I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.

Bottom trawl codend mesh selectivity for Bagrus docmac (Forskahl) from Lake Victoria, with some remarks on the proposed trawl fishery

The methods of estimating codend selectivity for alternate hauls are briefly described and applied to Bagrus docmac (forskahl) data from Lake Victoria. Certain inconsistencies are noted and it is postulated that they may be due to differential escapement of larger Bagrus docmac through the trawl meshes when the different codends were fitted. The calculated 50% selection lengths yielded a fair regression on measured weigh give a fork length selection factor of 2.33, consistent with data from the smallest codend mesh size used. The implications of the findings are briefly discussed in relation to the proposed Haplochromis trawl fishery 1971.

A new large mesh trawl for dermersal fishery

A new large mesh demersal trawl of 32 m head rope length is found more efficient for the exploitation of demersal fishes off Veraval. Increased catch with a proportionate increase of demersal fishes was obtained when compared to a standard bottom trawl of 32 m head rope length with small meshes, suggesting the possibility of increasing the mesh size of trawl nets in the forepart. This increases the mouth area of net which enhances the fishing power by covering a large area per tow. The net is simple in construction, easy to repair and maintain and fewer in the number of meshes.

Mesh selectivity for spotted seer, Scomberomorus guttatus (Bloch & Schneider)

An attempt has been made to evolve a suitable mesh size for the commercially significant size group of S. guttatus. To obtain maximum sustainable yield taking biological factors also into consideration, 52 mm mesh bar nets are suggested for exploiting the fishery on both the East coasts of India.

Mesh regulation in backwater prawn fishing gear

Size grade composition of different species of prawn caught by various back water fishing gear have been enumerated. 57 to 75% of P. indicus captured was less than 10 cm in length. M. dobsoni and M. monoceros captured were less than 10 cm in length. A cod end mesh size of 20-25 mm has been recommended for stake nets for the capture of P. indicus of 10 cm length along with other species.

Performance of 25m large mesh demersal trawl off Veraval, north west coast of India

Performance of a 25m large mesh demersal trawl, with 150mm mesh size in the fore parts of the trawl was evaluated in comparison with one boat high opening trawl of the Bay of Bengal Programme (BOBP) with 360 meshes of 160mm mesh size and 25.6m head rope length. An 8.2% increase in catch was obtained by 25m large mesh demersal trawl. The gear is comparatively cheaper, lighter in construction and offered better horizontal spread with significantly lower towing resistance. Commercial suitability of the gear for efficient harvesting of demersal fish resources of the region is discussed.

A decline in fish catches and the size of Oreochromis niloticus in Lake Wamala (Uganda) following human exploitation

The fishery of Lake Wamala has declined since the lake was stocked in 1956 and opened to fishing during the 1960s. Surveys were conducted on the lake during 1975/78 and 1988/92 to investigate the causes of declining fish catches. The lake produced an average of 4000 - 6000 tonnes of fish annually from 1960s through 1970s. Total fish catches decreased from a maximum of 7100 tonnes in 1967 to less than 500 tonnes by 1990s. Catch rates decreased from about 8 kg in the 1960s to less than 1 kg per net per night by 1975. During the 1970s the catch was dominated by Oreochromis niloticus (67%) followed by Clarias gariepinus (17%), and Protopterus aethiopicus (15.1 %). By 1990s the proportion of O. niloticus had decreased to 45.1% while that of P. aethiopicus had increased to 37.6%. These changes seem to have been caused by overfishing resulting from increased fishing effort from the recommended 250 to about 1000 boats and the additional increase in effort through driving fish into the nets by beating water. The maximum size of O. niloticus in the fishery decreased from 32 cm total length in 1975/78 to 22 cm in 1988/92 while the size at first maturity decreased from about 21 cm to 14 cm during the period. This has been concurrent with a shift in the mesh size of gillnet used from 127 mm (5") in 1960s to 64 mm by 1990s. Environmental changes, especially in lake level in 1980, may also have affected the fishery.

