Spherical Harmonics

The physalis method is suitable for the simulation of flows with suspended spherical particles. It differs from standard immersed boundary methods due to the use of a local spectral representation of the solution in the neighborhood of each particle, which is used to bridge the gap between the particle surface and the underlying fixed Cartesian grid. This analytic solution involves coefficients which are determined by matching with the finitedifference solution farther away from the particle. In the original implementation of the method this step was executed by solving an over-determined linear system via the singularvalue decomposition. Here a more efficient method to achieve the same end is described. The basic idea is to use scalar products of the finite-difference solutions with spherical harmonic functions taken over a spherical surface concentric with the particle. The new approach is tested on a number of examples and is found to posses a comparable accuracy to the original one, but to be significantly faster and to require less memory. An unusual test case that we describe demonstrates the accuracy with which the method conserves the fluid angular momentum in the case of a rotating particle.

Traditional QNLS delivers spectral snapshots; health, however, is a process unfolding through rhythms and cycles. This article introduces a temporal extension built on two complementary ideas. First, spherical-time harmonics (Sⁿ) model time as a compact manifold with orthogonal modes (analogous to spatial spherical harmonics), enabling decomposition of rhythms and quantification of entropy as scattering into higher-order modes. Second, temporal hydrodynamics (Chronoflux-inspired) treats changes in coherence as flows with velocity, acceleration, and turbulence. We outline a practical pipeline—capture → Sⁿ decomposition → flow modeling → visualization as temporal vector maps—and discuss benefits (early drift detection, differentiation of transient vs. persistent change) and challenges (longitudinal data, computational cost, operator-state regressors). The proposal is speculative but actionable: it reframes QNLS from static diagnosis to predictive, process-aware monitoring, aligning measurement with chronobiology and adaptive feedback protocols.

In vivo quantification of neuroanatomical shape variations is possible due to recent advances in medical imaging and has proven useful in the study of neuropathology and neurodevelopment. In this paper, we apply a spherical wavelet transformation to extract shape features of cortical surfaces reconstructed from magnetic resonance images (MRIs) of a set of subjects. The spherical wavelet transformation can characterize Manuscript received June 1, 2006.

A fundamental problem in signal processing is to design computationally efficient algorithms to filter signals. In many applications, the signals to filter lie on a sphere. Meaningful examples of data of this kind are weather data on the Earth, or images of the sky. It is then important to design filtering algorithms that are computationally efficient and capable of exploiting the rotational symmetry of the problem. In these applications, given a continuous signal f : S 2 → R on a 2-sphere S 2 ⊂ R 3 , we can only know the vector of its sampled values f ∈ R N : (f ) i = f (x i ) in a finite set of points P ⊂ S 2 , P = {x i } n-1 i=0 where our sensors are located. Perraudin et al. in [18] construct a sparse graph G on the vertex set P and then use a polynomial of the corresponding graph Laplacian matrix L ∈ R n×n to perform a computationally efficient -O(n) -filtering of the sampled signal f . In order to study how well this algorithm respects the symmetry of the problem -i.e it is equivariant to the rotation group SO(3) -it is important to guarantee 4.2 The Finite Element Method approximation of the Laplace-Beltrami

The distribution of the d electrons over the corresponding orbitals in transitionmetal complexes is a central concept in the theory of metal±ligand bonding. The description requires the assignment of an axis of quantization, which is unambiguous in symmetric environments but not clear-cut in the now commonly encountered case of a low-symmetry coordination environment. As the d-electron population can be derived from accurate diffraction data using the methods of charge-density analysis . Acta Cryst. A39, 377±387], the need for an appropriate procedure is relevant in this area of crystallography. Several criteria for the choice of coordinate system based on the resulting orbital populations are discussed. They are tested on a cobalt atom in a trigonal bipyramidal site and applied to transition-metal sites in Cu II -alanylvaline, and an open zirconocene. The population of the d-orbital cross terms for the different coordinate-system orientations is used to judge the results. In the cases examined, the intuitively most reasonable coordinate system corresponds to the one with smaller value of the sum of the populations of the d-orbital cross terms.

Predicting and quantifying the capability of mapping orbits in the vicinity of primitive bodies is challenging given the complex orbit geometries that exist and the irregular shape of the bodies themselves. This paper employs various quantitative metrics to characterize the performance and relative effectiveness of various types of mapping orbits including terminator, quasi-terminator, hovering, pingpong, and conic-like trajectories. Metrics of interest include surface area coverage, lighting conditions, and the variety of viewing angles achieved. The metrics discussed in this investigation are intended to enable mission designers and project stakeholders to better characterize candidate mapping orbits during preliminary mission formulation activities.

