse. The main objective of this study was to develop a fundamental understanding of the effect of CD on the transformations of nutrients in fixed-film biofilter systems subjected to alternating aeration I no-aeration cycles A fixed-film biofilter system consisting of three up-flow biofilters connected in series was developed and tested. The first and third biofilters were operated in a cyclic mode in which the biomass was subjected to aeration/no-aeration cycles. The influent wastewater was simulated aquaculture whose composition was based on actual water quality parameters of aquacuture wastewater from a prawn grow-out facility. The influent contained 8.5 - 9:3 mg!L a111monia-N, 8.5- 8.7 mg/L phosphate-P, and 45- 50 mg!L acetate. Two independent studies were conducted at two biofiltration rates to evaluate and confirm the effect of CD on nutrient transformations in the biofilter system for application in aquaculture: A third study was conducted to enhance denitrification in the system using an external carbon- source at a rate varying from 0-24 ml/min. The CD was varied in the range of0.25- 120 hours for the first two studies and fixed at 12 hours for the third study. This study identified the CD as an important process control parameter that can be used to optimise the performance of full-scale fixed-film systems for BNR which represents a novel contribution in this field of research. The CD resulted in environmental conditions that inhibited or enhanced nutrient transformations. The effect of CD on BNR in fixed-film systems in terms of phosphorus biomass saturation and depletion has been established. Short CDs did not permit the establishment of anaerobic activity in the un-aerated biofilter and, thus, inhibited phosphorus release. Long CDs resulted in extended anaerobic activity and, thus, resulted in active phosphorus release. Long CDs, however, resulted in depleting the biomass phosphorus reservoir in the releasing biofilter and saturating the biomass phosphorus reservoir in the up-taking biofilter in the cycle. This phosphorus biomass saturation/depletion phenomenon imposes a practical limit on how short or long the CD can be. The length of the CD should be somewhere just before saturation or depletion occur and for the system tested, the optimal CD was 12 hours for the biofiltration rates tested. The system achieved limited net phosphorus removal due to the limited sludge wasting and lack of external carbon supply during phosphorus uptake. The phosphorus saturation and depletion reflected the need to extract phosphorus from the phosphorus-rich micro-organisms, for example, through back-washing. The major challenges of achieving phosphorus removal in the system included: (I) overcoming the deterioration in the performance of the system during the transition period following the start of each new cycle; and (2) wasting excess phosphorus-saturated biomass following the aeration cycle. Denitrification occurred in poorly aerated sections of the third biofilter and generally declined as the CD increased and as the time progressed in the individual cycle. Denitrification and phosphorus uptake were supplied by an internal organic carbon source, and the addition of an external carbon source (acetate) to the third biofilter resulted in improved denitrification efficiency in the system from 18.4 without supplemental carbon to 88.7% when the carbon dose reached 24 mL/min The removal of TOC and nitrification improved as the CD increased, as a result of the reduction in the frequency of transition periods between the cycles. A conceptual design of an effective fixed-film BNR biofilter system for the treatment of the influent simulated aquaculture wastewater was proposed based on the findings of the study.

"Surviving" adolescence : apocalyptic and post-apocalyptic transformations in young adult fiction

This study, entitled "Surviving" Adolescence: Apocalyptic and post-apocalyptic transformations in young adult fiction‖, analyses how discourses surrounding the apocalyptic and post-apocalyptic are represented in selected young adult fiction published between 1997 and 2009. The term ―apocalypse‖ is used by current theorists to refer to an uncovering or disclosure (most often a truth), and ―post-apocalypse‖ means to be after a disclosure, after a revelation, or after catastrophe. This study offers a double reading of apocalyptic and post-apocalyptic discourses, and the dialectical tensions that are inherent in, and arise from, these discourses. Drawing on the current scholarship of children‘s and young adult literature this thesis uses post-structural theoretical perspectives to develop a framework and methodology for conducting a close textual analysis of exclusion, ‗un‘differentiation, prophecy, and simulacra of death. The combined theoretical perspectives and methodology offer new contributions to young adult fiction scholarship. This thesis finds that rather than conceiving adolescence as the endurance of a passing phase of a young person‘s life, there is a new trend emerging in young adult fiction that treats adolescence as a space of transformation essential to the survival of the young adult, and his/her community.

