Expansion Of Harmonic Functions Research Articles

Development of Precision Controllable Magnetic Field-Assisted Platform for Micro Electrical Machining.

In order to introduce the magnetic field into micro electrical machining technology to explore the influence of magnetic field on micro electrical machining, the development of a precision controllable magnetic field-assisted platform is particularly important. This platform needs to precisely control the spatial magnetic field. This study first completes the hardware design and construction of the magnetic field generation device, using electromagnetic coils with soft iron cores as the sources of the magnetic field. Mathematical models of the magnetic field are established and calibrated. Since the magnetic dipole model cannot effectively describe the magnetic field generated by the electromagnetic coil, this study adopts a more precise description method: the spherical harmonic function expansion model and the magnetic multipole superposition model. The calibration of the magnetic field model is based on actual excitation magnetic field data, so a magnetic field sampling device is designed to obtain the excitation magnetic field of the workspace. The model is calibrated based on a combination of the theoretical model and magnetic field data, and the performance of the constructed setup is analyzed. Finally, a magnetic field-assisted platform has been developed which can generate magnetic fields in any direction within the workspace with intensities ranging from 0 to 0.2 T. Its magnetic field model arrives at an error percentage of 2.986%, a variance of 0.9977, and a root mean square error (RMSE) of 0.71 mT, achieving precise control of the magnetic field in the workspace.

Rockfall simulation via spherical harmonic based discrete element method

The trajectory of falling rock is significantly affected by its shape. In order to capture the falling rock shape, spherical harmonic function expansion method is used. The reasonable order of spherical harmonic function expansion items is determined according to the error analysis. Spherical harmonic based discrete element method is utilized to simulate the rockfall process of cylindrical and irregular shape rocks. The influences of the rock and terrain shapes on the rockfall trajectory are analyzed. The results show that the shape of falling rock can be described precisely by spherical harmonic function expansion method, and only a little data should be stored. The predicted trajectory of cylindrical rockfall is close to the experimental result. Spherical and block falling rocks usually show more regular horizontal displacement, lateral offset and jump height during the motion process. Those motion quantities for long strip and flaky falling rocks are more sensitive to their initial rotations. Terrain has a strong influence on rockfall trajectory, falling rocks are prone to move along or rest at mountain valleys.

Research of integrated shimming Halbach magnet for High strength, compact Benchtop NMR device

Due to production and assembly errors of magnets resulting in reduced magnetic field homogeneity, passive shimming (PS) is necessary when Halbach magnets are used in high-resolution Benchtop Nuclear Magnetic Resonance (BNMR) spectrometer. The conventional PS technique, which places independent PS devices inside the magnet aperture, is no longer applicable to small-aperture compact high-field-strength Halbach magnet studies. In this paper, based on spherical harmonic function expansion, we improve the magnetic field homogeneity by optimizing the moving step of the magnet moving arrays composed of Halbach magnets to generate the corresponding harmonic terms to compensate for the magnetic field. With this approach, the homogeneity of a 1 T Halbach magnet was improved from the original 3913 ppm to 8 ppm in a L10 mm × R2.5 mm of 0.64% copper sulfate doped water sample. This work explores the PS mechanism based on the movement of magnetic blocks, which can be applied in BNMR and other compact high-field strength high-homogeneity Halbach magnets application circumstances.

Expansion of harmonic functions near the boundary of Dini domains

Let u be a harmonic function in a C^1 -Dini domain such that u vanishes on an open set of the boundary. We show that near every point in the open set, u can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that u has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.

