On the chaotic instability of a nonsliding liquid-filled top with a small spheroidal base via Melnikov-Holmes-Marsden integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations. (c) 2006 Elsevier Ltd. All rights reserved.

Studies on the universal parameters and onset of chaos in dissipative systems

It has become clear over the last few years that many deterministic dynamical systems described by simple but nonlinear equations with only a few variables can behave in an irregular or random fashion. This phenomenon, commonly called deterministic chaos, is essentially due to the fact that we cannot deal with infinitely precise numbers. In these systems trajectories emerging from nearby initial conditions diverge exponentially as time evolves)and therefore)any small error in the initial measurement spreads with time considerably, leading to unpredictable and chaotic behaviour The thesis work is mainly centered on the asymptotic behaviour of nonlinear and nonintegrable dissipative dynamical systems. It is found that completely deterministic nonlinear differential equations describing such systems can exhibit random or chaotic behaviour. Theoretical studies on this chaotic behaviour can enhance our understanding of various phenomena such as turbulence, nonlinear electronic circuits, erratic behaviour of heart and brain, fundamental molecular reactions involving DNA, meteorological phenomena, fluctuations in the cost of materials and so on. Chaos is studied mainly under two different approaches - the nature of the onset of chaos and the statistical description of the chaotic state.

Calculation of line integrals in boundary integral methods

We propose quadrature rules for the approximation of line integrals possessing logarithmic singularities and show their convergence. In some instances a superconvergence rate is demonstrated.

Bounds on integrals of the Wigner function

The integral of the Wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one. It is shown that for systems with 1 degree of freedom, the problem of determining the best possible upper and lower bounds on such an integral, over an possible states, reduces to the problem of finding the greatest and least eigenvalues of a Hermitian operator corresponding to the subregion. The problem is solved exactly in the case of an arbitrary elliptical region. These bounds provide checks on experimentally measured quasiprobability distributions.

Group theory and quasiprobability integrals of Wigner functions

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) approximate to so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

MHM assay: molecular sexing based on the sex-specific methylation pattern of the MHM region in chickens

Bird sex determination using molecular methods has proved to be a valuable tool in different studies. Although it is possible to sex most birds by coupling the CHD assay with others available methods, no sex-determining gene like SRY in mammalians has been identified in birds. The male hypermethylated (MHM) region on the Z chromosome has been found to be hypermethylated in males and hypomethylated in females in birds of the order Galliformes. We analyzed the DNA from feathers of 50 adult chickens to verify the methylation pattern of the MHM region by PCR and the restriction enzyme HpaII (a method named MHM assay). The results, visualized in agarose gel, were compared with PCR amplification of the CHD-Z and CHD-W genes (polyacrylamide gel) and with the birds` phenotype. All males (25) showed hypermethylation of the MHM region, and all females (25) showed hypomethylation. The sexing by MHM assay was in according with phenotype and CHD sexing. To our knowledge, this is the first study that uses the MHM region for sexing birds. Although the real role of the MHM region in the sex determination is still unclear, this could be a universal marker for sexing birds and may be involved in sex determination by its influence on transcriptional processes. The MHM assay could be a good alternative for CHD assay in developmental studies.

Transcripts of the MHM region on the chicken Z chromosome accumulate as non-coding RNA in the nucleus of female cells adjacent to the DMRT1 locus

The male hypermethylated (MHM) region, located near the middle of the short arm of the Z chromosome of chickens, consists of approximately 210 tandem repeats of a BamHI 2.2-kb sequence unit. Cytosines of the CpG dinucleotides of this region are extensively methylated on the two Z chromosomes in the male but much less methylated on the single Z chromosome in the female. The state of methylation of the MHM region is established after fertilization by about the 1-day embryonic stage. The MHM region is transcribed only in the female from the particular strand into heterogeneous, high molecular-mass, non-coding RNA, which is accumulated at the site of transcription, adjacent to the DMRT1 locus, in the nucleus. The transcriptional silence of the MHM region in the male is most likely caused by the CpG methylation, since treatment of the male embryonic fibroblasts with 5-azacytidine results in hypo-methylation and active transcription of this region. In ZZW triploid chickens, MHM regions are hypomethylated and transcribed on the two Z chromosomes, whereas MHM regions are hypermethylated and transcriptionally inactive on the three Z chromosomes in ZZZ triploid chickens, suggesting a possible role of the W chromosome on the state of the MHM region.

Path integrals for continuous-time Markov chains

This note presents a method of evaluating the distribution of a path integral for Markov chains on a countable state space.

Integrals for continuous-time Markov chains

This paper presents a method of evaluating the expected value of a path integral for a general Markov chain on a countable state space. We illustrate the method with reference to several models, including birth-death processes and the birth, death and catastrophe process. (C) 2002 Elsevier Science Inc. All rights reserved.

Invariant integrals applied to nematic liquid crystals with small Ericksen number and topological defects

Invariant integrals are derived for nematic liquid crystals and applied to materials with small Ericksen number and topological defects. The nematic material is confined between two infinite plates located at y = -h and y = h (h is an element of R+) with a semi-infinite plate at y = 0 and x < 0. Planar and homeotropic strong anchoring boundary conditions to the director field are assumed at these two infinite and semi-infinite plates, respectively. Thus, a line disclination appears in the system which coincides with the z-axis. Analytical solutions to the director field in the neighbourhood of the singularity are obtained. However, these solutions depend on an arbitrary parameter. The nematic elastic force is thus evaluated from an invariant integral of the energy-momentum tensor around a closed surface which does not contain the singularity. This allows one to determine this parameter which is a function of the nematic cell thickness and the strength of the disclination. Analytical solutions are also deduced for the director field in the whole region using the conformal mapping method. (C) 2013 Elsevier Ltd. All rights reserved.

Delay approximation of fractional integrals

This paper explores the calculation of fractional integrals by means of the time delay operator. The study starts by reviewing the memory properties of fractional operators and their relationship with time delay. Based on the time response of the Mittag-Leffler function an approximation of fractional integrals consisting of time delayed samples is proposed. The tuning of the approximation is optimized by means of a genetic algorithm. The results demonstrate the feasibility of the new perspective and the limits of their application.

Sherlock Holmes: from paper to the screen

Dissertação de Mestrado apresentado ao Instituto de Contabilidade e Administração do Porto para a obtenção do grau de Mestre em Tradução e Interpretação Especializadas, sob orientação da doutora Clara Sarmento Esta versão não contém as críticas e sugestões dos elementos do júri