As ligas Al-Si hipoeutéticas têm grande importância na indústria de fundição de peças devido às excelentes propriedades de fundição, como baixo ponto de fusão e alta fluidez. A análise térmica das curvas de resfriamento medidas durante a solidificação destas ligas pode ser utilizada para controlar a formação da macroestrutura de grãos. Esta análise envolve a determinação das temperaturas de início e final de solidificação, bem como a evolução da fração de sólido com o tempo a partir da chamada análise térmica de Fourier. Apesar desta técnica ter sido aplicada a diversas ligas comerciais, existem poucos dados referentes às ligas Al-Si binárias. Os dados são ainda mais escassos quando se deseja examinar o efeito do tratamento de inoculação do metal líquido para refino de grão. O objetivo do presente trabalho é investigar o efeito do tratamento de inoculação nas ligas binárias Al-3%Si, Al-7%Si e Al-11%Si através da análise térmica e metalográfica. Foram obtidos lingotes cilíndricos a partir do vazamento da liga Al-Si líquida com ou sem a adição de inoculante na forma da liga mãe Al-3%Ti-1%B, adicionada para se obter um teor nominal de 0,05%Ti. Curvas de resfriamento foram medidas a partir de termopares inseridos no interior da cavidade do molde...

This study sought to differentiate the species of skates encountered in Gulf St Vincent (GSV), South Australia using normalized elliptical Fourier analysis of body shape. Significant intraspecific variation was observed among whole body shapes. This was overcome by limiting subsequent analyses to the anterior snout region, where significant differences in shape were detected among the species examined and provided a high degree of classification success for the skates of GSV. More generally, this approach has the potential to provide a cost- and time-efficient means of discrimination among species of skates. Further research is required to investigate the potentially confounding effects of sexual dimorphism and ontogenetic variation in growth to improve the efficacy of the body shape analysis of the skates and batoids species in general. In addition, this approach requires considerable development to facilitate implementation in a fishery setting.; C. Izzo and E. B. Drew

Twins studies provide a powerful approach to determining the relative contribution of genetics and environment to observed variation. Such studies assume trait differences in monozygous (MZ) twins are due to environmental factors and those in dizygous (DZ) twins are due to both genetic and environmental factors. This study quantitated facial profiles of twins using Fourier equations, determining their value in profile analysis and the assessment of the genetic contribution to facial shape. Standardized profile slide photographs of 79 pairs of 4-6 year-old twins (37 MZ pairs, 42 DZ pairs) were scanned and x and y coordinates were extracted from each profile using sellion and Camper's plane as references. The coordinates were subjected to Fourier analysis and the normalised vertex projection coefficients were studied. The means of the differences between coefficients for MZ co-twins did not differ significantly from that of DZ co-twins, although the DZ group showed higher mean differences in the higher harmonics. Subjective examination of superimposed reconstructions showed wider variation between DZ co-twins than MZ co-twins. Correct classification of twins by discriminant function analysis using Fourier coefficients was similar for both groups (MZ: 70.3%; DZ: 73.8%). Fourier analysis could quantitate facial profiles of young children and differentiate some details...

In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.

This manuscript presents shortly the results obtained by participants of the scientific seminar which is held more than twenty years under leadership of the author at Donetsk University. In the list of references main publications are given. These results are published in serious scientific journals and reported at various conferences, including international ones at Moscow,ICM66; Kaluga,1975; Kiev,1983; Haifa,1994; Z\"urich,ICM94; Moscow,1995. The area of investigation is the Fourier analysis and the theory of approximation of functions. Used are methods of classical analysis including special functions, Banach spaces, etc., of harmonic analysis in finitedimensional Euclidean space, of Diophantine analysis, of random choice, etc. The results due to the author and active participants of the seminar, namely E. S. Belinskii, O. I. Kuznetsova, E. R. Liflyand, Yu. L. Nosenko, V. A. Glukhov, V. P. Zastavny, Val. V. Volchkov, V. O. Leontyev, and others, are given. Besides the participants of the seminar and other mathematicians from Donetsk, many mathematicians from other places were speakers at the seminar, in particular, A.A. Privalov, Z.A. Chanturia, Yu.A. Brudnyi, N.Ya. Krugljak, V.N. Temlyakov, B.D. Kotlyar, A.N. Podkorytov, M.A. Skopina...

