Orthogonal Zonal, Tesseral and Sectorial Wavelets on the Sphere for the Analysis of Satellite Data

The spherical harmonics Y _n,k } _{n=0,1,...;k=?n,...,n} represent a standard complete orthonormal system in ?²(Ω), where Ω is the unit sphere. In view of present and future satellite missions (e.g., for the determination of the Earth's gravity field) it is of particular importance to treat the different accuracies and sizes of data in dependence of the index pairs (n,k). It is, e.g., known that the GOCE mission yields essentially less accurate data in the zonal (k=0) case. Therefore, this paper presents new ways of constructing multiresolutions for a Sobolev space of functions on Ω allowing the separate treatment of certain classes of pairs (n,k) and, in particular, the separate treatment of different orders k. Orthogonal bandlimited as well as non-bandlimited detail and scale spaces adapted to certain (geo)scientific problems and to the character of the given data can now be used. Finally, an explicit representation of a non-bandlimited wavelet on Ω yielding an orthogonal decomposition of the function space is calculated for the first time.     

Oscillations of a gas in an open-ended tube near resonance

The pressure as function of time was measured near resonance in different axial locations of an open-ended tube. Flow visualisation showed that transition to turbulence was not influenced by the strong disturbance of the open end, except in a region near the open end which had a length of about three particle displacements. The pressure readings were decomposed into the first, second and third harmonic and compared with two different theories. In one case, the linearized theory for the oscillating flow in a tube was fitted to the boundary conditions, the obvious one at the piston and a model at the open end. In the second case, the nonlinear theory of Chester [1] was used. Both theories assume a relation between pressure and velocity at the open end that contains two free constants. The constants were determined by comparing the amplitude of the first and the second harmonic ofone pressure measurement with the theoretical predictions. Once the constants are fixed, the pressurep(ωt, x/L) is completely determined. For weak nonlinear effects, the pressure is essentially determined by one constantα(=k ₂) and the second constantβ(=k ₁) loses its significance. For the range of parameters given there isα=0.825±0.015. A very good approximation of the pressure near resonance can therefore be calculated with the following simple boundary condition at the open end $$p_E = \frac{{4\alpha }}{{3\pi }}\rho \hat u_E u_E = 0.350 \rho \hat u_E u_E .$$ Both theories predict a resonance frequency slightly above the experimental one. Changing Levine and Schwingers [2], end correction from 0.6133R to 1R eliminates the discrepancy for all tube lengths. For the first harmonic the variation of the amplitude and the phase of the pressure signal withω andx is very well predicted by both theories. The nonlinear theory describes also the small second and third harmonics fairly well while the linear theory predicts only the correct order of magnitude of these higher harmonics. The constantα that determines the energy loss at the open end shows an apparent increase if the boundary layer on the tube wall becomes turbulent. This occurs for \(A = 2\hat u/\sqrt {v\omega } \geqq 550\) to 750 which is close to the value observed in a tube with a closed end.     

Special Functions on the Sphere with Applications to Minimal Surfaces

A function which is homogeneous in x, y, z of degree n and satisfies V_xx + V_yy + V_zz = 0 is called a spherical harmonic. In polar coordinates, the spherical harmonics take the form rⁿf_n, where f_n is a spherical surface harmonic of degree n. On a sphere, f_n satisfies ▵ f_n + n(n + 1)f_n = 0, where ▵ is the spherical Laplacian. Bounded spherical surface harmonics are well studied, but in certain instances, unbounded spherical surface harmonics may be of interest. For example, if X is a parameterization of a minimal surface and n is the corresponding unit normal, it is known that the support function, w = X · n, satisfies ▵w + 2w = 0 on a branched covering of a sphere with some points removed. While simple in form, the boundary value problem for the support function has a very rich solution set. We illustrate this by using spherical harmonics of degree one to construct a number of classical genus-zero minimal surfaces such as the catenoid, the helicoid, Enneper's surface, and Hennenberg's surface, and Riemann's family of singly periodic genus-one minimal surfaces.     

On the Error Estimate of the Harmonic B_z Algorithm in MREIT from Noisy Magnetic Flux Field

Magnetic resonance electrical impedance tomography(MREIT, for short) is a new medical imaging technique developed recently to visualize the cross-section conductivity of biologic tissues. A new MREIT image reconstruction method called harmonic Bz algorithm was proposed in 2002 with the measurement of Bz that is a single component of an induced magnetic flux density subject to an injection current. The key idea is to solve a nonlinear integral equation by some iteration process. This paper deals with the convergence analysis as well as the error estimate for noisy input data Bz, which is the practical situation for MREIT. By analyzing the iteration process containing the Laplacian operation on the input magnetic field rigorously, the authors give the error estimate for the iterative solution in terms of the noisy level δ and the regularizing scheme for determiningΔBz approximately from the noisy input data. The regularizing scheme for computing the Laplacian from noisy input data is proposed with error analysis. Our results provide both the theoretical basis and the implementable scheme for evaluating the reconstruction accuracy using harmonic Bz algorithm with practical measurement data containing noise.     

