Boundary value problems for stochastic differential equations

A theory of two-point boundary value problems analogous to the theory of initial value problems for stochastic ordinary differential equations whose solutions form Markov processes is developed. The theory of initial value problems consists of three main parts: the proof that the solution process is markovian and diffusive; the construction of the Kolmogorov or Fokker-Planck equation of the process; and the proof that the transistion probability density of the process is a unique solution of the Fokker-Planck equation.

It is assumed here that the stochastic differential equation under consideration has, as an initial value problem, a diffusive markovian solution process. When a given boundary value problem for this stochastic equation almost surely has unique solutions, we show that the solution process of the boundary value problem is also a diffusive Markov process. Since a boundary value problem, unlike an initial value problem, has no preferred direction for the parameter set, we find that there are two Fokker-Planck equations, one for each direction. It is shown that the density of the solution process of the boundary value problem is the unique simultaneous solution of this pair of Fokker-Planck equations.

This theory is then applied to the problem of a vibrating string with stochastic density.

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade (HT-7U) when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode (m, n) helical perturbation can cause the formation of islands on rational surfaces with q = m/n and q = (m +/- 1, +/- 2, +/- 3,...)/n due to the toroidicity and plasma shape (i.e. elongation and triangularity), while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser (KAM) barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field B-r(r) and the toroidal magnetic field amplitude B(phi)0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.

Diffraction-limited imaging on the 200-inch telescope

We have used the technique of non-redundant masking at the Palomar 200-inch telescope and radio VLBI imaging software to make optical aperture synthesis maps of two binary stars, β Corona Borealis and σ Herculis. The dynamic range of the map of β CrB, a binary star with a separation of 230 milliarcseconds is 50:1. For σ Her, we find a separation of 70 milliarcseconds and the dynamic range of our image is 30:1. These demonstrate the potential of the non-redundant masking technique for diffraction-limited imaging of astronomical objects with high dynamic range.

We find that the optimal integration time for measuring the closure phase is longer than that for measuring the fringe amplitude. There is not a close relationship between amplitude errors and phase errors, as is found in radio interferometry. Amplitude self calibration is less effective at optical wavelengths than at radio wavelengths. Primary beam sensitivity correction made in radio aperture synthesis is not necessary in optical aperture synthesis.

The effects of atmospheric disturbances on optical aperture synthesis have been studied by Monte Carlo simulations based on the Kolmogorov theory of refractive-index fluctuations. For the non-redundant masking with τ_c-sized apertures, the simulated fringe amplitude gives an upper bound of the observed fringe amplitude. A smooth transition is seen from the non-redundant masking regime to the speckle regime with increasing aperture size. The fractional reduction of the fringe amplitude according to the bandwidth is nearly independent of the aperture size. The limiting magnitude of optical aperture synthesis with τ_c-sized apertures and that with apertures larger than τ_c are derived.

Monte Carlo simulations are also made to study the sensitivity and resolution of the bispectral analysis of speckle interferometry. We present the bispectral modulation transfer function and its signal-to-noise ratio at high light levels. The results confirm the validity of the heuristic interferometric view of image-forming process in the mid-spatial-frequency range. The signal-to- noise ratio of the bispectrum at arbitrary light levels is derived in the mid-spatial-frequency range.

The non-redundant masking technique is suitable for imaging bright objects with high resolution and high dynamic range, while the faintest limit will be better pursued by speckle imaging.

Towards understanding the mixing characteristics of turbulent buoyant flows

This work proposes a new simulation methodology in which variable density turbulent flows can be studied in the context of a mixing layer with or without the presence of gravity. Specifically, this methodology is developed to probe the nature of non-buoyantly-driven (i.e. isotropically-driven) or buoyantly-driven mixing deep inside a mixing layer. Numerical forcing methods are incorporated into both the velocity and scalar fields, which extends the length of time over which mixing physics can be studied. The simulation framework is designed to allow for independent variation of four non-dimensional parameters, including the Reynolds, Richardson, Atwood, and Schmidt numbers. Additionally, the governing equations are integrated in such a way to allow for the relative magnitude of buoyant energy production and non-buoyant energy production to be varied.