Strategising knowledge-based urban development : knowledge city transformations of Brisbane, Australia

In the global knowledge economy, knowledge-intensive industries and knowledge workers are extensively seen as the primary factors to improve the welfare and competitiveness of cities. To attract and retain such industries and workers, cities produce knowledge-based urban development strategies, and therefore such strategising has become an important development mechanism for cities and their economies. The paper discusses the critical connections between knowledge city foundations and integrated knowledge-based urban development mechanisms in both the local and regional level. In particular, the paper investigates Brisbane’s knowledge-based urban development strategies that support gentrification, attraction, and retention of investment and talent. Furthermore, the paper develops a knowledge-based urban development assessment framework to provide a clearer understanding of the local and regional policy frameworks, and relevant applications of Brisbane’s knowledge-based urban development experience, in becoming a prosperous knowledge city. The paper, with its knowledge-based urban development assessment framework, scrutinises Brisbane’s four development domains in detail: economy; society; institutional; built and natural environments. As part of the discussion of the case study findings, the paper describes the global orientation of Brisbane within the frame of regional and local level knowledge-based urban development strategies performing well. Although several good practices from Brisbane have already been internationally acknowledged, the research reveals that Brisbane is still in the early stages of its knowledge-based urban development implementation. Consequently, the development of a monitoring system for all knowledge-based urban development at all levels is highly crucial in accurately measuring the success and failure of specific knowledge-based urban development policies, and Brisbane’s progress towards a knowledge city transformation.

Linear models for endocytic transformations from live cell imaging

Endocytosis is the process by which cells internalise molecules including nutrient proteins from the extracellular media. In one form, macropinocytosis, the membrane at the cell surface ruffles and folds over to give rise to an internalised vesicle. Negatively charged phospholipids within the membrane called phosphoinositides then undergo a series of transformations that are critical for the correct trafficking of the vesicle within the cell, and which are often pirated by pathogens such as Salmonella. Advanced fluorescent video microscopy imaging now allows the detailed observation and quantification of these events in live cells over time. Here we use these observations as a basis for building differential equation models of the transformations. An initial investigation of these interactions was modelled with reaction rates proportional to the sum of the concentrations of the individual constituents. A first order linear system for the concentrations results. The structure of the system enables analytical expressions to be obtained and the problem becomes one of determining the reaction rates which generate the observed data plots. We present results with reaction rates which capture the general behaviour of the reactions so that we now have a complete mathematical model of phosphoinositide transformations that fits the experimental observations. Some excellent fits are obtained with modulated exponential functions; however, these are not solutions of the linear system. The question arises as to how the model may be modified to obtain a system whose solution provides a more accurate fit.

A typology for business transformations (research in progress)

The term “business transformation” is a buzzword, often used to signify fundamental changes undergone by organisations. Despite numerous works in enterprise transformation, IT-enabled business transformation and organizational transformation, there appears to be a lack of consensus on what actually constitutes a business transformation as opposed to other types of redesign or organisational improvement projects. Consequently, knowledge about which elements of a business system that are impacted by such an endeavour is largely inconsistent, and partially conflicting. We present a business transformation typology that considers 18 attributes pertaining to the transforming organisation and the transformation initiative. To explore our typology, we analysed 10 published case studies and classified them along two dimensions – one ranging from marginal to fundamental changes, and another on internal and external visibility. Our literature review reveals how the terminology has been misused, and we provide some directions to provide more clarity around transformation phenomena in IS research.

Data transformations in custom digital workflows : property graphs as a data model for user-defined mappings

This paper describes the use of property graphs for mapping data between AEC software tools, which are not linked by common data formats and/or other interoperability measures. The intention of introducing this in practice, education and research is to facilitate the use of diverse, non-integrated design and analysis applications by a variety of users who need to create customised digital workflows, including those who are not expert programmers. Data model types are examined by way of supporting the choice of directed, attributed, multi-relational graphs for such data transformation tasks. A brief exemplar design scenario is also presented to illustrate the concepts and methods proposed, and conclusions are drawn regarding the feasibility of this approach and directions for further research.