Oscillatory integral operators related to pointwise convergence of Schrödinger operators

In this thesis we consider smooth analogues of operators studied in connection with the pointwise convergence of the solution, u(x,t), (x,t) ∈ ℝ^n x ℝ, of the free Schrodinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we develop strategies to capture the oscillations and obtain sharp L^2 → L^2 bounds. We then consider, for fixed smooth t(x), the restriction of u to the surface (x,t(x)). We find that u(x,t(x)) ∈ L^2(D^n) when the initial data is in a suitable L^2-Sobolev space H^8 (ℝ^n), where s depends on conditions on t.

Completeness of certain function spaces

We give an example of a complete locally convex m-topology on the algebra of infinite differentiable functions on [0, 1] which is strictly coarser than the natural Frechet-topology but finer than the topology of pointwise convergence. A similar construction works on the algebra of continuous functions on [0, 1]. Using this examples we can separate different notions of diffotopy and homotopy.

Quelques théorèmes ergodiques pour des suites de fonctions

Le théorème ergodique de Birkhoff nous renseigne sur la convergence de suites de fonctions. Nous nous intéressons alors à étudier la convergence en moyenne et presque partout de ces suites, mais dans le cas où la suite est une suite strictement croissante de nombres entiers positifs. C’est alors que nous définirons les suites uniformes et étudierons la convergence presque partout pour ces suites. Nous regarderons également s’il existe certaines suites pour lesquelles la convergence n’a pas lieu. Nous présenterons alors un résultat dû en partie à Alexandra Bellow qui dit que de telles suites existent. Finalement, nous démontrerons une équivalence entre la notion de transformatiuon fortement mélangeante et la convergence d'une certaine suite qui utilise des “poids” qui satisfont certaines propriétés.

Ergodic theorems along sequences and Hardy fields.

Let a(x) be a real function with a regular growth as x --> infinity. [The precise technical assumption is that a(x) belongs to a Hardy field.] We establish sufficient growth conditions on a(x) so that the sequence ([a(n)])(infinity)(n=1) is a good averaging sequence in L2 for the pointwise ergodic theorem. A sequence (an) of positive integers is a good averaging sequence in L2 for the pointwise ergodic theorem if in any dynamical system (Omega, Sigma, m, T) for f [symbol, see text] in L2(Omega) the averages [equation, see text] converge for almost every omicron in. Our result implies that sequences like ([ndelta]), where delta > 1 and not an integer, ([n log n]), and ([n2/log n]) are good averaging sequences for L2. In fact, all the sequences we examine will turn out to be good averaging for Lp, p > 1; and even for L log L. We will also establish necessary and sufficient growth conditions on a(x) so that the sequence ([a(n)]) is good averaging for mean convergence. Note that for some a(x) (e.g., a(x) = log2 x), ([a(n)]) may be good for mean convergence without being good for pointwise convergence.

Some properties of g- and P-spaces

A γ-space with a strictly positive measure is separable. An example of a non-separable γ−space with c.c.c. is given. A P−space with c.c.c. is countable and discrete.

Research Monograph: Managing ubiquitous eco cities: telecommunication infrastructure networks, technology convergence and intelligent urban management systems

Efficient and effective urban management systems for Ubiquitous Eco Cities require having intelligent and integrated management mechanisms. This integration includes bringing together economic, socio-cultural and urban development with a well orchestrated, transparent and open decision making mechanism and necessary infrastructure and technologies. In Ubiquitous Eco Cities telecommunication technologies play an important role in monitoring and managing activities over wired, wireless or fibre-optic networks. Particularly technology convergence creates new ways in which the information and telecommunication technologies are used and formed the back bone or urban management systems. The 21st Century is an era where information has converged, in which people are able to access a variety of services, including internet and location based services, through multi-functional devices such as mobile phones and provides opportunities in the management of Ubiquitous Eco Cities. This research paper discusses the recent developments in telecommunication networks and trends in convergence technologies and their implications on the management of Ubiquitous Eco Cities and how this technological shift is likely to be beneficial in improving the quality of life and place of residents, workers and visitors. The research paper reports and introduces recent approaches on urban management systems, such as intelligent urban management systems, that are suitable for Ubiquitous Eco Cities.

Managing ubiquitous eco cities : the role of urban telecommunication infrastructure networks and convergence technologies

A successful urban management system for a Ubiquitous Eco City requires an integrated approach. This integration includes bringing together economic, socio-cultural and urban development with a well orchestrated, transparent and open decision making mechanism and necessary infrastructure and technologies. Rapidly developing information and telecommunication technologies and their platforms in the late 20th Century improves urban management and enhances the quality of life and place. Telecommunication technologies provide an important base for monitoring and managing activities over wired, wireless or fibre-optic networks. Particularly technology convergence creates new ways in which the information and telecommunication technologies are used. The 21st Century is an era where information has converged, in which people are able to access a variety of services, including internet and location based services, through multi-functional devices such as mobile phones and provides opportunities in the management of Ubiquitous Eco Cities. This paper discusses the recent developments in telecommunication networks and trends in convergence technologies and their implications on the management of Ubiquitous Eco Cities and how this technological shift is likely to be beneficial in improving the quality of life and place. The paper also introduces recent approaches on urban management systems, such as intelligent urban management systems, that are suitable for Ubiquitous Eco Cities.

Ourplace : the convergence of locative media and online participatory culture

The trans-locative potential of the Internet has driven the design of many online applications. Online communities largely cluster around topics of interest, which take precedence over participants’ geographical locations. The site of production is often disregarded when creative content appears online. However, for some, a sense of place is a defining aspect of creativity. Yet environments that focus on the display and sharing of regionally situated content have, so far, been largely overlooked. Recent developments in geo-technologies have precipitated the emergence of a new field of interactive media. Entitled locative media, it emphasizes the geographical context of media. This paper argues that we might combine practices of locative media (experiential mapping and geo-spatial annotation) with aspects of online participatory culture (uploading, file-sharing and search categorization) to produce online applications that support geographically ‘located’ communities. It discusses the design considerations and possibilities of this convergence,making reference to an example, OurPlace 3G to 3D, which has to date been developed as a prototype.1 It goes on to discuss the benefits and potential uses of such convergent applications, including the co-production of spatial- emporal narratives of place.

Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation

In this paper, we consider the variable-order nonlinear fractional diffusion equation View the MathML source where xRα(x,t) is a generalized Riesz fractional derivative of variable order View the MathML source and the nonlinear reaction term f(u,x,t) satisfies the Lipschitz condition |f(u1,x,t)-f(u2,x,t)|less-than-or-equals, slantL|u1-u2|. A new explicit finite-difference approximation is introduced. The convergence and stability of this approximation are proved. Finally, some numerical examples are provided to show that this method is computationally efficient. The proposed method and techniques are applicable to other variable-order nonlinear fractional differential equations.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.