An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed the for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newlydeveloped e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.

An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields

In this paper, an enriched radial point interpolation method (e-RPIM) is developed the for the determination of crack tip fields. In e-RPIM, the conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched RBF meshfree shape functions is firstly investigated to fit different surfaces. The surface fitting results have proven that, comparing with the conventional RBF shape function, the enriched RBF shape function has: (1) a similar accuracy to fit a polynomial surface; (2) a much better accuracy to fit a trigonometric surface; and (3) a similar interpolation stability without increase of the condition number of the RBF interpolation matrix. Therefore, it has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF shape function, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF meshfree shape function and the meshfree weak-form. Several problems of linear fracture mechanics are simulated using this newlydeveloped e-RPIM method. It has demonstrated that the present e-RPIM is very accurate and stable, and it has a good potential to develop a practical simulation tool for fracture mechanics problems.

An enriched radial point interpolation method based on weak-form and strong-form

In this article, an enriched radial point interpolation method (e-RPIM) is developed for computational mechanics. The conventional radial basis function (RBF) interpolation is novelly augmented by the suitable basis functions to reflect the natural properties of deformation. The performance of the enriched meshless RBF shape functions is first investigated using the surface fitting. The surface fitting results have proven that, compared with the conventional RBF, the enriched RBF interpolation has a much better accuracy to fit a complex surface than the conventional RBF interpolation. It has proven that the enriched RBF shape function will not only possess all advantages of the conventional RBF interpolation, but also can accurately reflect the deformation properties of problems. The system of equations for two-dimensional solids is then derived based on the enriched RBF shape function and both of the meshless strong-form and weak-form. A numerical example of a bar is presented to study the effectiveness and efficiency of e-RPIM. As an important application, the newly developed e-RPIM, which is augmented by selected trigonometric basis functions, is applied to crack problems. It has been demonstrated that the present e-RPIM is very accurate and stable for fracture mechanics problems.

On Szegö’s Type Theorems

This study is to look the effect of change in the ordering of the Fourier system on Szegö’s classical observations of asymptotic distribution of eigenvalues of finite Toeplitz forms.This is done by checking proofs and Szegö’s properties in the new set up.The Fourier system is unconditional [19], any arbitrary ordering of the Fourier system forms a basis for the Hilbert space L2 [-Π, Π].Here study about the classical Szegö’s theorem.Szegö’s type theorem for operators in L2(R+) and check its validity for certain multiplication operators.Since the trigonometric basis is not available in L2(R+) or in L2(R) .This study discussed about the classes of orderings of Haar System in L2 (R+) and in L2(R) in which Szegö’s Type TheoreT Am is valid for certain multiplication operators.It is divided into two sections. In the first section there is an ordering to Haar system in L2(R+) and prove that with respect to this ordering, Szegö’s Type theorem holds for general class of multiplication operators Tƒ with multiplier ƒ ε L2(R+), subject to some conditions on ƒ.Finally in second section more general classes of ordering of Haar system in L2(R+) and in L2(R) are identified in such a way that for certain classes of multiplication operators the asymptotic distribution of eigenvalues exists.

Cooperative greedy pursuit strategies for sparse signal representation by partitioning

Cooperative Greedy Pursuit Strategies are considered for approximating a signal partition subjected to a global constraint on sparsity. The approach aims at producing a high quality sparse approximation of the whole signal, using highly coherent redundant dictionaries. The cooperation takes place by ranking the partition units for their sequential stepwise approximation, and is realized by means of i)forward steps for the upgrading of an approximation and/or ii) backward steps for the corresponding downgrading. The advantage of the strategy is illustrated by approximation of music signals using redundant trigonometric dictionaries. In addition to rendering stunning improvements in sparsity with respect to the concomitant trigonometric basis, these dictionaries enable a fast implementation of the approach via the Fast Fourier Transform

A dedicated greedy pursuit algorithm for sparse spectral representation of music sound

A dedicated algorithm for sparse spectral representation of music sound is presented. The goal is to enable the representation of a piece of music signal as a linear superposition of as few spectral components as possible, without affecting the quality of the reproduction. A representation of this nature is said to be sparse. In the present context sparsity is accomplished by greedy selection of the spectral components, from an overcomplete set called a dictionary. The proposed algorithm is tailored to be applied with trigonometric dictionaries. Its distinctive feature being that it avoids the need for the actual construction of the whole dictionary, by implementing the required operations via the fast Fourier transform. The achieved sparsity is theoretically equivalent to that rendered by the orthogonal matching pursuit (OMP) method. The contribution of the proposed dedicated implementation is to extend the applicability of the standard OMP algorithm, by reducing its storage and computational demands. The suitability of the approach for producing sparse spectral representation is illustrated by comparison with the traditional method, in the line of the short time Fourier transform, involving only the corresponding orthonormal trigonometric basis.