The computational requirements needed to implement the proposed configuration are presented. They are justified in terms of grid resolution, order of accuracy, and transport scheme. Canonical features of turbulent buoyant flows are reproduced as validation of the proposed methodology. These features include the recovery of isotropic Kolmogorov scales under buoyant and non-buoyant conditions, the recovery of anisotropic one-dimensional energy spectra under buoyant conditions, and the preservation of known statistical distributions in the scalar field, as found in other DNS studies.

This simulation methodology is used to perform a parametric study of turbulent buoyant flows to discern the effects of varying the Reynolds, Richardson, and Atwood numbers on the resulting state of mixing. The effects of the Reynolds and Atwood numbers are isolated by looking at two energy dissipation rate conditions under non-buoyant (variable density) and constant density conditions. The effects of Richardson number are isolated by varying the ratio of buoyant energy production to total energy production from zero (non-buoyant) to one (entirely buoyant) under constant Atwood number, Schmidt number, and energy dissipation rate conditions. It is found that the major differences between non-buoyant and buoyant turbulent flows are contained in the transfer spectrum and longitudinal structure functions, while all other metrics are largely similar (e.g. energy spectra, alignment characteristics of the strain-rate tensor). Also, despite the differences noted between fully buoyant and non-buoyant turbulent fields, the scalar field, in all cases, is unchanged by these. The mixing dynamics in the scalar field are found to be insensitive to the source of turbulent kinetic energy production (non-buoyant vs. buoyant).

A theory of strong and weak scintillations with applications to astrophysics

The propagation of waves in an extended, irregular medium is studied under the "quasi-optics" and the "Markov random process" approximations. Under these assumptions, a Fokker-Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the characteristic functional. The derivation does not require Gaussian statistics of the random medium and the result can be applied to the time-dependent problem. We then solve the moment equations for the phase correlation function, angular broadening, temporal pulse smearing, intensity correlation function, and the probability distribution of the random waves. The necessary and sufficient conditions for strong scintillation are also given.

We also consider the problem of diffraction of waves by a random, phase-changing screen. The intensity correlation function is solved in the whole Fresnel diffraction region and the temporal pulse broadening function is derived rigorously from the wave equation.

The method of smooth perturbations is applied to interplanetary scintillations. We formulate and calculate the effects of the solar-wind velocity fluctuations on the observed intensity power spectrum and on the ratio of the observed "pattern" velocity and the true velocity of the solar wind in the three-dimensional spherical model. The r.m.s. solar-wind velocity fluctuations are found to be ~200 km/sec in the region about 20 solar radii from the Sun.

We then interpret the observed interstellar scintillation data using the theories derived under the Markov approximation, which are also valid for the strong scintillation. We find that the Kolmogorov power-law spectrum with an outer scale of 10 to 100 pc fits the scintillation data and that the ambient averaged electron density in the interstellar medium is about 0.025 cm^-3. It is also found that there exists a region of strong electron density fluctuation with thickness ~10 pc and mean electron density ~7 cm^-3 between the PSR 0833-45 pulsar and the earth.

Small scale turbulence in high Karlovitz number premixed flames

The purpose of this thesis is to characterize the behavior of the smallest turbulent scales in high Karlovitz number (Ka) premixed flames. These scales are particularly important in the two-way coupling between turbulence and chemistry and better understanding of these scales will support future modeling efforts using large eddy simulations (LES). The smallest turbulent scales are studied by considering the vorticity vector, ω, and its transport equation.

Due to the complexity of turbulent combustion introduced by the wide range of length and time scales, the two-dimensional vortex-flame interaction is first studied as a simplified test case. Numerical and analytical techniques are used to discern the dominate transport terms and their effects on vorticity based on the initial size and strength of the vortex. This description of the effects of the flame on a vortex provides a foundation for investigating vorticity in turbulent combustion.