A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^3$.; Comment: 31 pages, multiple figures

This is a survey on the subject of the title corresponding to three lectures I gave in June 2001 at the Workshop on Fourier Analysis and Convexity, at the Universita di Milano-Biccoca.; Comment: 52 pages, 15 figures 2 references added (Leptin and Mueller, Mackey) from first version

In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient for investigating evolutionary fluid mechanics equations in the whole space or in the torus. We here give an overview of results that we can get by Fourier analysis and paradifferential calculus, for the compressible Navier-Stokes equations. We focus on the Initial Value Problem in the case where the fluid domain is the whole space or the torus in dimension at least two, and also establish some asymptotic properties of global small solutions. The time decay estimates in the critical regularity framework that are stated at the end of the survey are new, to the best of our knowledge.

In this paper a local Fourier analysis for multigrid methods on tetrahedral grids is presented. Different smoothers for the discretization of the Laplace operator by linear finite elements on such grids are analyzed. A four-color smoother is presented as an efficient choice for regular tetrahedral grids, whereas line and plane relaxations are needed for poorly shaped tetrahedra. A novel partitioning of the Fourier space is proposed to analyze the four-color smoother. Numerical test calculations validate the theoretical predictions. A multigrid method is constructed in a block-wise form, by using different smoothers and different numbers of pre- and post-smoothing steps in each tetrahedron of the coarsest grid of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm.

The Helmholtz equation arises in many applications, such as seismic and medical imaging. These application are characterized by the need to propagate many wavelengths through an inhomogeneous medium. The typical size of the problems in 3D applications precludes the use of direct factorization to solve the equation and hence iterative methods are used in practice. For higher wavenumbers, the system becomes increasingly indefinite and thus good preconditioners need to be constructed. In this note we consider an accelerated Kazcmarz method (CGMN) and present an expression for the resulting iteration matrix. This iteration matrix can be used to analyze the convergence of the CGMN method. In particular, we present a Fourier analysis for the method applied to the 1D Helmholtz equation. This analysis suggests an optimal choice of the relaxation parameter. Finally, we present some numerical experiments.

A discrete Fourier analysis on the fundamental domain $\Omega_d$ of the $d$-dimensional lattice of type $A_d$ is studied, where $\Omega_2$ is the regular hexagon and $\Omega_3$ is the rhombic dodecahedron, and analogous results on $d$-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^d$. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.; Comment: 39 pages, 10 figures

We have developed a physically-intuitive method to calculate the local lattice constant as a function of position in a high-resolution transmission electron microscopy image by performing a two-dimensional fast Fourier transform. We apply a Gaussian filter with appropriate spatial full-width-half-max (FWHM) bandwidth to the image centered at the desired location to calculate the local lattice constant (as opposed to the average lattice constant). Fourier analysis of the filtered image yields the vertical and horizontal lattice constants at this location. The process is repeated by stepping the Gaussian filter across the image to produce a set of local lattice constants in the vertical and horizontal direction as a function of position in the image. The method has been implemented in a freely available tool on nanoHUB.; Comment: 5 pages, 3 figures

Let $H$ and $K$ be locally compact groups and also $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $\tau$. This paper presents a novel approach to the Fourier analysis of $G_\tau$, when $K$ is abelian. We define the $\tau$-dual group $G_{\hat{\tau}}$ of $G_\tau$ as the semi-direct product $H\ltimes_{\hat{\tau}}\hat{K}$, where $\hat{\tau}:H\to Aut(\hat{K})$ defined via (\ref{A}). We prove a Ponterjagin duality Theorem and also we study $\tau$-Fourier transforms on $G_\tau$. As a concrete application we show that how these techniques apply for the affine group and also we compute the $\tau$-dual group of Euclidean groups and the Weyl-Heisenberg groups.; Comment: This paper has been withdrawn by the author due to a crucial sign error in equations and references

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact $k$-step nilspaces. As a byproduct of our results we obtain a new type of limit theory for functions on abelian groups in the spirit of the so-called graph limit theory. Our proofs are based on an exact (non-approximative) version of higher order Fourier analysis which appears on ultra product groups.; Comment: arXiv admin note: substantial text overlap with arXiv:1010.6211

A general local Fourier analysis for overlapping block smoothers on triangular grids is presented. This analysis is explained in a general form for its application to problems with different discretizations. This tool is demonstrated for two different problems: a stabilized linear finite element discretization of Stokes equations and an edge-based discretization of the curl-curl operator by lowest-order N\'ed\'elec finite element method. In this latter, special Fourier modes have to be considered in order to perform the analysis. Numerical results comparing two- and three-grid convergence factors predicted by the local Fourier analysis to real asymptotic convergence factors are presented to confirm the predictions of the analysis and show their usefulness.