Transmission-line analysis of gravitational pulse detector response functions

Basic notations for measuring gravity waves are recalled, and a δ-pulse of areaJ ₀=10^?12 sec^?1 chosen as a convenient unit for sensitivity calculations. Pulse excitation of a long thin bar causes stress variations following trapezoidal patterns of time dependence, with a flat top duration proportional to distance from the bar's center of mass. A suitable arrangement of piezoelectric transducers (‘PX’) provides frequency-independent matching, both mechanical and electrical, allowing a straightforward and accurate calculation of electric signals. Signal shape is given explicitly for two extreme cases:

1. whole bar is piezoelectric (parabolic arcs resembling a sine wave),
2. very short PX sections (triangular or trapezoidal, depending on position along the bar).

A suitable combination ofn (=1, 3, 5, ...) transducers generates directly a triangular wave at then-th harmonic of the fundamental bar frequency, with amplitude proportional toL/n, so that the use of a long bar as a Fourier Analyser of gravity waves appears feasible. For an estimate of the sensitivity limit, preliminary results of a noise background analysis are quoted, according to which the PX-length can be optimized. A threshold energy relation \( \sim \sqrt[4]{{kT/M^2 Q_{{\text{PX}}} }}\) is obtained in the case where amplifier noise becomes important.     

To the theory of asymptotically stable second-order accurate two-stage scheme for an inhomogeneous parabolic initial-boundary value problem

An asymptotically stable two-stage difference scheme applied previously to a homogeneous parabolic equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial condition is extended to the case of an inhomogeneous parabolic equation with an inhomogeneous Dirichlet boundary condition. It is shown that, in the class of schemes with two stages (at every time step), this difference scheme is uniquely determined by ensuring that high-frequency spatial perturbations are fast damped with time and the scheme is second-order accurate and has a minimal error. Comparisons reveal that the two-stage scheme provides certain advantages over some widely used difference schemes. In the case of an inhomogeneous equation and a homogeneous boundary condition, it is shown that the extended scheme is second-order accurate in time (for individual harmonics). The possibility of achieving second-order accuracy in the case of an inhomogeneous Dirichlet condition is explored, specifically, by varying the boundary values at time grid nodes by O(τ ²), where τ is the time step. A somewhat worse error estimate is obtained for the one-dimensional heat equation with arbitrary sufficiently smooth boundary data, namely, $O\left( {\tau ^2 \ln \frac{T} {\tau }} \right) $ , where T is the length of the time interval.     

The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients $a_{l}^{m}$ of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form ?? ^p or |????? ₀| ^p respectively, where ?? is the co-latitude and p>?1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, $E(l) = \sqrt{\sum_{m} (a_{l}^{m})^{2}}$ where l and m are the spherical harmonic degree and order, of l ^?(p+3/2) or l ^?(p+1) respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of E(l) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface.     

Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.     

Levi Harmonic Maps of Contact Riemannian Manifolds

We study Levi harmonic maps, i.e., C ^∞ solutions f:M→M′ to \(\tau_{\mathcal{H}} (f) \equiv \operatorname{trace}_{g} ( \varPi_{\mathcal{H}}\beta_{f} ) = 0\) , where (M,η,g) is an (almost) contact (semi) Riemannian manifold, M′ is a (semi) Riemannian manifold, β _f is the second fundamental form of f, and \(\varPi_{\mathcal{H}} \beta_{f}\) is the restriction of β _f to the Levi distribution \({\mathcal{H}} = \operatorname{Ker}(\eta)\) . Many examples are exhibited, e.g., the Hopf vector field on the unit sphere S ²ⁿ⁺¹, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannian manifold of constant curvature 1 are Levi harmonic maps. A CR map f of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if f is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map.     

Stable direction recovery in single-index models with a diverging number of predictors

Large dimensional predictors are often introduced in regressions to attenuate the possible modeling bias. We consider the stable direction recovery in single-index models in which we solely assume the response Y is independent of the diverging dimensional predictors X when βτ 0 X is given, where β 0 is a p n × 1 vector, and p n →∞ as the sample size n →∞. We first explore sufficient conditions under which the least squares estimation β n0 recovers the direction β 0 consistently even when p n = o(√ n). To enhance the model interpretability by excluding irrelevant predictors in regressions, we suggest an e1-regularization algorithm with a quadratic constraint on magnitude of least squares residuals to search for a sparse estimation of β 0 . Not only can the solution β n of e1-regularization recover β 0 consistently, it also produces sufficiently sparse estimators which enable us to select "important" predictors to facilitate the model interpretation while maintaining the prediction accuracy. Further analysis by simulations and an application to the car price data suggest that our proposed estimation procedures have good finite-sample performance and are computationally efficient.     

