A geometric approach to stationary defect solutions in one space dimension

In this manuscript, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial dierential equations in one spatial dimension...

On Selection of Search Space Dimension in Compressive Sampling Matching Pursuit

Compressive Sampling Matching Pursuit (CoSaMP) is one of the popular greedy methods in the emerging field of Compressed Sensing (CS). In addition to the appealing empirical performance, CoSaMP has also splendid theoretical guarantees for convergence. In this paper, we propose a modification in CoSaMP to adaptively choose the dimension of search space in each iteration, using a threshold based approach. Using Monte Carlo simulations, we show that this modification improves the reconstruction capability of the CoSaMP algorithm in clean as well as noisy measurement cases. From empirical observations, we also propose an optimum value for the threshold to use in applications.

A dimensão espacial e o proceder pedagógico

This research was conducted at the Post Graduate Program in Education at the Federal University of Rio Grande do Norte. It had the aim to understand the spatial conception associated with the pedagogical action. It promoted a historical-scientific approach to concepts of space to emphasize the spatiality from the geographical and educational theoretical referential. In it empiria situated in a university student universe, has utilized a quantitative measuring instrument as a subsidy to a qualitative dialogical analysis. It showed a synthetic framework arising from the interaction and the reflection focused into problems, challenges and potential of a pedagogical vision that considers the spatiality of teaching education. It seeks to promote delineations of paths pedagogical to a integrate approach of heterotopic space conception, as a necessary critical condition to the appropriation of social reality without doing the epistemological dichotomy, enhancing it signification to the learning academic teaching ambience; as well the scientific and teaching formation in the geography course

Quantum scattering in one dimension

A self-contained discussion of non-relativistic quantum scattering is presented in the case of central potentials in one space dimension, which will facilitate the understanding of the more complex scattering theory in two and three dimensions. The present discussion illustrates in a simple way the concepts of partial-wave decomposition, phase shift, optical theorem and effective-range expansion.

Tecnologias de informação geográfica e promoção do pensamento espacial crítico: estratégias transdisciplinares em educação para o desenvolvimento sustentável no 3º CEB

Perante a fragilidade da atual conjuntura mundial nos contextos económico, social e ambiental, urge a necessidade de educar os novos cidadãos para a sustentabilidade, pelo conhecimento interativo do local onde vivem e do mundo que os integra. “Ensinar a pensar” e a encontrar soluções criativas sustentáveis, torna-se incontornável. Reconhecendo-se que este propósito carece da integração do conhecimento presencial no território, e a importância das competências de saber pensar o espaço e intervir no meio, partilhadas pela Educação Geográfica (EG) e pela Educação para o Desenvolvimento Sustentável (EDS), onde a dimensão “Espaço” é crítica e aglutinadora das aprendizagens, no presente trabalho propõe-se dar resposta à questão de como desenvolver o Pensamento Espacial Crítico, com recurso a Tecnologias de Informação Geográfica (TIG), de forma a promover aprendizagens significativas em EDS, ao nível do 3.º Ciclo do Ensino Básico (CEB) e no contexto da Escola Básica de Campia. Tendo como grande finalidade a inovação das práticas educativas, espera-se com esta investigação contribuir para o alargamento de fronteiras do conhecimento em Multimédia em Educação e da EG, assumindo-se a importância desta no domínio da EDS, estimulando o Pensamento Espacial (PE), o Pensamento Crítico (PC) e a forma como os alunos e restante comunidade educativa olham e atuam sobre o meio. Face à finalidade apresentada e dado o caráter inovador da presente investigação, adotou-se a metodologia de investigação-ação (I/A), no contexto do paradigma sóciocrítico, de pendor qualitativo. Este estudo foi desenvolvido por intermédio de uma Oficina de Formação, em dois ciclos de I/A, tendo como objetivo a conceção e implementação de estratégias transdisciplinares de ensino e aprendizagem (E/A) em EDS, visando o desenvolvimento de capacidades de PEC e sendo suportadas por TIG. Foram concebidos diversos instrumentos de recolha de dados, para cada ciclo de I/A, e o corpo de dados foi analisado essencialmente através da técnica de análise de conteúdo e pontualmente através da análise estatística de cariz descritivo. O modelo de análise de dados centrou-se numa análise SWOT para identificação das forças, fraquezas, oportunidades e ameaças, e numa matriz TOWS para identificação das ações a empreender entre os ciclos de I/A. Propõe-se, para o efeito, um referencial teórico didático para o conceito de PEC, através de uma taxonomia de capacidades e competências resultante da implementação dos ciclos de I/A. Os resultados obtidos permitem observar que: i) a EG, pelas competências que preconiza e pelo enfoque da dimensão espacial, é potencialmente aglutinadora das aprendizagens, no currículo do 3.º CEB, e pode favorecer a transdisciplinaridade, essencial na EDS; ii) as estratégias de ensino e aprendizagem assentes na EG e com recurso a TIG são promotoras de aprendizagens significativas em EDS pelo desenvolvimento de capacidades de PEC. Contudo, as limitações evidenciadas na investigação suscitaram, através dos ciclos de I/A, a redefinição do percurso formativo proposto e dos instrumentos de recolha de dados concebidos, bem como a introdução de melhorias à taxonomia de PEC desenvolvida no âmbito da presente tese. Entre outras limitações discutidas (como a resistência à utilização de tecnologia em contexto de E/A), salientamos uma limitação sistémica, inerente aos atuais contextos educativos formais (currículo, distribuição do serviço docente, etc.), que desincentiva uma efetiva implementação de estratégias transdisciplinares e do trabalho colaborativo entre professores. Apesar das limitações elencadas consideramos que este estudo contribui para um aprofundamento do conhecimento sobre a potencialidade das TIG na promoção de competências e capacidades de PEC dos alunos, nomeadamente pelo avanço na clarificação das mesmas, plasmadas no instrumento desenvolvido no âmbito desta tese (taxonomia de PEC). Como disseminação desta investigação, salienta-se que o referido instrumento integrará o referencial teórico de um projeto europeu Erasmus + (ENAbLE), para suporte à conceção dos dispositivos didáticos que acompanharão uma aplicação de TIG que será concebida especificamente para o contexto de E/A.