Interest in studying small bodies has grown significantly in the last two decades, and there are a number of past, present, and future missions. These small body missions challenge navigators with significantly different kinds of problems than the planets and moons do. The small bodies' shape is often irregular and their gravitational field significantly weak, which make the designing of a stable orbit a complex dynamical problem. In the initial phase of spacecraft rendezvous with a small body, the determination of the gravitational parameter and lower-degree spherical harmonics are of crucial importance for safe navigation purposes. This motivates studying how well one can determine the total mass and lower-degree spherical harmonics in a relatively short time in the initial phase of the spacecraft rendezvous via flybys. A quick turnaround for the gravity data is of high value since it will facilitate the subsequent mission design of the main scientific observation campaign. We will present how one can approach the problem to determine a desirable flyby geometry for a general small body. We will work in the non-dimensional formulation since it will generalize our results across different size/mass bodies and the rotation rate for a specific combination of gravitational coefficients.

We propose a method to reproduce 3D auditory scenes captured by spherical microphone arrays over headphones. This algorithm employs expansions of the captured sound and the head related transfer function over the sphere and uses the orthonormality of the spherical harmonics. Using a spherical microphone array, we first record the 3D auditory scene, then the recordings are spatially filtered and reproduced through headphones in the orthogonal beam-space of the head related transfer functions (HRTFs). We use the KEMAR HRTF measurements to verify our algorithm. In experiments, we use a hemispherical array for recording. The reproduction results are posted online.

The Gravity field and steady-state Ocean Circulation Explorer (GOCE) is one of the flagships in ESA’s Living Planet Programme. With the help of the on-board, very precise gravitational gradiometer the Earth’s gravity field is to be determined with unprecedented accuracy. The gradiometer measures gravity field differences that are contained in six gravity gradient (GG) tensor components. Because of the instrument characteristics, four out of six tensor elements are very accurate, whereas the other two are less accurate. We will concentrate on the noise characteristics of the accurate GGs by a spectral analysis of the residuals of a semi-analytical gravity field solution for the first measurement phase (November - December 2009). Formal errors of spherical harmonic coefficients are estimated in a semi-analytical way by means of a GOCE combined gravity field solution. In addition they are then compared with formal errors of recent GOCE-only models. As in the measured GG tensor there ar...

PURPOSE: Multiparametric MRI (mpMRI) is becoming an increasingly important tool for localizing prostate cancer. Common mpMRI sequences include T2-weighted (T2W), dynamic contrast-enhanced (DCE), and diffusion-weighted (DW) MRI. Recent work has identified DW-MRI as helpful for improving prostate cancer detection. Additionally, several studies have established a correlation between apparent diffusion coefficient (ADC) values and tissue morphology of prostate cancer. Prostate tissue diffusion properties do not follow a mono-exponential diffusion decay model at b-values above 600-700 s/mm. Therefore, more complex models such as biexponential, intravoxel incoherent motion (IVIM), kurtosis, and statistical models have been proposed for better characterization of DW-MRI signal attenuation in the prostate. Studies exploring these models reported different values for diffusion parameters in cancerous and normal tissue regions of interest (ROIs). However, none of these studies used an unsuper...

Many point cloud classification methods are developed under the assumption that all point clouds in the dataset are well aligned with the canonical axes so that the 3D Cartesian point coordinates can be employed to learn features. When input point clouds are not aligned, the classification performance drops significantly. In this work, we focus on a mathematically transparent point cloud classification method called PointHop, analyze its reason for failure due to pose variations, and solve the problem by replacing its pose dependent modules with rotation invariant counterparts. The proposed method is named SO(3)-Invariant PointHop (or S3I-PointHop in short). We also significantly simplify the PointHop pipeline using only one single hop along with multiple spatial aggregation techniques. The idea of exploiting more spatial information is novel. Experiments on the ModelNet40 dataset demonstrate the superiority of S3I-PointHop over traditional PointHop-like methods.

This article presents a comprehensive study of special functions and spherical harmonics, with applications in mathematical physics and spectral theory. It begins with the Legendre polynomials, derived from the angular part of the Laplacian in spherical coordinates, and explores their properties, including orthogonality, recurrence relations, and Rodrigues' formula. The associated Legendre functions are introduced to handle azimuthal dependence, forming the basis for spherical harmonics.

