Knot theory and its applications

Nella tesi verranno presi in considerazione tre aspetti: si descriverà come la teoria dei nodi si sia sviluppata nel corso degli anni in relazione alle diverse scoperte scientifiche avvenute. Si potrà quindi subito avere una idea di come questa teoria sia estremamente connessa a diverse altre. Nel secondo capitolo ci si occuperà degli aspetti più formali di questa teoria. Si introdurrà il concetto di nodi equivalenti e di invariante dei nodi. Si definiranno diversi invarianti, dai più elementari, le mosse di Reidemeister, il numero di incroci e la tricolorabilità, fino ai polinomi invarianti, tra cui il polinomio di Alexander, il polinomio di Jones e quello di Kaufman. Infine si spiegheranno alcune applicazioni della teoria dei nodi in chimica, fisica e biologia. Sulla chimica, si definirà la chiralità molecolare e si mostrerà come la chiralità dei nodi possa essere utile nel determinare quella molecolare. In campo fisico, si mostrerà la relazione che esiste tra l'equazione di Yang-Baxter e i nodi. E in conclusione si mostrerà come modellare un importante processo biologico, la recombinazione del DNA, grazie alla teoria dei nodi.

Trenzas y nudos

El objetivo de este trabajo es mostrar al lector la fuerte relación entre la teoría de trenzas y la teoría de nudos, dos ramas de la topología de baja de dimensión más próximas de lo que parece. La primera parte de este estudio está dedicada a los conceptos y resultados básicos acerca de teoría de trenzas, que nos ayudarán a cumplir nuestro objetivo. En segundo lugar, introduciremos la noción de nudo matemático y explicaremos cómo formarlos a partir de trenzas. En esta sección daremos dos grandes resultados: los Teoremas de Alexander y de Markov. Por último, nos aproximaremos a la construcción original del polinomio de Jones, un invariante de nudos, a partir de grupos de trenzas, lo que pone de manifiesto la estrecha relación entre estas disciplinas.

Stripping Sight Bare : Alison Jones’ ‘Portraits by Proxy’

In this paper I analyse UK artist Alison Jones’ sonic interventions Portrait of the Artist by Proxy (2008), Voyeurism by Proxy (2008) and Art, Lies and Audio Tapes (2009). In Portrait of the Artist by Proxy, Jones – who, due to deteriorating vision, has not seen her reflection in a mirror in years – asks and trusts participants to audio-describe her own image back to her. In Voyeurism by Proxy, Jones asks participants to audio-describe erotic drawings by Gustav Klimt. In Art, Lies and Audio Tapes, Jones asks participants to audio-describe other artworks, such as W.F. Yeames’ And When Did You Last see Your Father?. In these portraits by proxy, Jones opens her image, and other images, to interpretation. In doing so, Jones draws attention to the way sight is privileged as a mode of access to fixed, fundamental truths in Western culture – a mode assumed to be untainted by filters that skew perception of the object. “In a culture where vision is by far the dominant sense,” Jones says, “and as a visual artist with a visual impairment, I am reliant on audio-description …Inevitably, there are limitations imposed by language, time and the interpreter’s background knowledge of the subject viewed, as well as their personal bias of what is deemed important to impart in their description” . In these works, Jones strips these background knowledges, biases and assumptions bare. She reveals different perceptions, as well as tendencies or censor, edit or exaggerate descriptions. In this paper, I investigate how, by revealing unconscious biases, Jones’ works renders herself and her participants vulnerable to a change of perception. I also examine how Jones’ later editing of the audio-descriptions allows her to show the instabilities of sight, and, in Portrait of the Artist by Proxy, to reclaim authorship of her own image.

More Move, More Loss: Lloyd Jones's Mister Pip [A Review]

A review of Lloyd Jones's Mister Pip; winner of the 2007 Commonwealth Writer's Prize and shortlisted for the 2007 Man Booker Prize.