Connected Hopf algebras of finite Gelfand-Kirillov dimension

Following the seminal work of Zhuang, connected Hopf algebras of finite GK-dimension over algebraically closed fields of characteristic zero have been the subject of several recent papers. This thesis is concerned with continuing this line of research and promoting connected Hopf algebras as a natural, intricate and interesting class of algebras. We begin by discussing the theory of connected Hopf algebras which are either commutative or cocommutative, and then proceed to review the modern theory of arbitrary connected Hopf algebras of finite GK-dimension initiated by Zhuang. We next focus on the (left) coideal subalgebras of connected Hopf algebras of finite GK-dimension. They are shown to be deformations of commutative polynomial algebras. A number of homological properties follow immediately from this fact. Further properties are described, examples are considered and invariants are constructed. A connected Hopf algebra is said to be "primitively thick" if the difference between its GK-dimension and the vector-space dimension of its primitive space is precisely one . Building on the results of Wang, Zhang and Zhuang,, we describe a method of constructing such a Hopf algebra, and as a result obtain a host of new examples of such objects. Moreover, we prove that such a Hopf algebra can never be isomorphic to the enveloping algebra of a semisimple Lie algebra, nor can a semisimple Lie algebra appear as its primitive space. It has been asked in the literature whether connected Hopf algebras of finite GK-dimension are always isomorphic as algebras to enveloping algebras of Lie algebras. We provide a negative answer to this question by constructing a counterexample of GK-dimension 5. Substantial progress was made in determining the order of the antipode of a finite dimensional pointed Hopf algebra by Taft and Wilson in the 1970s. Our final main result is to show that the proof of their result can be generalised to give an analogous result for arbitrary pointed Hopf algebras.

Application of a continuum theory to vertical vibrations of a layer of granular material

Many interesting phenomena have been observed in layers of granular materials subjected to vertical oscillations; these include the formation of a variety of standing wave patterns, and the occurrence of isolated features called oscillons, which alternately form conical heaps and craters oscillating at one-half of the forcing frequency. No continuum-based explanation of these phenomena has previously been proposed. We apply a continuum theory, termed the double-shearing theory, which has had success in analyzing various problems in the flow of granular materials, to the problem of a layer of granular material on a vertically vibrating rigid base undergoing vertical oscillations in plane strain. There exists a trivial solution in which the layer moves as a rigid body. By investigating linear perturbations of this solution, we find that at certain amplitudes and frequencies this trivial solution can bifurcate. The time dependence of the perturbed solution is governed by Mathieu’s equation, which allows stable, unstable and periodic solutions, and the observed period-doubling behaviour. Several solutions for the spatial velocity distribution are obtained; these include one in which the surface undergoes vertical velocities that have sinusoidal dependence on the horizontal space dimension, which corresponds to the formation of striped standing waves, and is one of the observed patterns. An alternative continuum theory of granular material mechanics, in which the principal axes of stress and rate-of-deformation are coincident, is shown to be incapable of giving rise to similar instabilities.