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.; Comment: 29 pages

To understand methodological features of the detrended fluctuation analysis (DFA) using a higher-order polynomial fitting, we establish the direct connection between DFA and Fourier analysis. Based on an exact calculation of the single-frequency response of the DFA, the following facts are shown analytically: (1) in the analysis of stochastic processes exhibiting a power-law scaling of the power spectral density (PSD), $S(f) \sim f^{-\beta}$, a higher-order detrending in the DFA has no adverse effect in the estimation of the DFA scaling exponent $\alpha$, which satisfies the scaling relation $\alpha = (\beta+1)/2$; (2) the upper limit of the scaling exponents detectable by the DFA depends on the order of polynomial fit used in the DFA, and is bounded by $m + 1$, where $m$ is the order of the polynomial fit; (3) the relation between the time scale in the DFA and the corresponding frequency in the PSD are distorted depending on both the order of the DFA and the frequency dependence of the PSD. We can improve the scale distortion by introducing the corrected time scale in the DFA corresponding to the inverse of the frequency scale in the PSD. In addition, our analytical approach makes it possible to characterize variants of the DFA using different types of detrending. As an application...

Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a Fourier series and to use this representation to study certain properties of the particular numerical method, like the convergence rate or an error estimate. In the process of applying a local Fourier analysis, it is typically necessary to determine the supremum of a more or less complicated term with respect to all frequencies and, potentially, other variables. The problem of computing such a supremum can be rewritten as a quantifier elimination problem, which can be solved with cylindrical algebraic decomposition, a well-known tool from symbolic computation. The combination of local Fourier analysis and cylindrical algebraic decomposition is a machinery that can be applied to a wide class of problems. In the present paper, we will discuss two examples. The first example is to compute the convergence rate of a multigrid method. As second example we will see that the machinery can also be used to do something rather different: We will compare approximation error estimates for different kinds of discretizations.; Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N25

10 pages, 6 figures, 5 tables.; Bivalves are excellent candidates for geographically based studies of the morphological variation in individuals of different populations based on the analysis of their shape profiles. In this study, we quantified the overall shell shape differences in individuals of different populations of Ruditapes decussatus and Ruditapes philippinarum in relation to their geographical and genetic distances. A total of 395 and 124 individuals of R. decussatus (nine populations) and R. philippinarum (four populations), respectively, were sampled in different Mediterranean and Atlantic coastal locations. Pictures of the left valve were taken from all individuals. Their profiles were analysed using elliptic Fourier analysis. Mean outlines were computed. In order to classify different individuals for species, the coefficients of harmonic equations were analysed by partial least square discriminant analysis and soft independent modelling of class analogy. The results showed a high percentage of correct classification (99%) between the two species in the independent test. We found that the morphological distance between R. philippinarum and R. decussatus is higher than the morphological distance among populations of the same species. The absence of correspondence between the geographical location and the pattern of morphological and genetic variation indicates the occurrence of a reaction norm in the morphological adaptation of shell shapes to different local environmental conditions.; We would like to thank Davide Cascione for his help during the process of image acquisition. Jacopo Aguzzi is a Fellow of the ‘Juan de la Cierva’ Postdoctoral Program (MECSpain).; Peer reviewed

In order to ensure that electrical energy reaches consumers uninterrupted, researchers constantly try to improve power transmission lines. To realize this improvement, probable faults should be analysed through every known method, and new methods should also be implemented. In this study, firstly, the Keban power transmission line located in the Eastern Anatolia region of Turkey was modelled. After that, probable short circuit scenarios were applied on the model, and the short circuit faults in the scenarios were analysed by using the Fourier analysis. The Fourier analysis is a mathematical method that is used as an effective way to determine the sudden changes in the frequency and time band. The study was successful in determining phase and grounding faults through the analyses of the scenarios using Fourier analysis. The fact that the mathematical method was applied on the probable scenarios on a physical model increases the importance of the study.