The second section develops the theory of spherical harmonics as eigenfunctions of the Laplacian on the unit sphere, establishing their completeness and orthonormality. These functions are shown to generalize Fourier series to the sphere and are used to expand arbitrary functions in terms of angular modes.

The third section introduces Bessel functions, arising from the radial part of the Helmholtz equation in cylindrical and spherical coordinates. Their series representations, recurrence relations, and integral forms are discussed, along with their role in wave propagation and scattering problems.

Finally, the article explores the confluent hypergeometric function, its differential equation, series expansion, integral representations, and special cases such as Laguerre polynomials. These functions are essential in quantum mechanics and other areas involving radial equations.

Overall, the paper provides a unified framework for understanding classical special functions and their role in solving partial differential equations in physics.

This work presents a quaternion-based formulation of vector analysis, with particular emphasis on the angular part of the Laplacian operator and its spectral decomposition. By leveraging the algebraic structure of quaternions and the Lie group of rotations SO(3), the author derives a simplified expression for the angular component of the Laplacian, The spectral analysis reveals that the eigenfunctions of
L are vector spherical harmonics, constructed from classical scalar spherical harmonics.

The paper further develops a basis of vector functions on the sphere using these harmonics and applies the formalism to quantum wave equations. Specifically, it incorporates electromagnetic corrections from Pauli and Dirac into the Klein-Gordon equation using the van der Waerden–Kramers method, with a return to two-component wave functions as suggested by Weyl. The wave functions are valued in the quaternionic space and are represented via 2×2 complex matrices.

Finally, the study explores the fine structure of the quantum system by solving the second-order wave equation under the influence of electromagnetic potentials. The spectral properties of the associated operators are analyzed, leading to a refined understanding of the energy levels and their dependence on angular momentum and spin interactions.

Optimization problems involving eigenvalues arise in many applications. Let x be a vector of real parameters and let A(x) be a continuously differentiable symmetric matrix function of x. We consider a particular problem that occurs frequently: the minimization of the maximum eigenvalue of A(x), subject to linear constraints and bounds on x. The eigenvalues of A(x) are not differentiable at points x where they coalesce, so the optimization problem is said to be nonsmooth. Furthermore, it is typically the case that the optimization objective tends to make eigenvalues coalesce at a solution point. There are three main purposes of the paper. The first is to present a clear and self-contained derivation of the Clarke generalized gradient of the max eigenvalue function in terms of a "dual matrix." The second purpose is to describe a new algorithm, based on the ideas of a previous paper by the author [SIAM J. Matrix Anal. Appl., 9 (1988), pp. 256-268], which is suitable for solving large- scale eigenvalue optimization problems. The algorithm uses a "successive partial linear programming" formulation that should be useful for other large-scale structured nonsmooth optimization problems as well as large-scale nonlinear programming with a relatively small number of nonlinear constraints. The third purpose is to report on our extensive numerical experience with the new algorithm, solving problems that arise in the following application areas: the optimal design of columns against buckling; the construction of optimal preconditioners for numerical linear equation solvers; the bounding of the Shannon capacity of a graph. We emphasize the role of the dual matrix, whose dimension is equal to the multiplicity of the minimal max eigenvalue. The dual matrix is computed by the optimization algorithm and used for verification of optimality and sensitivity analysis.

The authors construct self-similar solutions for an N -dimensional transport equation, where the velocity is given by the Riezs transform. These solutions imply nonuniqueness of weak solution. In addition, self-similar solution for a one-dimensional conservative equation involving the Hilbert transform is obtained.

