Eigenvalues of the Laplace-Beltrami operator on a large spherical cap under the Robin problem

In order to make the operator freely walk in virtual environment (VE), a new prototype of VE system for assembly is presented in this paper. In this system, a special machine is designed and it enables the operator to walk inside the VE in any directions over a long distance without actually leaving the physical device. Images produced by the computer are projected upon the surface of a large spherical cap screen surrounding the operator by means of a group of high power projectors. Signals provided by the position sensors attached to the operator are used by the computer to update the projected images, which provides the operator the illusion of walking freely through the virtual scenes. This VE system can acquire wide field of view and form panorama surrounding the operator, which may enhance the realistic sense of simulation.

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Gelfand problem on a large spherical cap

Journal of Elliptic and Parabolic Equations ◽

10.1007/s41808-020-00091-9 ◽

2020 ◽

Author(s):

Yoshitsugu Kabeya ◽

Vitaly Moroz

Keyword(s):

Spherical Cap ◽

Gelfand Problem ◽

Large Spherical

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Electron-diffraction studies of amorphous carbon thin films

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100151337 ◽

1993 ◽

Vol 51 ◽

pp. 1100-1101

Author(s):

David A. Muller

Keyword(s):

Thin Films ◽

Electron Diffraction ◽

Amorphous Carbon ◽

Glassy Carbon ◽

Spherical Structure ◽

Local Ordering ◽

Carbon Arc ◽

Similar Appearance ◽

Carbon Thin Films ◽

Large Spherical

The sp2 rich amorphous carbons have a wide variety of microstructures ranging from flat sheetlike structures such as glassy carbon to highly curved materials having similar local ordering to the fullerenes. These differences are most apparent in the region of the graphite (0002) reflection of the energy filtered diffracted intensity obtained from these materials (Fig. 1). All these materials consist mainly of threefold coordinated atoms. This accounts for their similar appearance above 0.8 Å-1. The fullerene curves (b,c) show a string of peaks at distance scales corresponding to the packing of the large spherical and oblate molecules. The beam damaged C60 (c) shows an evolution to the sp2 amorphous carbons as the spherical structure is destroyed although the (220) reflection in fee fcc at 0.2 Å-1 does not disappear completely. This 0.2 Å-1 peak is present in the 1960 data of Kakinoki et. al. who grew films in a carbon arc under conditions similar to those needed to form fullerene rich soots.

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Random Assignment Problems on 2d Manifolds

Journal of Statistical Physics ◽

10.1007/s10955-021-02768-4 ◽

2021 ◽

Vol 183 (2) ◽

Author(s):

D. Benedetto ◽

E. Caglioti ◽

S. Caracciolo ◽

M. D’Achille ◽

G. Sicuro ◽

...

Keyword(s):

Assignment Problem ◽

Unit Area ◽

Constant Term ◽

Beltrami Operator ◽

Explicit Calculation ◽

Dimensional Manifold ◽

Two Dimensional ◽

Random Assignment ◽

Assignment Problems ◽

Theoretical Formulation

AbstractWe consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold $$\Omega $$ Ω of unit area. It is known that the average cost scales as $$E_{\Omega }(N)\sim {1}/{2\pi }\ln N$$ E Ω ( N ) ∼ 1 / 2 π ln N with a correction that is at most of order $$\sqrt{\ln N\ln \ln N}$$ ln N ln ln N . In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first $$\Omega $$ Ω -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on $$\Omega $$ Ω . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

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Linear evolution equations on the half-line with dynamic boundary conditions

European Journal of Applied Mathematics ◽

10.1017/s0956792521000103 ◽

2021 ◽

pp. 1-33

Author(s):

D. A. SMITH ◽

W. Y. TOH

Keyword(s):

Boundary Conditions ◽

Heat Equation ◽

Evolution Equations ◽

Half Line ◽

Robin Problem ◽

Dynamic Boundary Conditions ◽

Transform Method ◽

Linear Evolution ◽

Robin Condition ◽

Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

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An enumerative formula for the spherical cap discrepancy

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2021.113409 ◽

2021 ◽

Vol 390 ◽

pp. 113409

Author(s):

Holger Heitsch ◽

René Henrion

Keyword(s):

Spherical Cap

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Research and Fabrication of Broadband Ring Flextensional Underwater Transducer

Sensors ◽

10.3390/s21041548 ◽

2021 ◽

Vol 21 (4) ◽

pp. 1548

Author(s):

Jiuling Hu ◽

Lianjin Hong ◽

Lili Yin ◽

Yu Lan ◽

Hao Sun ◽

...

Keyword(s):

High Speed ◽

Structural Parameters ◽

Low Frequency ◽

Flexural Vibration ◽

Underwater Acoustic Communication ◽

Voltage Response ◽

Water Tank ◽

Spherical Cap ◽

Finite Element Software ◽

Underwater Transducer

At present, high-speed underwater acoustic communication requires underwater transducers with the characteristics of low frequency and broadband. The low-frequency transducers also are expected to be low-frequency directional for realization of point-to-point communication. In order to achieve the above targets, this paper proposes a new type of flextensional transducer which is constructed of double mosaic piezoelectric ceramic rings and spherical cap metal shells. The transducer realizes broadband transmission by means of the coupling between radial vibration of the piezoelectric rings and high-order flexural vibration of the spherical cap metal shells. The low-frequency directional transmission of the transducer is realized by using excitation signals with different amplitude and phase on two mosaic piezoelectric rings. The relationship between transmitting voltage response (TVR), resonance frequency and structural parameters of the transducer is analyzed by finite element software COMSOL. The broadband performance of the transducer is also optimized. On this basis, the low-frequency directivity of the transducer is further analyzed and the ratio of the excitation signals of the two piezoelectric rings is obtained. Finally, a prototype of the broadband ring flextensional underwater transducer is fabricated according to the results of simulation. The electroacoustic performance of the transducer is tested in an anechoic water tank. Experimental results show that the maximum TVR of the transducer is 147.2 dB and the operation bandwidth is 1.5–4 kHz, which means that the transducer has good low-frequency, broadband transmission capability. Meanwhile, cardioid directivity is obtained at 1.4 kHz and low-frequency directivity is realized.

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Approximating pointwise products of quasimodes

Forum Mathematicum ◽

10.1515/forum-2019-0208 ◽

2020 ◽

Vol 32 (3) ◽

pp. 541-552

Author(s):

Mei Ling Jin

Keyword(s):

Vector Space ◽

Harmonic Analysis ◽

Riemannian Manifolds ◽

Spectral Projection ◽

Beltrami Operator ◽

Singular Potentials ◽

Link Type ◽

Right Hand ◽

Low Degree ◽

The Right

AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

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