Damage profile evolution model based on the Boltzmann transport equation for silicon micromachining with the focused helium ion beam

In this work, a damage profile evolution model based on the Boltzmann transport equation was developed for silicon micromachining using the focused helium ion beam. The energy grouping method, the spherical harmonic function expansion method, and the discrete ordinate method were used to solve the model numerically to obtain the implanted ion distribution and the amorphous damage profile. The influence of the ion beam energy and the ion dose on the evolution of the amorphous damage profile and the helium ion distribution in the silicon substrate were investigated. The implantation depth of the ions was positively correlated with the beam energy, and the influence of the ion dose on the implantation depth of the ions was very slight. The amorphous damage profiles predicted by the model were in good agreement with the experimental transmission electron microscope images. The results showed that the amorphous damage profile evolved into an approximately cylindrical shape for low beam energy (<10 keV), and an approximate “vase” shape for high beam energy (>10 keV). For the specified beam energy, the maximum amorphous width increased linearly and the amorphous depth first increased rapidly and then slowly increased with the increasing ion dose, and there was a limit on the amorphous depth. The computational efficiency of the model was not limited by the number of particles, which made up for a common shortcoming of the Monte Carlo and molecular dynamics methods.

Overconvergence of polynomial expansions of harmonic functions

ABSTRACTThis paper concerns homogeneous polynomial expansions of harmonic functions. It presents new results on the relationship between their gap structure and their overconvergence properties outside the ball of convergence.

Analog of the classical borel theorem for entire harmonic functions in ℝn and generalized orders

The article describes research on the growth of functions that are harmonic in the whole space ℝ n , n≥3, and thus they are called entire harmonic. A relation has been established between the maximum terms of entire functions of finite order in the plane, which are given by power series whose coefficients are somewhat connected. Also, the maximum modulus of a harmonic function in the space ℝ n is evaluated through the maximum modulus of some entire function in the plane, the coefficients which are expressed in terms of the coefficients of the expansion of the harmonic function in a series by Laplace spherical functions. These results made it possible to obtain an analog of the classical Borel theorem for entire harmonic functions of finite order in ℝ n . Besides, the study has revealed the most general characteristics of the growth of entire harmonic functions in ℝ n in terms of the uniform norm of Laplace spherical functions in the expansion of harmonic functions in series. Slow growth of the harmonic functions in the space has also been studied. The obtained results are analogous to the classical results that are known for entire functions of one complex variable. The research findings are important because harmonic functions occupy a special place not only in many mathematical studies but also in the application of mathematical analysis to physics and mechanics, where these functions often describe various stationary processes.

Straightforward approximation of the translating and pulsating free surface Green function

The translating and pulsating free surface Green function represents the velocity potential of a three-dimensional free surface source advancing in waves. This function involves singular wave integral, which is troublesome in numerical computation. In the present study, a regular wave integral approach is developed for the discretisation of the singular wave integral in a whole space harmonic function expansion, which permits the free surface wave produced by the fluid motion to be decomposed by plane regular propagation waves. This approximation gives rise to a simple and straightforward evaluation of the Green function. The algorithm is validated from comparisons between present numerical results and existing numerical data.

Harmonic function expansion for translating Green functions and dissipative free-surface waves

Three-dimensional gravity free-surface waves involving energy dissipation are examined. The singular wave integral from a non-dissipative translating Green function is approximated as a regular wave integral of a dissipative Green function. The regular wave integral method enables a dissipative free-surface wave to be derived from an expansion of whole-space harmonic functions. The new approximation technique, applicable to dissipative and non-dissipative free-surface wave problems, simplifies the numerical simulations of three-dimensional Green functions and free-surface waves.

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ℝN such that \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is connected. It is known that, for each w ∈ Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on \(\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega\) is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.