A procedure used for expanding the height of a pressure surface over the Northern Hemisphere in a series of spherical-harmonic components is described. The corresponding spherical-harmonic representation of the stream function is obtained by utilizing the geostrophic balance condition. From this representation mean values for the spectral distribution of the kinetic energy at the 500 mb level for January 1957 is presented. The behaviour of the components with the largest horizontal scales is considered a t the 500 mb and the 1000 mb levels during the 90 days period from 1 December 1956 to 28 February 1957. I n general each of these components exhibits smaller or larger fluctuations, and it is attempted to investigate the character of the shorter periodic fluctuations by eliminating the constant and the long periodic parts of the stream field. For the components with wavenumber 1, 2, and 3, and with the most large-scale meridional variation, the 24 hours tendency fields show a more or less regular westward propagation with mean values for the velocity of propagation, corresponding to periods about 5 days. For the components with the same wavenumbers but with the second largest meridional scales we find for the daily deviations from the 15 days mean flow displacements also mainly towards the west and also with mean values for the velocity of propagation, decreasing with decreasing horizontal scale. The mean values of the velocity of propagation obtained in this way for the different components are nearly in accordance with the velocities determined by the Rossby effect, especially if this effect is reduced somewhat by a weak divergence effect. On the basis of the spectral form of the barotropic vorticity equation the time derivatives for the expansion coefficients as well as for the amplitude and phase angle of the stream function at the 500 mb level have been computed for each day in January 1957, for some components of the zonal flow and some of the components with wavenumber 1, 2, 5, and 6. From these time derivatives 48 hours tendencies have been evaluated and compared with the corresponding observed ones. In general it is found that the agreement is better for components with moderately large scales than for components with very large scales. To some extent this may be explained by the neglect of quasi-stationary effects, and as a very simple attempt, these are represented by a constant term. Finally the contributions from the barotropic interactions between different groups of components to the change of kinetic energy of individual components are considered.

We develop a wavelet transform on the sphere, based on the spherical HEALPix coordinate system (Hierarchical Equal Area iso-Latitude Pixelization). HEALPix is heavily used for astronomical data processing applications; it is intrinsically multiscale and locally euclidean, hence appealing for building multiscale system. Furthermore, the equal-area pixelization enables us to employ average-interpolating refinement, giving wavelets of local support. HEALPix wavelets have numerous applications in geopotential modeling. A statistical analysis demonstrates wavelet compressibility of the geopotential field and shows that geopotential wavelet coefficients have many of the statistical properties that were previously observed with wavelet coefficients of natural images. The HEALPix wavelet expansion allows to evaluate a gravimetric quantity over a local region far more rapidly than the classic approach based on spherical harmonics. Our software tools are specifically tailored to demonstrate these advantages.

We develop a wavelet transform on the sphere, based on the spherical HEALPix coordinate system (Hierarchical Equal Area iso-Latitude Pixelization). HEALPix is heavily used for astronomical data processing applications; it is intrinsically multiscale and locally euclidean, hence appealing for building multiscale system. Furthermore, the equal-area pixelization enables us to employ average-interpolating refinement, giving wavelets of local support. HEALPix wavelets have numerous applications in geopotential modeling. A statistical analysis demonstrates wavelet compressibility of the geopotential field and shows that geopotential wavelet coefficients have many of the statistical properties that were previously observed with wavelet coefficients of natural images. The HEALPix wavelet expansion allows to evaluate a gravimetric quantity over a local region far more rapidly than the classic approach based on spherical harmonics. Our software tools are specifically tailored to demonstrate these advantages.

There are numerous applications in geodesy and other geo-sciences in which the gravitational potential effect or other functions of the potential are computed by forward modelling from a given mass distribution. Different volume discretisations, e.g. prisms, tesseroids or mass layers are used. In order to control the numerical realisation of the forward calculation in the practical application, e.g. in reduction tasks, these evaluation programs should be verified against rigorous analytical solutions. In this contribution, a closed analytical solution for the potential of an ellipsoidal shell as a test body is presented. Furthermore, we derive the respective closed formulae for the gravity vector and the gravity gradient tensor. Program implementations of the tesseroid approach are compared on the basis of this ellipsoidal mass arrangement. For the practical usage, fast-converging expansions in spherical harmonics are provided in addition. The derivation of the formulae is based on ...

Bending elasticity of vesicle membranes studied by Monte Carlo simulations of vesicle thermal shape fluctuations Samo Penic ˇ,a Ales ˇIglic ˇ,b Isak Bivas c and Miha Fos ˇnaric ˇ*b The membrane bending stiffness of nearly spherical lipid vesicles can be deduced from the analysis of their thermal shape fluctuations. The theoretical basis of this analysis [Milner and Safran, Phys. Rev. A: At., Mol., Opt. Phys., 1987, 36, 4371-4379] uses the mean field approximation. In this work we apply Monte Carlo simulations and estimate the error in the determination of the bending stiffness due to the approximations applied in the theory. It is less than 10%. The method presented in this work can also be used to determine the changes of the bending stiffness of biological membranes due to their chemical and/or structural modifications.