Visual front-end wars : Viola-Jones face detector vs Fourier Lucas-Kanade

The performance of visual speech recognition (VSR) systems are significantly influenced by the accuracy of the visual front-end. The current state-of-the-art VSR systems use off-the-shelf face detectors such as Viola- Jones (VJ) which has limited reliability for changes in illumination and head poses. For a VSR system to perform well under these conditions, an accurate visual front end is required. This is an important problem to be solved in many practical implementations of audio visual speech recognition systems, for example in automotive environments for an efficient human-vehicle computer interface. In this paper, we re-examine the current state-of-the-art VSR by comparing off-the-shelf face detectors with the recently developed Fourier Lucas-Kanade (FLK) image alignment technique. A variety of image alignment and visual speech recognition experiments are performed on a clean dataset as well as with a challenging automotive audio-visual speech dataset. Our results indicate that the FLK image alignment technique can significantly outperform off-the shelf face detectors, but requires frequent fine-tuning.

The connection between Mueller and Jones matrices of polarization optics

A necessary and sufficient condition for the 4 × 4 Mueller matrix to be derivable from the 2 × 2 Jones matrix is obtained. This condition allows one to determine if a given Mueller matrix describes a totally polarized system or a partially polarized (depolarizing) system. The result of Barakat is analysed in the light of this condition. A recently reported experimentally measured Mueller matrix is examined using this condition and is shown to represent a partially polarized system.

Who is the South African retail giant behind the David Jones bid?

Yesterday morning came the news South African retailer Woolworths had offered $4 a share to acquire David Jones, a proposal that has the approval of the department store’s board. This offer, worth an estimated A$2.15 billion, represents a 25% premium over the current share price value. An earlier offer, from rival Myer, was rejected earlier this year. But who is Woolworths? Not to be confused with Australia’s largest supermarket, Woolworths SA is one of South Africa’s largest retailers.

David Jones seeks to exploit top end gap in Australian grocery

While Australia’s supermarkets continue to battle on price, the South African owner of David Jones, Woolworths Holdings, has found a gap in the Australian grocery market and is preparing to exploit it. Reports that the group has appointed Pieter de Wet to overhaul the David Jones food business signals once again the ever-changing face of the Australian food and grocery landscape. A re-energised David Jones food offer will provide positive outcomes for both shoppers and suppliers, while potentially becoming another headache for Coles and Woolworths.

Translational and rotational motion in molecular liquids: A computer simulation study of Lennard-Jones ellipsoids

In order to understand the translational and rotational motion in dense molecular liquids, detailed molecular dynamics simulations of Lennard-Jones ellipsoids have been carried out for three different values of the aspect ratio kappa. For ellipsoids with an aspect ratio equal to 2, the product of the translational diffusion coefficient (D-T) and the average orientational correlation time of the l-th rank harmonics (tau(lR)), converges to a nearly constant value at high density. Surprisingly, this density independent value of D-T tau(lR) is within 5% of the hydrodynamic prediction with the slip boundary condition. This is despite the fact that both D-T and tau(lR) themselves change nearly by an order of magnitude in the density range considered, and the rotational correlation function itself is strongly nonexponential. For small aspect ratios (kappa less than or equal to 1.5), the rotational correlation function remains largely Gaussian even at a very large density, while for a large aspect ratio (kappa greater than or equal to 3), the transition to the nematic liquid-crystalline phase precludes the hydrodynamic regime. Thus, the rotational dynamics of ellipsoids show great sensitivity to the aspect ratio. At low density, tau(lR) goes through a minimum value, indicating the role of interactions in enhancing the rate of orientational relaxation. (C) 1997 American Institute of Physics. [S0021-9606(97)50142-5].

Dependence of diffusivity on density and solute diameter in liquid phase: A molecular dynamics study of Lennard-Jones system