Signal space separation beamformer

Event Abstract Back to Event Signal space separation beamformer Samu Taulu1*, Jiri Vrba1, Jukka Nenonen1 and Antti Ahonen1 1 Elekta Oy, Finland We have combined signal space separation and beamformers (SSS beamformer). The SSS beamformer differs from the conventional beamformer by operating on the magnetostatic multipole moments instead of physical MEG sensors. SSS represents the MEG signals by a truncated spherical harmonic function expansion that has spatially different components and truncation orders for brain signals and external interference. We tested the method by simulation in the presence of simulated brain noise. The SSS beamformer performs at least as well as the conventional beamformer, provided that the order of the internal expansion is sufficiently high. For beamformer outputs which depend on power or power difference normalized by the projected noise, the spatial resolution of the SSS beamformer is significantly better than that of the conventional beamformers if the sources are deep, and about the same as that of the conventional beamformer when the sources are superficial. For dual-state beamformer outputs which depend on the ratio of powers, the SSS and conventional beamfomers have the same spatial resolution. These observations are consistent with the different characteristics of the signal vectors composed of the multipole moments or the outputs of the physical sensors. The signal vector comprised of multipoles tends to exhibit a larger change with respect to a modification of the source configuration than the corresponding signal vector of physical sensors, especially for relatively deep sources. The sensor noise covariance matrix in the SSS basis is non-diagonal. The SSS beamformers with diagonalized noise covariance matrix exhibit better spatial resolution than that with non-diagonal noise covariance matrix. In summary, the SSS beamformers are computationally more efficient than the conventional beamformers and provide at least the same or better spatial resolution as compared to conventional beamformers. Conference: Biomag 2010 - 17th International Conference on Biomagnetism , Dubrovnik, Croatia, 28 Mar - 1 Apr, 2010. Presentation Type: Poster Presentation Topic: MEG Modeling Citation: Taulu S, Vrba J, Nenonen J and Ahonen A (2010). Signal space separation beamformer. Front. Neurosci. Conference Abstract: Biomag 2010 - 17th International Conference on Biomagnetism . doi: 10.3389/conf.fnins.2010.06.00061 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 21 Mar 2010; Published Online: 21 Mar 2010. * Correspondence: Samu Taulu, Elekta Oy, Helsinki, Finland, samu.taulu@neuromag.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Samu Taulu Jiri Vrba Jukka Nenonen Antti Ahonen Google Samu Taulu Jiri Vrba Jukka Nenonen Antti Ahonen Google Scholar Samu Taulu Jiri Vrba Jukka Nenonen Antti Ahonen PubMed Samu Taulu Jiri Vrba Jukka Nenonen Antti Ahonen Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

The Steady‐State Analytical Solution of the Global Atmospheric Electricity Model on the Charging of Thunderstorms in Lower Atmosphere

AbstractGlobal atmospheric electricity model is the circuit between atmosphere and surface. In the lower atmosphere, the main sources of electric current within the model are thunderstorms, which are considered as vertical electric dipoles. Based on the current continuity equation, using spherical harmonic function expansion, with the lower boundary as the surface atmospheric electric potential and the upper boundary condition as the electric potential of the bottom of the ionosphere, and non‐homogeneous current source function of the vertical dipole assumptions for the charging of thunderstorms, we derive the steady‐state analytical solution of the global atmospheric electricity model in the lower atmosphere. It is proved that the solution can be written as the sum of limited polynomial series. The conclusions are consistent with the previous theoretical analysis and experimental observation. It is an improvement to the previous approximate solution, and the formula in this article can be used in numerical calculation and simulation of atmospheric electric parameters in the future.

Universal overconvergence of homogeneous expansions of harmonic functions

Universal overconvergence of homogeneous expansions of harmonic functions

Universal overconvergence of homogeneous expansions of harmonic functions

On certain domains of Euclidean space, we establish the existence of harmonic functions, which are universal in the sense that their Taylor polynomials approximate all plausibly approximable functions in the complement of the domain.