Prostate cancer, which is also known as prostatic adenocarcinoma, is an unconstrained growth of epithelial cells in the prostate and has become one of the leading causes of cancer-related death worldwide. The survival of patients with prostate cancer relies on detection at an early, treatable stage. In this paper, we introduce a new comprehensive framework to precisely differentiate between malignant and benign prostate cancer. This framework proposes a noninvasive computer-aided diagnosis system that integrates two imaging modalities of MR (diffusion-weighted (DW) and T2-weighted (T2W)). For the first time, it utilizes the combination of functional features represented by apparent diffusion coefficient (ADC) maps estimated from DW-MRI for the whole prostate in combination with texture features with its first- and second-order representations, extracted from T2W-MRIs of the whole prostate, and shape features represented by spherical harmonics constructed for the lesion inside the pr...

Most of the temporal lobe epilepsy detection approaches are based on hippocampus deformation and use complicated features, resulting, detection is done with complicated features extraction and pre-processing task. In this paper, a new detection method based on shape-based features and spherical harmonics is proposed which can analysis the hippocampus shape anomaly and detection asymmetry. This method consisted of two main parts; (1) shape feature extraction, and (2) image classification. For evaluation, HFH database is used which is publicly available in this field. Nine different geometry and 256 spherical harmonic features are introduced then selected Eighteen of them that detect the asymmetry in hippocampus significantly in a randomly selected subset of the dataset. Then a support vector machine (SVM) classifier was employed to classify the remaining images of the dataset to normal and epileptic images using our selected features. On a dataset of 25 images, 12 images were used fo...

The Kelvin-inverted ellipsoid, with the center of inversion at the center of the ellipsoid, is a nonconvex biquadratic surface that is the image of a triaxial ellipsoid under the Kelvin mapping. It is the most general nonconvex 3-D body for which the Kelvin inversion method can be used to obtain analytic solutions for low-frequency scattering problems. We consider Rayleigh scattering by such a fourth-degree surface and provide all relevant analytical calculations possible within the theory of ellipsoidal harmonics. It is shown that only ellipsoidal harmonics of even degree are needed to express the capacity of the inverted ellipsoid. Special cases of prolate or oblate spheroids and that of the sphere are recovered through appropriate limiting processes. The crucial calculations of the norm integrals, which are expressible in terms of known ellipsoidal harmonics, are outlined in Appendix B.

With the improvement of the VLBI system and the increase of number of VLBI observations, it is possible to identify systematic effects on radio source positions.  The current celestial reference frame only takes into account the positions of the radio sources. However, some radio sources show apparent proper motions, thus it is not possible to consider their positions stable in time.  The estimation and analysis of the systematic effects can provide us the physical reasons of these variations and an alternative way to correct the positions.  Such a study can be used to improve the next realization of the celestial reference frame: ICRF3. The study  Estimate the systematics in apparent proper motions with Vievs.  Compare results obtained with VieVS against results of Calc/Solve  comparison using same approach used by Titov and Lambert (2013)  Approach applied by Titov and Lambert (2013) based on a the three-step procedure. The steps are: 1. Estimation of source coordinate time series from analysis of VLBI sessions. 2. Least square method to fit proper motions to the source coordinate time series. 3. Fit of spherical harmonics to the proper motions of the sources T&L (2013) -Calc/Solve Present study -VieVS

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The progress of geomagnetic field modelling is traced from 1600 to the present day, with particular emphasis on the method of spherical harmonic analysis, illustrated by examples taken mainly from British sources. No attempt is made to be comprehensive (a catalogue of models has been given elsewhere by Barraclough) or to include models of temporal variations of the geomagnetic field.

The Chapman-Miller method is the one most widely used for the determination of lunar daily variations in geophysical data. The joint IAGA/ IAMAP committee on lunar variations has suggested that the method would be still more widely applied if a simple up-to-date account were available. This note is an attempt to provide such an account. The Chapman-Miller method is described with minimum mathematical detail, the method of determination of probable errors is improved, and a computer program is appended.

With a view to encouraging magnetic observatory staff to make fuller use of their own data, a simple but rigorous method is presented for the determination of coherent, periodic variations from long series of observatory data. The method is readily adaptable to the different types of data published by observatories, is able to cope with gaps in data, is tuneable to any desired frequency and can run on a modest PC. As an illustration of the method, it has been applied to data from the Kandilli Observatory. Key word geomagnetism.