Internal nanoparticle structure of temperature-responsive self-assembled PNIPAM-b-PEG-b-PNIPAM triblock copolymers in aqueous solutions: NMR, SANS and light scattering studies

In this study we report detailed information on the internal structure of PNIPAM-b-PEG-b-PNIPAM nanoparticles formed from self-assembly in aqueous solutions upon increase in temperature. NMR spectroscopy, light scattering and small-angle neutron scattering (SANS) were used to monitor different stages of nanoparticle formation as a function of temperature, providing insight into the fundamental processes involved. The presence of PEG in a copolymer structure significantly affects the formation of nanoparticles, making their transition to occur over a broader temperature range. The crucial parameter that controls the transition is the ratio of PEG/PNIPAM. For pure PNIPAM, the transition is sharp; the higher the PEG/PNIPAM ratio results in a broader transition. This behavior is explained by different mechanisms of PNIPAM block incorporation during nanoparticle formation at different PEG/PNIPAM ratios. Contrast variation experiments using SANS show that the structure of nanoparticles above cloud point temperatures for PNIPAM-b-PEG-b-PNIPAM copolymers is drastically different from the structure of PNIPAM mesoglobules. In contrast with pure PNIPAM mesoglobules, where solid-like particles and chain network with a mesh size of 1-3 nm are present; nanoparticles formed from PNIPAM-b-PEG-b-PNIPAM copolymers have non-uniform structure with “frozen” areas interconnected by single chains in Gaussian conformation. SANS data with deuterated “invisible” PEG blocks imply that PEG is uniformly distributed inside of a nanoparticle. It is kinetically flexible PEG blocks which affect the nanoparticle formation by prevention of PNIPAM microphase separation.

A model of road effect using line integrals and a test of the performance of two new road indices using the distribution of small mammals in an Atlantic Forest landscape

Effects of roads on wildlife and its habitat have been measured using metrics, such as the nearest road distance, road density, and effective mesh size. In this work we introduce two new indices: (1) Integral Road Effect (IRE), which measured the sum effects of points in a road at a fixed point in the forest; and (2) Average Value of the Infinitesimal Road Effect (AVIRE), which measured the average of the effects of roads at this point. IRE is formally defined as the line integral of a special function (the infinitesimal road effect) along the curves that model the roads, whereas AVIRE is the quotient of IRE by the length of the roads. Combining tools of ArcGIS software with a numerical algorithm, we calculated these and other road and habitat cover indices in a sample of points in a human-modified landscape in the Brazilian Atlantic Forest, where data on the abundance of two groups of small mammals (forest specialists and habitat generalists) were collected in the field. We then compared through the Akaike Information Criterion (AIC) a set of candidate regression models to explain the variation in small mammal abundance, including models with our two new road indices (AVIRE and IRE) or models with other road effect indices (nearest road distance, mesh size, and road density), and reference models (containing only habitat indices, or only the intercept without the effect of any variable). Compared to other road effect indices, AVIRE showed the best performance to explain abundance of forest specialist species, whereas the nearest road distance obtained the best performance to generalist species. AVIRE and habitat together were included in the best model for both small mammal groups, that is, higher abundance of specialist and generalist small mammals occurred where there is lower average road effect (less AVIRE) and more habitat. Moreover, AVIRE was not significantly correlated with habitat cover of specialists and generalists differing from the other road effect indices, except mesh size, which allows for separating the effect of roads from the effect of habitat on small mammal communities. We suggest that the proposed indices and GIS procedures could also be useful to describe other spatial ecological phenomena, such as edge effect in habitat fragments. (C) 2012 Elsevier B.V. All rights reserved.

Interpolation estimate for a finite-element space with embedded discontinuities

We consider a recently proposed finite-element space that consists of piecewise affine functions with discontinuities across a smooth given interface Γ (a curve in two dimensions, a surface in three dimensions). Contrary to existing extended finite element methodologies, the space is a variant of the standard conforming Formula space that can be implemented element by element. Further, it neither introduces new unknowns nor deteriorates the sparsity structure. It is proved that, for u arbitrary in Formula, the interpolant Formula defined by this new space satisfies Graphic where h is the mesh size, Formula is the domain, Formula, Formula, Formula and standard notation has been adopted for the function spaces. This result proves the good approximation properties of the finite-element space as compared to any space consisting of functions that are continuous across Γ, which would yield an error in the Formula-norm of order Graphic. These properties make this space especially attractive for approximating the pressure in problems with surface tension or other immersed interfaces that lead to discontinuities in the pressure field. Furthermore, the result still holds for interfaces that end within the domain, as happens for example in cracked domains.