Subsequently, enstrophy, ω² = ω • ω, and its transport equation are investigated in premixed turbulent combustion. For this purpose, a series of direct numerical simulations (DNS) of premixed n-heptane/air flames are performed, the conditions of which span a wide range of unburnt Karlovitz numbers and turbulent Reynolds numbers. Theoretical scaling analysis along with the DNS results support that, at high Karlovitz number, enstrophy transport is controlled by the viscous dissipation and vortex stretching/production terms. As a result, vorticity scales throughout the flame with the inverse of the Kolmogorov time scale, τ_η, just as in homogeneous isotropic turbulence. As τ_η is only a function of the viscosity and dissipation rate, this supports the validity of Kolmogorov’s first similarity hypothesis for sufficiently high Ka numbers (Ka ≳ 100). These conclusions are in contrast to low Karlovitz number behavior, where dilatation and baroclinic torque have a significant impact on vorticity within the flame. Results are unaffected by the transport model, chemical model, turbulent Reynolds number, and lastly the physical configuration.

Next, the isotropy of vorticity is assessed. It is found that given a sufficiently large value of the Karlovitz number (Ka ≳ 100) the vorticity is isotropic. At lower Karlovitz numbers, anisotropy develops due to the effects of the flame on the vortex stretching/production term. In this case, the local dynamics of vorticity in the strain-rate tensor, S, eigenframe are altered by the flame. At sufficiently high Karlovitz numbers, the dynamics of vorticity in this eigenframe resemble that of homogeneous isotropic turbulence.

Combined, the results of this thesis support that both the magnitude and orientation of vorticity resemble the behavior of homogeneous isotropic turbulence, given a sufficiently high Karlovitz number (Ka ≳ 100). This supports the validity of Kolmogorov’s first similarity hypothesis and the hypothesis of local isotropy under these condition. However, dramatically different behavior is found at lower Karlovitz numbers. These conclusions provides/suggests directions for modeling high Karlovitz number premixed flames using LES. With more accurate models, the design of aircraft combustors and other combustion based devices may better mitigate the detrimental effects of combustion, from reducing CO₂ and soot production to increasing engine efficiency.

Formação in vitro de biofilme de Candida albicans em materiais usados no preenchimento dos acessos aos parafusos das próteses sobre implantes

Os orifícios de acesso aos parafusos de retenção devem ser preenchidos para que o parafuso não seja danificado caso seja necessária a remoção da prótese. Dentre os materiais mais utilizados estão o algodão, a fita de politetrafluoretileno e a guta percha. O objetivo deste estudo é avaliar a formação de biofilme de Candida albicans nos materiais anteriormente descritos, buscando estabelecer um parâmetro que contribua para a escolha do tipo de material mais adequado a ser utilizado clinicamente. Foram utilizados UCLAs, análogos e parafusos sextavados, todos de titânio. Os conjuntos foram montados com torque de 32N. Os materiais foram condensados no interior dos UCLAs e colocados em meio de cultura com uma suspensão de 3x106 células/ml de Candida albicans. O sistema foi armazenado à 37C com agitação, por 15 dias e o meio foi renovado a cada 48 horas. A quantificação de biofilme foi realizada pelo ensaio de MTT e leitura à 490nm, resultando em diferentes valores de densidade óptica. A normalidade (p=0,304 - Kolmogorov-Smirnov) e a igualdade de variâncias (p=0,721 - Scheffe) foram testadas primeiramente. O teste de análise de variância demonstrou diferença significativa entre os grupos (p<0,001) e com o Holm-Sidak foi observada diferença significativa entre os grupos algodão e guta (p<0,05) e algodão e fita de politetrafluoetileno (p<0,05); não houve diferença significativa entre os grupos guta e fita de politatrafluoretileno (p>0,05), apesar dos valores da fita de politetrafluoetileno terem sido maiores. Considerando-se as limitações deste estudo in vitro, podemos concluir que tanto a guta-percha quanto a fita de politetrafluoretileno apresentaram menor formação de biofilme, não havendo diferença estatisticamente significativa entre os materiais. O algodão apresentou um nível de formação de biofilme significativamente maior que a fita de politetrafluoretileno e a guta percha. Diante disso, serão necessários novos estudos para confirmar as limitações que este tipo de material pode apresentar quando usado como material de preenchimento do acesso do parafuso da prótese sobre implante.