The method of rectangular polynomial analysis (RPA) is developed and refined to represent a curl-free potential field of internal origin. It is applied to annual mean values of the geomagnetic field from 42 European observatories. RPA is found to be an efficient means of representing the regional field, though less suitable for modelling the anomaly field.

Various methods that take account of the potential nature of the field have been proposed for modelling geomagnetic data on a regional scale. Several of these have been applied to a standard data set based on annual mean values from observatories in Europe. Here, we examine some of the properties of spherical cap harmonic analysis when applied to this data set, and compare the quality of fit with that of the other models. It is found that, for this data set, rectangular polynomial analysis provides a compact fit to main field data, but that in most other cases, for both main field and anomaly data, spherical cap harmonic analysis provides the better fit. Although relatively insensitive to chosen cap size, spherical cap harmonic analysis deteriorates more rapidly than the other methods when the number of coefficients is reduced.

A spherical harmonic model of the second time-derivative of the geomagnetic field is determined, for the first time, directly from measures of the secular acceleration based on observatory annual mean data. The data span the interval 1964.5-1975.5, and 165 observatories are included. The model comprises the 32 coefficients of degree and order up to 6 that are significant at the 5 per cent level. Its primary purpose is to aid in the reduction of data to epoch for the 1980 series of navigational charts. The model is compared with earlier estimates of secular acceleration, derived by less direct methods.

Using a very large body of post-1955 data, a spherical harmonic model of the geomagnetic field and its secular variation is derived for 1965.0. This model is compared with the original International Geomagnetic Reference Field (IGRF) and with individual models used, or proposed for use, in producing the IGRF. Positions of the dip-poles, the geomagnetic poles and the eccentric dipole are derived from the model, together with their rates of change, and comparisons are made with other estimates of these positions.

We have solved a variety of axisymetric induction problems by a functional analytic method due to Hutson, Kendall and Malin which extends to high frequency the radius of convergence of Price's first method. In the present paper the choice of acceleration parameter and the rate of convergence are discussed in greater detail. The problems treated have involved a cap of a sphere of variable electrical surface conductivity to model a shelving ocean, with a superconducting sphere below to model the conducting Earth. Electric currents are induced in both by an external magnetic field, uniform in space, and varying sinusoidally in time. This problem has previously been extensively studied but the following new points have emerged from our computations: 1. When a superconducting sphere is placed beneath an otherwise isolated spherical cap the distribution of electric current over the cap is modified as though the conductivity of the cap had been reduced. A physical explanation is suggested for this effect. 2. Let the line across which a reversal of the vertical component of induced magnetic field occurs be t. Suppose that the ocean bed begins to shelve upwards to its margin along a line V. Then the position o f t , even at moderate (4h) frequencies, may be only very loosely related to the position of V. In general, the position o f t will vary with time, and will also depend upon a number of other parameters. An alternative solution in one case is presented by using Legendre series. The results for series of various lengths are compared with the functional analytic solution, thus enabling an estimate to be made of the number of terms which must be included to give any required accuracy. This comparison shows that a large number of terms must often be taken, and it follows that the Legendre polynomial method must be used with caution fcr induction problems even for moderately high frequency.

A general method of computing the solution of a wide class of geomagnetic induction problems is given. The method is based on an integral equation for the electric current J induced in a surface conductor by a time-varying imposed electromagnetic field. The standard iterative solution of this equation is not convergent for large frequencies. However, an alternative iterative scheme, believed to be new in the context of geomagnetism, is suggested by an analogy from functional analysis, and this formulation allows a rigorous argument for its convergence to be developed. In the special case of axial symmetry the iterative method is shown to lead to a continuous solution for all finite frequencies and conductivities. This appears to clarify a problem which has been much discussed in the literature. Trial calculations are also presented for the case of axial symmetry. In the general case, physical reasoning indicates that the method can be expected to converge. The axisymmetric computations are made in a way which simulate the eventual computation of general problems, showing that the method is feasible. The method also applies to problems in three dimensions and to more general surfaces and surface geometries.

The Bipolar Spherical Harmonics (BipoSH) form a natural basis to study the CMB two point correlation function in a non-statistically isotropic (non-SI) universe. The coefficients of expansion in this basis are a generalisation of the well known CMB angular power spectrum and contain complete information of the statistical properties of a non-SI but Gaussian random CMB sky. We use these coefficients to describe the weak lensing of CMB photons in a non-SI universe. Finally we show that the results reduce to the standard weak lensing results in the isotropic limit.

