Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations

We present and analyze a novel wavelet-Fourier technique for the numerical treatment of multidimensional advection-diffusion-reaction equations based on the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin technique with the compressed sensing approach, the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the PDE. In this paper, we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multi-dimensional setting. Our theoretical analysis relies on new estimates for the local a-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme is shown by numerical illustrations in the one-, two-, and three-dimensional case

Measurable Cones and Stable, Measurable Functions

We define a notion of stable and measurable map between cones endowed with measurability tests and show that it forms a cpo-enriched cartesian closed category. This category gives a denotational model of an extension of PCF supporting the main primitives of probabilistic functional programming, like continuous and discrete probabilistic distributions, sampling, conditioning and full recursion. We prove the soundness and adequacy of this model with respect to a call-by-name operational semantics and give some examples of its denotations

Multiresolution approximation of the vector fields on T^3

Multiresolution approximation (MRA) of the vector fields on T^3 is studied. We introduced in the Fourier space a triad of vector fields called helical vectors which derived from the spherical coordinate system basis. Utilizing the helical vectors, we proved the orthogonal decomposition of L^2(T^3) which is a synthesis of the Hodge decomposition of the differential 1- or 2-form on T^3 and the Beltrami decomposition that decompose the space of solenoidal vector fields into the eigenspaces of curl operator. In the course of proof, a general construction procedure of the divergence-free orthonormal complete basis from the basis of scalar function space is presented. Applying this procedure to MRA of L^2(T^3), we discussed the MRA of vector fields on T^3 and the analyticity and regularity of vector wavelets. It is conjectured that the solenoidal wavelet basis must break r-regular condition, i.e. some wavelet functions cannot be rapidly decreasing function because of the inevitable singularities of helical vectors. The localization property and spatial structure of solenoidal wavelets derived from the Littlewood-Paley type MRA (Meyer's wavelet) are also investigated numerically.Comment: LaTeX, 33 Pages, 3 figures. submitted to J. Math. Phy

Complexity and biourbanism: thermodynamical architectural and urban models integrated in modern geographic mapping

The paper was presented on 5th April 2012 by Eleni Tracada in Theoretical Currents II conference in the University of Lincoln.Abstract Vital elements in urban fabric have been often suppressed for reasons of ‘style’. Recent theories, such as Biourbanism, suggest that cities risk becoming unstable and deprived of healthy social interactions. Our paper aims at exploring the reasons for which, fractal cities, which have being conceived as symmetries and patterns, can have scientifically proven and beneficial impact on human fitness of body and mind. During the last few decades, modern urban fabric lost some very important elements, only because urban design and planning turned out to be stylistic aerial views or new landscapes of iconic technological landmarks. Biourbanism attempts to re-establish lost values and balance, not only in urban fabric, but also in reinforcing human-oriented design principles in either micro or macro scale. Human life in cities and beyond emerges during ‘connectivity’ via geometrical continuity of grids and fractals, via path connectivity among highly active nodes, via exchange/movement of people and, finally via exchange of information (networks). All these elements form a hypercomplex system of several interconnected layers of a dynamic structure, all influencing each other in a non-linear manner. Sometimes networks of communication at all levels may suffer from sudden collapse of dynamic patterns, which have been proved to be vital for a long time either to landscapes and cityscapes. We are now talking about negotiating boundaries between human activities, changes in geographic mapping and, mainly about sustainable systems to support continuous growth of communities. We are not only talking about simple lives (‘Bios’) as Urban Syntax (bio and socio-geometrical synthesis), but also about affinities between developing topographies created by roadways and trajectories and the built environment. We shall also have the opportunity to show recent applications of these theories in our postgraduate students’ work, such as a 3D model as a new method of cartography of the Island of Mauritius, with intend to highlight developments in topography and architecture through a series of historical important events and mutating socio-political and economical geographies. This model may be able to predict failures in proposed and/or activated models of expansion, which do not follow strictly morphogenetic and physiological design processes. The same kind of modelling is capable to enable recognition of ‘optimal forms’ at different feedback scales, which, through morphogenetic processes, guarantee an optimal systemic efficiency, and therefore quality of life.ADT funds, university of Derby

The fractal urban coherence in biourbanism: the factual elements of urban fabric

This article is available online and will be inserted in also printed format in the Journal in October 2013.During the last few decades, modern urban fabric lost some very important elements, only because urban design and planning turned out to be stylistic aerial views or new landscapes of iconic technological landmarks. Biourbanism attempts to re-establish lost values and balance, not only in urban fabric, but also in reinforcing human-oriented design principles in either micro or macro scale. Biourbanism operates as a catalyst of theories and practices in both architecture and urban design to guarantee high standards in services, which are currently fundamental to the survival of communities worldwide. Human life in cities emerges during connectivity via geometrical continuity of grids and fractals, via path connectivity among highly active nodes, via exchange/movement of people and, finally via exchange of information (networks). In most human activities taking place in central areas of cities, people often feel excluded from design processes in the built environment. This paper aims at exploring the reasons for which, fractal cities, which have being conceived as symmetries and patterns, can have scientifically proven and beneficial impact on human fitness of body and mind; research has found that, brain traumas caused by visual agnosia become evident when patterns disappear from either 2D or 3D emergences in architectural and urban design.ADT Fund