Postrelease survival, vertical and horizontal movements, and thermal habitats of five species of pelagic sharks in the central Pacific Ocean

From 2001 to 2006, 71 pop-up satellite archival tags (PSATs) were deployed on five species of pelagic shark (blue shark [Prionace glauca]; shortfin mako [Isurus oxyrinchus]; silky shark [Carcharhinus falciformis]; oceanic whitetip shark [C. longimanus]; and bigeye thresher [Alopias superciliosus]) in the central Pacific Ocean to determine species-specific movement patterns and survival rates after release from longline fishing gear. Only a single postrelease mortality could be unequivocally documented: a male blue shark which succumbed seven days after release. Meta-analysis of published reports and the current study (n=78 reporting PSATs) indicated that the summary effect of postrelease mortality for blue sharks was 15% (95% CI, 8.5–25.1%) and suggested that catch-and-release in longline fisheries can be a viable management tool to protect parental biomass in shark populations. Pelagic sharks displayed species-specific depth and temperature ranges, although with significant individual temporal and spatial variability in vertical movement patterns, which were also punctuated by stochastic events (e.g., El Niño-Southern Oscillation). Pelagic species can be separated into three broad groups based on daytime temperature preferences by using the unweighted pair-group method with arithmetic averaging clustering on a Kolmogorov-Smirnov Dmax distance matrix: 1) epipelagic species (silky and oceanic whitetip sharks), which spent >95% of their time at temperatures within 2°C of sea surface temperature; 2) mesopelagic-I species (blue sharks and shortfin makos, which spent 95% of their time at temperatures from 9.7° to 26.9°C and from 9.4° to 25.0°C, respectively; and 3) mesopelagic-II species (bigeye threshers), which spent 95% of their time at temperatures from 6.7° to 21.2°C. Distinct thermal niche partitioning based on body size and latitude was also evident within epipelagic species.

Evaluation of a Pound Net Leader Designed to Reduce Sea Turtle Bycatch

Offshore pound net leaders in the southern portion of Chesapeake Bay in Virginia waters were documented to incidentally take protected loggerhead, Caretta caretta, and Kemp’s ridley, Lepidochelys kempii, sea turtles. Because of these losses, NOAA’s National Marine Fisheries Service (NMFS) in 2004 closed the area to offshore pound net leaders annually from 6 May to 15 July and initiated a study of an experimental leader design that replaced the top two-thirds of the traditional mesh panel leader with vertical ropes (0.95 cm) spaced 61 cm apart. This experimental leader was tested on four pound net sites on the eastern shore of Chesapeake Bay in 2004 and 2005. During the 2 trial periods, 21 loggerhead and Kemp’s ridley sea turtles were found interacting with the control leader and 1 leatherback turtle, Dermochelys coriacea, was found interacting with the experimental leader. Results of a negative binomial regression analysis comparing the two leader designs found the experimental leader significantly reduced sea turtle interactions (p=0.03). Finfish were sampled from the pound nets in the study to assess finfish catch performance differences between the two leader designs. Although the conclusions from this element of the experiment are not robust, paired t-test and Wilcoxon signed rank test results determined no significant harvest weight difference between the two leaders. Kolmogorov-Smirnov tests did not reveal any substantive size selectivity differences between the two leaders.