Gaussianity of temperature fluctuations in the Cosmic Microwave Background(CMB) implies that the statistical properties of the temperature field can be completely characterized by its two point correlation function. The two point correlation function can be expanded in full generality in the bipolar spherical harmonic(BipoSH) basis. Looking for significant deviations from zero for Bipolar Spherical Harmonic(BipoSH) Coefficients derived from observed CMB maps forms the basis of the strategy used to detect isotropy violation. In order to quantify "significant deviation" we need to understand the distributions of these coefficients. We analytically evaluate the moments and the distribution of the coefficients of expansion(A LM l 1 l 2 ), using characteristic function approach. We show that for BipoSH coefficients with M = 0 an analytical form for the moments up to any arbitrary order can be derived. For the remaining BipoSH coefficients with M = 0, the moments derived using the characteristic function approach need to be supplemented with a correction term. The correction term is found to be important particularly at low multipoles. We provide a general prescription for calculating these corrections, however we restrict the explicit calculations only up to kurtosis. We confirm our results with measurements of BipoSH coefficients on numerically simulated statistically isotropic CMB maps.

The statistical expectation values of the temperature fluctuations of cosmic microwave background (CMB) are assumed to be preserved under rotations of the sky. We investigate the statistical isotropy of the CMB anisotropy maps recently measured by the Wilkinson Microwave Anisotropy Probe (WMAP) using bipolar spherical harmonic power spectrum proposed in Hajian & Souradeep 2003. The Bipolar Power Spectrum (BiPS) is estimated for the full sky CMB anisotropy maps of the first year WMAP data. The method allows us to isolate regions in multipole space and study each region independently. This search shows no evidence for violation of statistical isotropy in the first-year WMAP data on angular scales larger than that corresponding to l ≈ 60 .

Figure 1: First 3 principal components of our statistical diffuse (left) and specular (middle) albedo models. Both are visualised in linear sRGB space. Right: rendering of the combined model under frontal illumination in nonlinear sRGB space.

The scheme leading to the computation of the normal mode eigenfrequencies of the laterally inhomogeneous Earth is derived. Rayleigh's variational principle is used to calculate the first-order corrections to the eigenfrequencies of a spherically symmetric, non-rotating and elastic Earth model due to the lateral variations in the real Earth. The Wigner-Eckart theorem is used, whenever applicable, to simplify many complicated angular integrals involving the strain deviator tensor to more basic and manageable integrals involving three spherical harmonics. In particular, near-resonance multiplet coupling for spheroidal-spheroidal, spheroidaltoroidal and toroidal-toroidal multiplets is presented along with the uncoupled case of a single arbitrary spheroidal or toroidal multiplet.

The scheme leading to the computation of the normal mode eigenfrequencies of the laterally inhomogeneous Earth is derived. Rayleigh's variational principle is used to calculate the first-order corrections to the eigenfrequencies of a spherically symmetric, non-rotating and elastic Earth model due to the lateral variations in the real Earth. The Wigner-Eckart theorem is used, whenever applicable, to simplify many complicated angular integrals involving the strain deviator tensor to more basic and manageable integrals involving three spherical harmonics. In particular, near-resonance multiplet coupling for spheroidal-spheroidal, spheroidaltoroidal and toroidal-toroidal multiplets is presented along with the uncoupled case of a single arbitrary spheroidal or toroidal multiplet.

A numerical solution for electromagnetic scattering from a two‐dimensional earth model of arbitrary conductivity distribution has been developed and compared with analog model results. A frequency‐domain variational integral is Fourier transformed in the strike direction, and a solution is obtained using the finite‐element method for each of a finite number of harmonics or wavenumbers in transform space. The solution is obtained in terms of the secondary electric fields. Principally due to the inaccuracy associated with numerical derivatives of electric fields, the secondary magnetic field is computed by integrating over the scattering currents in harmonic space and is then inverse Fourier transformed.

The classical problem of extrapolation of a bandlimited signal from limited time domain data is revisited for signals defined on the sphere. That is, given limited or incomplete measurements of an isotropic low pass signal on the unit sphere, S 2 , find the unique extrapolation to the complete unit sphere. Signals defined on the unit sphere arise in a number of applications, such as beampatterns in azimuth and elevation and head related transfer functions. Our investigations explore the role of integral equation operators in characterizing the extrapolation problem which leads to an iterative algorithm analogous to that obtained in the time-frequency case.