Membrane associated transporter protein gene (SLC45A2) and the genetic basis of normal human pigmentation variation

This work is concerned with the genetic basis of normal human pigmentation variation. Specifically, the role of polymorphisms within the solute carrier family 45 member 2 (SLC45A2 or membrane associated transporter protein; MATP) gene were investigated with respect to variation in hair, skin and eye colour ― both between and within populations. SLC45A2 is an important regulator of melanin production and mutations in the gene underly the most recently identified form of oculocutaneous albinism. There is evidence to suggest that non-synonymous polymorphisms in SLC45A2 are associated with normal pigmentation variation between populations. Therefore, the underlying hypothesis of this thesis is that polymorphisms in SLC45A2 will alter the function or regulation of the protein, thereby altering the important role it plays in melanogenesis and providing a mechanism for normal pigmentation variation. In order to investigate the role that SLC45A2 polymorphisms play in human pigmentation variation, a DNA database was established which collected pigmentation phenotypic information and blood samples of more than 700 individuals. This database was used as the foundation for two association studies outlined in this thesis, the first of which involved genotyping two previously-described non-synonymous polymorphisms, p.Glu272Lys and p.Phe374Leu, in four different population groups. For both polymorphisms, allele frequencies were significantly different between population groups and the 272Lys and 374Leu alleles were strongly associated with black hair, brown eyes and olive skin colour in Caucasians. This was the first report to show that SLC45A2 polymorphisms were associated with normal human intra-population pigmentation variation. The second association study involved genotyping several SLC45A2 promoter polymorphisms to determine if they also played a role in pigmentation variation. Firstly, the transcription start site (TSS), and hence putative proximal promoter region, was identified using 5' RNA ligase mediated rapid amplification of cDNA ends (RLM-RACE). Two alternate TSSs were identified and the putative promoter region was screened for novel polymorphisms using denaturing high performance liquid chromatography (dHPLC). A novel duplication (c.–1176_–1174dupAAT) was identified along with other previously described single nucleotide polymorphisms (c.–1721C>G and c.–1169G>A). Strong linkage disequilibrium ensured that all three polymorphisms were associated with skin colour such that the –1721G, +dup and –1169A alleles were associated with olive skin in Caucasians. No linkage disequilibrium was observed between the promoter and coding region polymorphisms, suggesting independent effects. The association analyses were complemented with functional data, showing that the –1721G, +dup and –1169A alleles significantly decreased SLC45A2 transcriptional activity. Based on in silico bioinformatic analysis that showed these alleles remove a microphthalmia-associated transcription factor (MITF) binding site, and that MITF is a known regulator of SLC45A2 (Baxter and Pavan, 2002; Du and Fisher, 2002), it was postulated that SLC45A2 promoter polymorphisms could contribute to the regulation of pigmentation by altering MITF binding affinity. Further characterisation of the SLC45A2 promoter was carried out using luciferase reporter assays to determine the transcriptional activity of different regions of the promoter. Five constructs were designed of increasing length and their promoter activity evaluated. Constitutive promoter activity was observed within the first ~200 bp and promoter activity increased as the construct size increased. The functional impact of the –1721G, +dup and –1169A alleles, which removed a MITF consensus binding site, were assessed using electrophoretic mobility shift assays (EMSA) and expression analysis of genotyped melanoblast and melanocyte cell lines. EMSA results confirmed that the promoter polymorphisms affected DNA-protein binding. Interestingly, however, the protein/s involved were not MITF, or at least MITF was not the protein directly binding to the DNA. In an effort to more thoroughly characterise the functional consequences of SLC45A2 promoter polymorphisms, the mRNA expression levels of SLC45A2 and MITF were determined in melanocyte/melanoblast cell lines. Based on SLC45A2’s role in processing and trafficking TYRP1 from the trans-Golgi network to stage 2 melanosmes, the mRNA expression of TYRP1 was also investigated. Expression results suggested a coordinated expression of pigmentation genes. This thesis has substantially contributed to the field of pigmentation by showing that SLC45A2 polymorphisms not only show allele frequency differences between population groups, but also contribute to normal pigmentation variation within a Caucasian population. In addition, promoter polymorphisms have been shown to have functional consequences for SLC45A2 transcription and the expression of other pigmentation genes. Combined, the data presented in this work supports the notion that SLC45A2 is an important contributor to normal pigmentation variation and should be the target of further research to elucidate its role in determining pigmentation phenotypes. Understanding SLC45A2’s function may lead to the development of therapeutic interventions for oculocutaneous albinism and other disorders of pigmentation. It may also help in our understanding of skin cancer susceptibility and evolutionary adaptation to different UV environments, and contribute to the forensic application of pigmentation phenotype prediction.