Qualidade do debridamento obtido pela técnica de instrumentação da lima única de NiTi em canais ovais e circulares

O presente avalia a qualidade da remoção de tecido pulpar após o preparo químico-cirúrgico realizado com a técnica da lima única, descrita por Ghassan Yared. Ainda não há publicado pesquisa sobre os resultados desta técnica. Este estudo compara o percentual de tecido pulpar remanescente em canais radiculares ovais e circulares de incisivos inferiores recém-extraídos que possuíssem polpa viva e armazenados em formol a 10%. Foram comparadas duas técnicas: ProTaper Universal e a técnica da lima única F2. Após uma rigorosa seleção, quarenta e oito dentes com polpa viva que possuíam indicação de extração, foram preparados, classificados em canais ovais e circulares, separados aleatoriamente em 4 grupos e instrumentados com as duas técnicas. O grupo controle, com 12 espécimes, não recebeu nenhum tipo de intervenção. G1 (n=12), canais ovais, instrumentados com a técnica ProTaper Universal; G2 (n=12), canais ovais instrumentados com a técnica da lima única F2; G3 (n=12), canais circulares instrumentados com a técnica ProTaper Universal; G4 (n=12), canais circulares instrumentados com a técnica da lima única F2. Então, seções transversais foram preparadas para avaliação histológica. A análise da quantidade de tecido pulpar remanescente foi avaliada digitalmente. A análise preliminar dos dados brutos em conjunto de todos os grupos experimentais revelou um padrão de distribuição normal por meio do uso do teste Kolmogorov-Smirnov. A análise foi realizada, e os dados brutos foram avaliados através de métodos não-paramétricos: Teste H Kruskal-Wallis. O valor percentual mínimo de tecido remanescente foi de 0% e o máximo de 37,78% entre todos os grupos. Os valores relativos a quantidade de tecido pulpar remanescente variaram entre 0 a 43.47% m2. Os resultados do Teste H Kruskal-Wallis não revelaram diferenças entre as seções mais apicais (p > 0.05). Entretanto, foi encontrada diferença significante entre as seções mais apicais e a seção do terço médio (p < 0.05). Também foram encontradas diferenças significantes quando canais circulares foram comparados com canais ovais independente da técnica de instrumentação utilizada (p < 0.05). Porém, entre as duas técnicas de instrumentação estudadas, tanto nos canais ovais quanto para os os canais circulares, não houve diferença estatística significante (p > 0.05). A proposta deste estudo é a de fazer uma reflexão sobre a real necessidade de um grande número de instrumentos para o total preparo de canais radiculares, uma vez que nenhuma das técnicas foi capaz de debridar por completo o espaço do canal radicular.

The non-local nature of structure functions

Kolmogorov's two-thirds, ((Δv) 2) ∼ e 2/ 3r 2/ 3, and five-thirds, E ∼ e 2/ 3k -5/ 3, laws are formally equivalent in the limit of vanishing viscosity, v → 0. However, for most Reynolds numbers encountered in laboratory scale experiments, or numerical simulations, it is invariably easier to observe the five-thirds law. By creating artificial fields of isotropic turbulence composed of a random sea of Gaussian eddies whose size and energy distribution can be controlled, we show why this is the case. The energy of eddies of scale, s, is shown to vary as s 2/ 3, in accordance with Kolmogorov's 1941 law, and we vary the range of scales, γ = s max/s min, in any one realisation from γ = 25 to γ = 800. This is equivalent to varying the Reynolds number in an experiment from R λ = 60 to R λ = 600. While there is some evidence of a five-thirds law for g > 50 (R λ > 100), the two-thirds law only starts to become apparent when g approaches 200 (R λ ∼ 240). The reason for this discrepancy is that the second-order structure function is a poor filter, mixing information about energy and enstrophy, and from scales larger and smaller than r. In particular, in the inertial range, ((Δv) 2) takes the form of a mixed power-law, a 1+a 2r 2+a 3r 2/ 3, where a 2r 2 tracks the variation in enstrophy and a 3r 2/ 3 the variation in energy. These findings are shown to be consistent with experimental data where the polution of the r 2/ 3 law by the enstrophy contribution, a 2r 2, is clearly evident. We show that higherorder structure functions (of even order) suffer from a similar deficiency.