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.

Entire Solutions of Hydrodynamical Equations with Exponential Dissipation

We consider a modification of the three-dimensional Navier-Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as e(vertical bar k vertical bar/kd) at high wavenumbers vertical bar k vertical bar. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than e-(C(k/kd) ln(vertical bar k vertical bar/kd)) for any C < 1/(2 ln 2). The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with C = C-* = 1/ ln 2. The same behavior with a universal constant C-* is conjectured for the Navier-Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier-Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.

On methods of calculating successive positions of a shock front

In this paper we have discussed limits of the validity of Whitham's characteristic rule for finding successive positions of a shock in one space dimension. We start with an example for which the exact solution is known and show that the characteristic rule gives correct result only if the state behind the shock is uniform. Then we take the gas dynamic equations in two cases: one of a shock propagating through a stratified layer and other down a nonuniform tube and derive exact equations for the evolution of the shock amplitude along a shock path. These exact results are then compared with the results obtained by the characteristic rule. The characteristic rule not only incorrectly accounts for the deviation of the state behind the shock from a uniform state but also gives a coefficient in the equation which differ significantly from the exact coefficients for a wide range of values of the shock strength.

Model exact low-lying states and spin dynamics in ferric wheels: Fe-6 to Fe-12

Using an efficient numerical scheme that exploits spatial symmetries and spin parity, we have obtained the exact low-lying eigenstates of exchange Hamiltonians for ferric wheels up to Fe-12. The largest calculation involves the Fe-12 ring which spans a Hilbert space dimension of about 145x10(6) for the M-S=0 subspace. Our calculated gaps from the singlet ground state to the excited triplet state agree well with the experimentally measured values. Study of the static structure factor shows that the ground state is spontaneously dimerized for ferric wheels. The spin states of ferric wheels can be viewed as quantized states of a rigid rotor with the gap between the ground and first excited states defining the inverse of the moment of inertia. We have studied the quantum dynamics of Fe-10 as a representative of ferric wheels. We use the low-lying states of Fe-10 to solve exactly the time-dependent Schrodinger equation and find the magnetization of the molecule in the presence of an alternating magnetic field at zero temperature. We observe a nontrivial oscillation of the magnetization which is dependent on the amplitude of the ac field. We have also studied the torque response of Fe-12 as a function of a magnetic field, which clearly shows spin-state crossover.

On the explicit determination of the polar decomposition in n-dimensional vector spaces

A method for the explicit determination of the polar decomposition (and the related problem of finding tensor square roots) when the underlying vector space dimension n is arbitrary (but finite), is proposed. The method uses the spectral resolution, and avoids the determination of eigenvectors when the tensor is invertible. For any given dimension n, an appropriately constructed van der Monde matrix is shown to play a key role in the construction of each of the component matrices (and their inverses) in the polar decomposition.

Full spin and spatial symmetry adapted technique for correlated electronic hamiltonians: Application to an icosahedral cluster

One of the long standing problems in quantum chemistry had been the inability to exploit full spatial and spin symmetry of an electronic Hamiltonian belonging to a non-Abelian point group. Here, we present a general technique which can utilize all the symmetries of an electronic (magnetic) Hamiltonian to obtain its full eigenvalue spectrum. This is a hybrid method based on Valence Bond basis and the basis of constant z-component of the total spin. This technique is applicable to systems with any point group symmetry and is easy to implement on a computer. We illustrate the power of the method by applying it to a model icosahedral half-filled electronic system. This model spans a huge Hilbert space (dimension 1,778,966) and in the largest non-Abelian point group. The C60 molecule has this symmetry and hence our calculation throw light on the higher energy excited states of the bucky ball. This method can also be utilized to study finite temperature properties of strongly correlated systems within an exact diagonalization approach. (C) 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

A finite difference method for a time-dependent singularly perturbed convection-diffusion-reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin-Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin-Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.

Local exponential divergence plot and optimal embedding of a chaotic time-series

we propose here a local exponential divergence plot which is capable of providing a new means of characterizing chaotic time series. The suggested plot defines a time dependent exponent LAMBDA and a ''plus'' exponent LAMBDA+ which serves as a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time and the largest Lyapunov exponent.