Mesozooplankton size data from the fjord branch Kapisigdlit located in the Godthaabsfjord system, West Greenland, 2010

Sampling was conducted from March 24 to August 5 2010, in the fjord branch Kapisigdlit located in the inner part of the Godthåbsfjord system, West Greenland. The vessel "Lille Masik" was used during all cruises except on June 17-18 where sampling was done from RV Dana (National Institute for Aquatic Resources, Denmark). A total of 15 cruises (of 1-2 days duration) 7-10 days apart was carried out along a transect composed of 6 stations (St.), spanning the length of the 26 km long fjord branch. St. 1 was located at the mouth of the fjord branch and St. 6 was located at the end of the fjord branch, in the middle of a shallower inner creek . St. 1-4 was covering deeper parts of the fjord, and St. 5 was located on the slope leading up to the shallow inner creek. Mesozooplankton was sampled by vertical net tows using a Hydrobios Multinet (type Mini) equipped with a flow meter and 50 µm mesh nets or a WP-2 net 50 µm mesh size equipped with a non-filtering cod-end. Sampling was conducted at various times of day at the different stations. The nets were hauled with a speed of 0.2-0.3 m s**-1 from 100, 75 and 50 m depth to the surface at St. 2 + 4, 5 and 6, respectively. The content was immediately preserved in buffered formalin (4% final concentration). All samples were analyzed in the Plankton sorting and identification center in Szczecin (www.nmfri.gdynia.pl). Samples containing high numbers of zooplankton were split into subsamples. All copepods and other zooplankton were identified down to lowest possible taxonomic level (approx. 400 per sample), length measured and counted. Copepods were sorted into development stages (nauplii stage 1 - copepodite stage 6) using morphological features and sizes, and up to 10 individuals of each stage was length measured.

Biomass and stable isotope composition of plankton in five size fractions along a zonal transect (24°N) from the Canary Islands to Florida in January-March 2011

The spatial variability of biomass and stable isotopes in plankton size fractions in the upper 200 m was studied in a high spatial resolution transect along 24°N from Canary Islands to Florida (January - March 2011) during Leg 8 of the Malaspina-2010 expedition (http://www.expedicionmalaspina.es) to determine nitrogen and carbon sources. Plankton samples were collected by vertical tows of a microplankton net (40 mm mesh size) and a mesoplankton net (200 mm mesh size) through the upper 200 m of the water column. Sampling was between 10:00 and 16:00 h GMT. Plankton was separated into five size fractions (40 - 200, 200 - 500, 500 - 1000, 1000 - 2000 and > 2000 mm) by gentle filtration of the samples by a graded series of nylon sieves (2000, 1000, 500, 200 and 40 mm). Large gelatinous organisms were removed before filtration. Aliquots for each size fraction were collected on pre-weighed glass-fibre filters, dried (60°C, 48 h) and stored in a desiccator before determination of biomass (dry weight), carbon and nitrogen content and natural abundance of stable carbon and nitrogen isotopes ashore. Vertical advection of waters predominated in lateral zones while the central Atlantic (30-70°W) was characterized by a strong stratification and oligotrophic surface waters. Plankton biomass was low in the central zone and high in both eastern and western sides, with most of the variability due to either large (>2000 µm) and small plankton (<500 µm). Carbon isotopes reflected mainly the advection the deep water in lateral zones. Stable nitrogen isotopes showed a nearly symmetrical spatial distribution in all fractions, with the lowest values (delta15N <1per mill) in the central zone, and were inversely correlated to carbon stable isotopes (delta13C) and to the abundance of the nitrogen-fixer Trichodesmium. Diazotrophy was estimated to account for >50% of organic nitrogen in the central zone, and even >30% in eastern and western zones. The impact of diazotrophy increased with the size of the organisms, supporting the wide participation of all trophic levels in the processing of recently fixed nitrogen. These results indicate that atmospheric sources of carbon and nitrogen prevail over deep water sources in the subtropical North Atlantic and that the zone influenced by diazotrophy is much larger than reported in previous studies.

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion

A coupled elastoplastic-damage constitutive model with Lode angle dependent failure criterion for high strain and ballistic applications is presented. A Lode angle dependent function is added to the equivalent plastic strain to failure definition of the Johnson–Cook failure criterion. The weakening in the elastic law and in the Johnson–Cook-like constitutive relation implicitly introduces the Lode angle dependency in the elastoplastic behaviour. The material model is calibrated for precipitation hardened Inconel 718 nickel-base superalloy. The combination of a Lode angle dependent failure criterion with weakened constitutive equations is proven to predict fracture patterns of the mechanical tests performed and provide reliable results. Additionally, the mesh size dependency on the prediction of the fracture patterns was studied, showing that was crucial to predict such patterns

A study of the mechanical behaviour of the human aortic artery by means of non-linear finite element models