Investigations into the variation of self-diffusivity with solute radius, density, and degree of disorder of the host medium is explored. The system consists of a binary mixture of a relatively smaller sized solute, whose size is varied and a larger sized solvent interacting via Lennard-Jones potential. Calculations have been performed at three different reduced densities of 0.7, 0.8, and 0.933. These simulations show that diffusivity exhibits a maximum for some intermediate size of the solute when the solute diameter is varied. The maximum is found at the same size of the solute at all densities which is at variance with the prediction of the levitation effect. In order to understand this anomaly, additional simulations were carried out in which the degree of disorder has been varied while keeping the density constant. The results show that the diffusivity maximum gradually disappears with increase in disorder. Disorder has been characterized by means of the minimal spanning tree. Simulations have also been carried out in which the degree of disorder is constant and only the density is altered. The results from these simulations show that the maximum in diffusivity now shifts to larger distances with decrease in density. This is in agreement with the changes in void and neck distribution with density of the host medium. These results are in excellent agreement with the predictions of the levitation effect. They suggest that the effect of disorder is to shift the maximum in diffusivity towards smaller solute radius while that of the decrease in density is to shift it towards larger solute radius. Thus, in real systems where the degree of disorder is lower at higher density and vice versa, the effect due to density and disorder have opposing influences. These are confirmed by the changes seen in the velocity autocorrelation function, self part of the intermediate scattering function and activation energy. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.3701619]

Solid-liquid transition in polydisperse Lennard-Jones systems

We study melting of a face-centered crystalline solid consisting of polydisperse Lennard-Jones spheres with Gaussian polydispersity in size. The phase diagram reproduces the existence of a nearly temperature invariant terminal polydispersity (delta(t) similar or equal to 0.11), with no signature of reentrant melting. The absence of reentrant melting can be attributed to the influence of the attractive part of the potential upon melting. We find that at terminal polydispersity the fractional density change approaches zero, which seems to arise from vanishingly small compressibility of the disordered phase. At constant temperature and volume fraction the system undergoes a sharp transition from crystalline solid to the disordered amorphous or fluid state with increasing polydispersity. This has been quantified by second- and third-order rotational invariant bond orientational order, as well as by the average inherent structure energy. The translational order parameter also indicates a similar sharp structural change at delta similar or equal to 0.09 in case of T* = 1.0, phi = 0.58. The free energy calculation further supports the sharp nature of the transition. The third-order rotationally invariant bond order shows that with increasing polydispersity, the local cluster favors a more icosahedral arrangement and the system loses its local crystalline symmetry. Interestingly, the value of structure factor S(k) of the amorphous phase at delta similar or equal to 0.10 (just beyond the solid-liquid transition density at T* = 1) becomes 2.75, which is below the value of 2.85 required for freezing given by the empirical Hansen-Verlet rule of crystallization, well known in the theory of freezing.

Use of polydispersity index as control parameter to study melting/freezing of Lennard-Jones system: Comparison among predictions of bifurcation theory with Lindemann criterion, inherent structure analysis and Hansen-Verlet rule

Using polydispersity index as an additional order parameter we investigate freezing/melting transition of Lennard-Jones polydisperse systems (with Gaussian polydispersity in size), especially to gain insight into the origin of the terminal polydispersity. The average inherent structure (IS) energy and root mean square displacement (RMSD) of the solid before melting both exhibit quite similar polydispersity dependence including a discontinuity at solid-liquid transition point. Lindemann ratio, obtained from RMSD, is found to be dependent on temperature. At a given number density, there exists a value of polydispersity index (delta (P)) above which no crystalline solid is stable. This transition value of polydispersity(termed as transition polydispersity, delta (P) ) is found to depend strongly on temperature, a feature missed in hard sphere model systems. Additionally, for a particular temperature when number density is increased, delta (P) shifts to higher values. This temperature and number density dependent value of delta (P) saturates surprisingly to a value which is found to be nearly the same for all temperatures, known as terminal polydispersity (delta (TP)). This value (delta (TP) similar to 0.11) is in excellent agreement with the experimental value of 0.12, but differs from hard sphere transition where this limiting value is only 0.048. Terminal polydispersity (delta (TP)) thus has a quasiuniversal character. Interestingly, the bifurcation diagram obtained from non-linear integral equation theories of freezing seems to provide an explanation of the existence of unique terminal polydispersity in polydisperse systems. Global bond orientational order parameter is calculated to obtain further insights into mechanism for melting.

A Lennard-Jones Embedded-Atom Potential and Its Application to the Study of Melting

The nearest-neighbour Lennard-Jones potential from the embedded-atom method is extended to a form that includes more than nearest neighbours. The model has been applied to study melting with molecular dynamics. The calculated melting point, fractional volume change on melting, heat of fusion and linear coefficients of thermal expansion are in good agreement with experimental data. We have found that the second and third neighbours influence the melting point distinctly.