An H1 function with non-summable Fourier expansion

1983 ◽  
pp. 193-200 ◽  
Author(s):  
E. M. Stein
Keyword(s):  
Fourier Expansion

scholarly journals Gravitational Interaction in the Chimney Lattice Universe

Universe ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 101
Author(s):  
Maxim Eingorn ◽  
Andrew McLaughlin ◽  
Ezgi Canay ◽  
Maksym Brilenkov ◽  
Alexander Zhuk
Keyword(s):  
Helmholtz Equation ◽  
Gravitational Potential ◽  
Fourier Expansion ◽  
Gravitational Interaction ◽  
Alternative Solution ◽  
The Third ◽  
Present Universe ◽  
Delta Functions ◽  
Yukawa Potentials ◽  
The Universe

We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.

scholarly journals A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Niklas Ericsson
Keyword(s):  
Fourier Expansion ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Countable Family ◽  
Angular Variable ◽  
Two Dimensional ◽  
Sobolev Norms ◽  
Well Posed ◽  
Variational Formulations

Abstract We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems.

Chaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microfluidic device

2016 ◽  
Vol 5 (1) ◽  
pp. 1-8
Author(s):  
Yasser Aboelkassem
Keyword(s):  
Mathematical Modeling ◽  
Stream Function ◽  
Microfluidic Device ◽  
Fourier Expansion ◽  
Expansion Method ◽  
Small Oscillation ◽  
Chaotic Mixing ◽  
Mixing Efficiency ◽  
Induced Flow ◽  
Mixing Patterns

AbstractChaotic mixing by oscillating a Stokeslet in a circular Hele-Shaw microffluidic device is presented in this article. Mathematical modeling for the induced flow motions by moving a Stokeslet along the x-axis is derived using Fourier expansion method. The solution is formulated in terms of the velocity stream function. The model is then used to explore different stirring dynamics as function of the Stokeslet parameters. For instance, the effects of using various oscillation amplitudes and force strengths are investigated. Mixing patterns using Poincaré maps are obtained numerically and have been used to characterize the mixing efficiency. Results have shown that, for a given Stokeslet’s strength, efficient mixing can be obtained when small oscillation amplitudes are used. The present mixing platform is expected to be useful for many of biomicrofluidic applications.

AVERAGE SIZE OF GAPS IN THE FOURIER EXPANSION OF MODULAR FORMS

2007 ◽  
Vol 03 (02) ◽  
pp. 207-215 ◽  
Author(s):  
EMRE ALKAN
Keyword(s):  
Elliptic Curve ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Complex Multiplication ◽  
Gap Function ◽  
Quantitative Results ◽  
Average Size

We prove that certain powers of the gap function for the newform associated to an elliptic curve without complex multiplication are "finite" on average. In particular we obtain quantitative results on the number of large values of the gap function.

scholarly journals On Dirichlet series attached to cusp forms and the Siegel-zero

1984 ◽  
Vol 25 (1) ◽  
pp. 107-119 ◽  
Author(s):  
F. Grupp
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Fourier Expansion ◽  
Dirichlet Character ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Siegel Zero ◽  
Image Position

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansionFor every Dirichlet character xmod Q we define

scholarly journals Anchoring Energy for Nematic Liquid Crystals an Analysis of the Proposed Forms

1984 ◽  
Vol 39 (11) ◽  
pp. 1066-1076 ◽  
Author(s):  
G. Barbero ◽  
N. V. Madhusudana ◽  
G. Durand
Keyword(s):  
Liquid Crystal ◽  
Infinite Series ◽  
Tilt Angle ◽  
Legendre Polynomials ◽  
Fourier Expansion ◽  
Easy Axis ◽  
Practical Interest ◽  
Surface Anchoring ◽  
Functional Forms ◽  
Anchoring Energy

We analyze the proposed functional forms describing the surface anchoring energy of MBBA nematic liquid crystal. Measurements of the surface torque versus tilt angle suggest that the simple cosinus-square form is not adequate for MBBA. The generalized form in an infinite series in cosinus-square is not useful since the set of functions is not orthogonal. Analyses using the orthogonal Legendre polynomials and Fourier expansion are given. Finally we analyse the data in terms of small tilt angles compared to the easy axis, and very far from it, which are the two limits of practical interest.

Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile

1996 ◽  
Vol 35 (1) ◽  
pp. 61 ◽  
Author(s):  
B. V. Dorrío ◽  
J. Blanco-García ◽  
C. López ◽  
A. F. Doval ◽  
R. Soto ◽  
...  
Keyword(s):  
Fourier Expansion ◽  
Phase Error ◽  
Intensity Profile ◽  
Fizeau Interferometer ◽  
Error Calculation

A note on Riesz potential

1951 ◽  
Vol 47 (2) ◽  
pp. 436-442 ◽  
Author(s):  
F. C. Auluck ◽  
L. S. Kothari
Keyword(s):  
Electromagnetic Field ◽  
Recent Work ◽  
Quantum Electrodynamics ◽  
Fourier Expansion ◽  
Riesz Potential ◽  
Arbitrary Parameter ◽  
Electromagnetic Potentials ◽  
Longitudinal Part ◽  
Definition Of

The object of the present paper is to discuss the Fourier expansion of the Riesz potential. For this purpose a new definition of the electromagnetic potentials, depending upon an arbitrary parameter α is given. It is shown that this definition is a generalization of the Wentzel potentials in the α-plane, whereas that given by Fremberg (3) is a generalization of the Maxwell potentials. The analysis is applied to the problem of eliminating, in a straightforward way, the longitudinal part of the potential describing the electromagnetic field. The problem of the quantization of the field, based on its Fourier expansion, will be considered in another paper. The recent work of Tomonaga, Schwinger and Dyson, and the regularization process of Pauli has lifted the theory of quantum electrodynamics to a much higher level of rigour and fruitful applicability. All the same, a further study of Riesz potential seems to us of some interest in this field.

Stokes’ first problem for an electro-conducting micropolar fluid with thermoelectric properties

2010 ◽  
Vol 88 (1) ◽  
pp. 35-48 ◽  
Author(s):  
Magdy A. Ezzat ◽  
Hamdy M. Youssef
Keyword(s):  
Thermoelectric Properties ◽  
Micropolar Fluid ◽  
Inversion Formula ◽  
Figure Of Merit ◽  
Fourier Expansion ◽  
Fourier Transforms ◽  
Heat Sources ◽  
Layer Model ◽  
Thermoelectric Figure ◽  
Coupled Equations

This work is related to the flow of an electro-conducting micropolar fluid presenting thermoelectric properties effect in the presence of a magnetic field. The electro-conducting thermofluid equation of heat transfer with one relaxation time is derived. The flow of an electro-conducting micropolar fluid over a plate that is moved suddenly is considered. The governing coupled equations in the frame of the boundary-layer model are applied to Stokes' first problem with heat sources. Laplace-transform and Fourier-transform techniques are used to obtain the solution. The inverses of the Fourier transforms are obtained analytically. The Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier-expansion techniques. Numerical results for the temperature distribution, the velocity, and the microrotation components are represented graphically. Thermoelectric figure-of-merit, Seebeck and Peltier effects on a micropolar fluid are studied.

Sign in / Sign up

Export Citation Format

Share Document