En la presente tesis desarrollamos una estrategia para la simulación numérica del comportamiento mecánico de la aorta humana usando modelos de elementos finitos no lineales. Prestamos especial atención a tres aspectos claves relacionados con la biomecánica de los tejidos blandos. Primero, el análisis del comportamiento anisótropo característico de los tejidos blandos debido a las familias de fibras de colágeno. Segundo, el análisis del ablandamiento presentado por los vasos sanguíneos cuando estos soportan cargas fuera del rango de funcionamiento fisiológico. Y finalmente, la inclusión de las tensiones residuales en las simulaciones en concordancia con el experimento de apertura de ángulo. El análisis del daño se aborda mediante dos aproximaciones diferentes. En la primera aproximación se presenta una formulación de daño local con regularización. Esta formulación tiene dos ingredientes principales. Por una parte, usa los principios de la teoría de la fisura difusa para garantizar la objetividad de los resultados con diferentes mallas. Por otra parte, usa el modelo bidimensional de Hodge-Petruska para describir el comportamiento mesoscópico de los fibriles. Partiendo de este modelo mesoscópico, las propiedades macroscópicas de las fibras de colágeno son obtenidas a través de un proceso de homogenización. En la segunda aproximación se presenta un modelo de daño no-local enriquecido con el gradiente de la variable de daño. El modelo se construye a partir del enriquecimiento de la función de energía con un término que contiene el gradiente material de la variable de daño no-local. La inclusión de este término asegura una regularización implícita de la implementación por elementos finitos, dando lugar a resultados de las simulaciones que no dependen de la malla. La aplicabilidad de este último modelo a problemas de biomecánica se estudia por medio de una simulación de un procedimiento quirúrgico típico conocido como angioplastia de balón. In the present thesis we develop a framework for the numerical simulation of the mechanical behaviour of the human aorta using non-linear finite element models. Special attention is paid to three key aspects related to the biomechanics of soft tissues. First, the modelling of the characteristic anisotropic behaviour of the softue due to the collagen fibre families. Secondly, the modelling of damage-related softening that blood vessels exhibit when subjected to loads beyond their physiological range. And finally, the inclusion of the residual stresses in the simulations in accordance with the opening-angle experiment The modelling of damage is addressed with two major and different approaches. In the first approach a continuum local damage formulation with regularisation is presented. This formulation has two principal ingredients. On the one hand, it makes use of the principles of the smeared crack theory to avoid the mesh size dependence of the structural response in softening. On the other hand, it uses a Hodge-Petruska bidimensional model to describe the fibrils as staggered arrays of tropocollagen molecules, and from this mesoscopic model the macroscopic material properties of the collagen fibres are obtained using an homogenisation process. In the second approach a non-local gradient-enhanced damage formulation is introduced. The model is built around the enhancement of the free energy function by means of a term that contains the referential gradient of the non-local damage variable. The inclusion of this term ensures an implicit regularisation of the finite element implementation, yielding mesh-objective results of the simulations. The applicability of the later model to biomechanically-related problems is studied by means of the simulation of a typical surgical procedure, namely, the balloon angioplasty.

Modelling the size distribution of cosmic voids

The Standard Cosmological Model is generally accepted by the scientific community, there are still an amount of unresolved issues. From the observable characteristics of the structures in the Universe,it should be possible to impose constraints on the cosmological parameters. Cosmic Voids (CV) are a major component of the LSS and have been shown to possess great potential for constraining DE and testing theories of gravity. But a gap between CV observations and theory still persists. A theoretical model for void statistical distribution as a function of size exists (SvdW) However, the SvdW model has been unsuccesful in reproducing the results obtained from cosmological simulations. This undermines the possibility of using voids as cosmological probes. The goal of our thesis work is to cover the gap between theoretical predictions and measured distributions of cosmic voids. We develop an algorithm to identify voids in simulations,consistently with theory. We inspecting the possibilities offered by a recently proposed refinement of the SvdW (the Vdn model, Jennings et al., 2013). Comparing void catalogues to theory, we validate the Vdn model, finding that it is reliable over a large range of radii, at all the redshifts considered and for all the cosmological models inspected. We have then searched for a size function model for voids identified in a distribution of biased tracers. We find that, naively applying the same procedure used for the unbiased tracers to a halo mock distribution does not provide success- full results, suggesting that the Vdn model requires to be reconsidered when dealing with biased samples. Thus, we test two alternative exten- sions of the model and find that two scaling relations exist: both the Dark Matter void radii and the underlying Dark Matter density contrast scale with the halo-defined void radii. We use these findings to develop a semi-analytical model which gives promising results.