Conditions for the completeness of the system of polynomials

We consider the space A₂(K,γ) of functions which are analytic in the unit disc K and squaresummable in K with respect to plane Lebesgue measureσ with weightγ=¦D¦², D∈ A₂(K, 1), D(z) ≠ 0, z ∈ K. We establish the inequality $$\smallint _K |Dg|^2 u d\sigma \leqslant \smallint _K u d\sigma ,$$ where g represents the distance from 1/D to the closure of the polynomials [in the metric of A₂(K,γ)] and u is any function which is harmonic and nonnegative in K. By means of this inequality we obtain sufficient conditions for the completeness of the system of polynomials in A₂(K,γ) in terms of membership of certain functions of D in the class H₂ (Hardy-2).     

The harmonic Bergman kernels for punctured domains

We obtain the explicit formulae for the harmonic Bergman kernels of Bn／{0} and Rn／Bn and study the connection between harmonic Bergman kernel and weighted harmonic Bergman kernel.We also get the explicit formula for the weighted harmonic Bergman kernel of Bn／{0} with the weight 1/｜x｜4.     

Automorphism Group of Representation Ring of the Weak Hopf Algebra $$\widetilde{H_8 }$$

Let H₈ be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra \(\widetilde{H_8 }\)based on H₈, then we investigate the structure of the representation ring of \(\widetilde{H_8 }\). Finally, we prove that the automorphism group of \(r\left( {\widetilde{H_8 }} \right)\)is just isomorphic to D₆, where D₆ is the dihedral group with order 12.     

Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ _ij , b _i and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ _ij $\tilde y$ , b _i $\tilde y$ are assumed to be locally Lipshitz.Here denotes convolution and $\tilde y$ is the distribution which on functions, is realised by the formula $\tilde y\left( r \right): = y\left( { - r} \right)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.     

On the finite dimensionality of a K3 surface

For a smooth projective surface X the finite dimensionality of the Chow motive h(X), as conjectured by Kimura, has several geometric consequences. For a complex surface of general type with p _g = 0 it is equivalent to Bloch’s conjecture. The conjecture is still open for a K3 surface X which is not a Kummer surface. In this paper we prove some results on Kimura’s conjecture for complex K3 surfaces. If X has a large Picard number ρ = ρ(X), i.e. ρ = 19,20, then the motive of X is finite dimensional. If X has a non-symplectic group acting trivially on algebraic cycles then the motive of X is finite dimensional. If X has a symplectic involution i, i.e. a Nikulin involution, then the finite dimensionality of h(X) implies ${h(X) \simeq h(Y)}$ , where Y is a desingularization of the quotient surface ${X/\langle i \rangle }$ . We give several examples of K3 surfaces with a Nikulin involution such that the isomorphism ${h(X) \simeq h(Y)}$ holds, so giving some evidence to Kimura’s conjecture in this case.     

Implicit Sampling, with Application to Data Assimilation

There are many computational tasks, in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i = 1, 2, ··· ), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space, where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markovchain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction in the context of numerical integration is explained, and then it is applied to data assimilation.     

Some Remarks to the Formal and Local Theory of the Generalized Dhombres Functional Equation

We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation ${f(zf(z))=\varphi(f(z))}$ in the complex domain. First we give two proofs of the existence theorem about solutions f with f(0) = w ₀ and ${w_0 \in \mathbb{C}^\star {\setminus}\mathbb{E}}$ where ${\mathbb{E}}$ denotes the group of complex roots of 1. Afterwards we represent solutions f by means of infinite products where we use on the one hand the canonical convergence of complex analysis, on the other hand we show how solutions converge with respect to the weak topology. In this section we also study solutions where the initial value z ₀ is different from zero.     

3G-inequality for planar domains

The 3G-inequality for Green functions g _D on arbitrary bounded domains in ${{\mathbb{R}}^2}$ , which Bass and Burdzy (Probab Theory Relat Fields 101(4):479–493, 1995) obtained by a genuinely probabilistic proof (using loops of Brownian motion around the origin), is proven (in a more precise form) employing elementary properties of harmonic measures only. Since harmonic measures are hitting distributions of Brownian motion, this purely analytic proof can be viewed as well as being probabilistic. A spin-off is an upper estimate of g _D on subdisks B′ of an open disk B in terms of g _B divided by the capacity of ${B'\setminus D}$ with respect to B.     

Sums with convolutions of Dirichlet characters

We bound short sums of the form ${\sum_{n\le X}(\chi_1{*}\chi_2)(n)}$ , where χ ₁*χ ₂ is the convolution of two primitive Dirichlet characters χ ₁ and χ ₂ with conductors q ₁ and q ₂, respectively.     

Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dy _t ?=?Δy _t dt?+ dX _t (y _t ), where Δ stands for the Laplace operator on a space of the form ${L^p(\mathbb R^n)}$ , and X is a finite dimensional noisy nonlinearity whose typical form is given by ${X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}$ , where each x?=?(x ⁽¹⁾, … , x ^(N)) is a γ-H?lder function generating a rough path and each f _i is a smooth enough function defined on ${L^p(\mathbb R^n)}$ . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.