Construction of nonparametric Bayesian models from parametric Bayes equations

We consider the general problem of constructing nonparametric Bayesian models on infinite-dimensional random objects, such as functions, infinite graphs or infinite permutations. The problem has generated much interest in machine learning, where it is treated heuristically, but has not been studied in full generality in non-parametric Bayesian statistics, which tends to focus on models over probability distributions. Our approach applies a standard tool of stochastic process theory, the construction of stochastic processes from their finite-dimensional marginal distributions. The main contribution of the paper is a generalization of the classic Kolmogorov extension theorem to conditional probabilities. This extension allows a rigorous construction of nonparametric Bayesian models from systems of finite-dimensional, parametric Bayes equations. Using this approach, we show (i) how existence of a conjugate posterior for the nonparametric model can be guaranteed by choosing conjugate finite-dimensional models in the construction, (ii) how the mapping to the posterior parameters of the nonparametric model can be explicitly determined, and (iii) that the construction of conjugate models in essence requires the finite-dimensional models to be in the exponential family. As an application of our constructive framework, we derive a model on infinite permutations, the nonparametric Bayesian analogue of a model recently proposed for the analysis of rank data.

Geometry and interaction of structures in homogeneous isotropic turbulence

A strategy to extract turbulence structures from direct numerical simulation (DNS) data is described along with a systematic analysis of geometry and spatial distribution of the educed structures. A DNS dataset of decaying homogeneous isotropic turbulence at Reynolds number Reλ = 141 is considered. A bandpass filtering procedure is shown to be effective in extracting enstrophy and dissipation structures with their smallest scales matching the filter width, L. The geometry of these educed structures is characterized and classified through the use of two non-dimensional quantities, planarity' and filamentarity', obtained using the Minkowski functionals. The planarity increases gradually by a small amount as L is decreased, and its narrow variation suggests a nearly circular cross-section for the educed structures. The filamentarity increases significantly as L decreases demonstrating that the educed structures become progressively more tubular. An analysis of the preferential alignment between the filtered strain and vorticity fields reveals that vortical structures of a given scale L are most likely to align with the largest extensional strain at a scale 3-5 times larger than L. This is consistent with the classical energy cascade picture, in which vortices of a given scale are stretched by and absorb energy from structures of a somewhat larger scale. The spatial distribution of the educed structures shows that the enstrophy structures at the 5η scale (where η is the Kolmogorov scale) are more concentrated near the ones that are 3-5 times larger, which gives further support to the classical picture. Finally, it is shown by analysing the volume fraction of the educed enstrophy structures that there is a tendency for them to cluster around a larger structure or clusters of larger structures. Copyright © 2012 Cambridge University Press.

Nucleation, growth, and control of ferroelectric-ferroelastic domains in thin polycrystalline films

The unique response of ferroic materials to external excitations facilitates them for diverse technologies, such as nonvolatile memory devices. The primary driving force behind this response is encoded in domain switching. In bulk ferroics, domains switch in a two-step process: nucleation and growth. For ferroelectrics, this can be explained by the Kolmogorov-Avrami-Ishibashi (KAI) model. Nevertheless, it is unclear whether domains remain correlated in finite geometries, as required by the KAI model. Moreover, although ferroelastic domains exist in many ferroelectrics, experimental limitations have hindered the study of their switching mechanisms. This uncertainty limits our understanding of domain switching and controllability, preventing thin-film and polycrystalline ferroelectrics from reaching their full technological potential. Here we used piezoresponse force microscopy to study the switching mechanisms of ferroelectric-ferroelastic domains in thin polycrystalline Pb 0.7Zr0.3TiO3 films at the nanometer scale. We have found that switched biferroic domains can nucleate at multiple sites with a coherence length that may span several grains, and that nucleators merge to form mesoscale domains, in a manner consistent with that expected from the KAI model. © 2012 American Physical Society.

动物能量密度(能值)分布规律的研究

为了考察动物能量密度的分布规律 ,考虑到目前动物能量密度研究多限于陆生动物 ,补充测定了 72种常见水生无脊椎动物的能量密度 ,对近 2 3 0种动物的能量密度的分布规律进行了分析研究 ,并经Kolmogorov smirno检验 ,首次证实了动物能量密度分布规律为正态分布