Presentation of electromagnetic multichannel data: The signal space separation method

Measurement of external magnetic fields provides information on electric current distribution inside an object. For example, in magnetoencephalography modern measurement devices sample the magnetic field produced by the brain in several hundred distinct locations around the head. The signal space separation (SSS) method creates a fundamental linear basis for all measurable multichannel signal vectors of magnetic origin. The SSS basis is based on the fact that the magnetic field can be expressed as a combination of two separate and rapidly converging expansions of harmonic functions with one expansion for signals arising from inside of the measurement volume of the sensor array and another for signals arising from outside of this volume. The separation is based on the different convergence volumes of the two expansions and on the fact that the sensors are located in a source current-free volume between the interesting and interfering sources. Individual terms of the expansions are shown to contain uncorrelated information of the underlying source distribution. SSS provides a stable decomposition of the measurement into a fundamental device-independent form when used with an accurately calibrated multichannel device. The external interference signals are elegantly suppressed by leaving the interference components out from the reconstruction based on the decomposition. Representation of multichannel data with the SSS basis is shown to provide a large variety of applications for improved analysis of multichannel data.

Universal Overconvergence of Polynomial Expansions of Harmonic Functions

For each compact subset K of R N let H ( K) denote the space of functions that are harmonic on some neighbourhood of K. The space H ( K) is equipped with the topology of uniform convergence on K. Let Ω be an open subset of R N such that 0∈ Ω and R N \\ Ω is connected. It is shown that there exists a series ∑ H n , where H n is a homogeneous harmonic polynomial of degree n on R N , such that (i) ∑ H n converges on some ball of centre 0 to a function that is continuous on Ω and harmonic on Ω, (ii) the partial sums of ∑ H n are dense in H ( K) for every compact subset K of R N \\ Ω with connected complement. Some refinements are given and our results are compared with an analogous theorem concerning overconvergence of power series.

Rational approximation to harmonic functions

Pade-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Pade-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied.

Harmonic function expansion of nearly oblate systems

We show how to develop an expansion of nearly oblate systems in terms of a set of potential-density pairs. A harmonic (multipole) structure is imposed on the potential set at infinity, and the density can be made everywhere regular. We concentrate on a set whose zeroth order functions describe the perfect oblate spheroid of de Zeeuw (1985). This set is not bi-orthogonal, but it can be shown to be complete in a weak sense. Poisson's equation can be solved approximately by truncating the expansion of the potential in such a set. A simple example of a potential which is not one of the basis functions is expanded using the symmetric members of the basis set up to fourth order. The basis functions up to first order are reconstructed approximately using 10,000 particles to show that this set could be used as part of an $N$-body code.

Kac's solution of the telegrapher's equation for tunneling time analysis: An application of the wavelet formalism.

A plausible description of traversal time was given, both in classically allowed and forbidden regions, through a path-integral solution of the telegrapher's equation. This analysis was applied to a simulation based on microwave propagation in a waveguide considered as a one-dimensional system. An extension of the analysis has been performed in order to compare the traversal (or delay) time results relative to a beat-envelope signal with those as deduced from the distribution function of the randomized time and its analytical continuation in imaginary time. Subsequently, in tight analogy with a step signal in an electronic circuit (zero-dimensional system), we have searched for a simulation of traversal processes in real time, even for classically forbidden (tunneling) processes. First we have considered a finite series expansion of harmonic functions of the signal in the neighborhood of its rise, and applied the above mentioned procedure to each harmonic, implying analytical continuation in imaginary time and an arbitrary truncation in the range of the signal. Then, in order to avoid these shortcomings, we have considered a waveletlike description of the signal.

Motions of a fluid drop in linear and quadratic flows

A theoretical analysis is presented of the flow field near a spherical fluid drop immersed in an incompressible Newtonian fluid which, at large distances from the drop, is undergoing an undisturbed flow. The undisturbed flows considered here are relevant to studies of drop motions near a phase boundary, and to some aspects of the coalescence of liquid drops. Exacl solutions in closed form have been found using the harmonic function expansion in spherical coordinates. Calculation of the hydrodynamic force on the drop leads to a generalization of Faxen’s law to afluid particle in anarbitrary undisturbed creeping-flow. The solutions are then expressed in terms of she fundamental singularity solutions for Stokes flow in anticipation of future analysis of the drop coalescence. In addition, the deformed shapes are determined for a fluid drop freely suspended in an axisymmetric Poiseuillian flow.