scholarly journals Volume Preserving Maps Between p-Balls

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1404
Author(s):  
Adrian Holhoş ◽  
Daniela Roşca
Keyword(s):  
Regular Octahedron ◽  
Volume Preserving ◽  
Explicit Expressions

We construct a volume preserving map U p from the p-ball B p ( r ) = x ∈ R 3 , ∥ x ∥ p ≤ r to the regular octahedron B 1 ( r ′ ) , for arbitrary p > 0 . Then we calculate the inverse U p − 1 and we also deduce explicit expressions for U ∞ and U ∞ − 1 . This allows us to construct volume preserving maps between arbitrary balls B p ( r ) and B p ′ ( r ˜ ) , and also to map uniform and refinable grids between them. Finally we list some possible applications of our maps.

scholarly journals Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 994 ◽  
Author(s):  
Adrian Holhoş ◽  
Daniela Roşca
Keyword(s):  
Unit Ball ◽  
Multiresolution Analysis ◽  
Uniform Grid ◽  
Orthonormal Wavelet ◽  
Wavelet Bases ◽  
Piecewise Constant ◽  
Regular Octahedron ◽  
Piecewise Constant Functions ◽  
Local Support ◽  
Volume Preserving

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

Volume preserving immersed boundary methods for two-phase fluid flows

2011 ◽  
Vol 69 (4) ◽  
pp. 842-858 ◽  
Author(s):  
Yibao Li ◽  
Eunok Jung ◽  
Wanho Lee ◽  
Hyun Geun Lee ◽  
Junseok Kim
Keyword(s):  
Fluid Flows ◽  
Immersed Boundary ◽  
Immersed Boundary Methods ◽  
Two Phase ◽  
Boundary Methods ◽  
Phase Fluid ◽  
Volume Preserving

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

The use of Riemann problems in solving a class of transcendental equations

1973 ◽  
Vol 73 (1) ◽  
pp. 111-118 ◽  
Author(s):  
E. E. Burniston ◽  
C. E. Siewert
Keyword(s):  
Boundary Value Problem ◽  
Closed Form ◽  
Boundary Value ◽  
Explicit Solutions ◽  
Riemann Boundary Value Problem ◽  
Riemann Problems ◽  
Riemann Boundary ◽  
Canonical Solution ◽  
Transcendental Equations ◽  
Explicit Expressions

AbstractA method of finding explicit expressions for the roots of a certain class of transcendental equations is discussed. In particular it is shown by determining a canonical solution of an associated Riemann boundary-value problem that expressions for the roots may be derived in closed form. The explicit solutions to two transcendental equations, tan β = ωβ and β tan β = ω, are discussed in detail, and additional specific results are given.

Volume-preserving integrators have linear error growth

1998 ◽  
Vol 242 (1-2) ◽  
pp. 25-30 ◽  
Author(s):  
G.R.W Quispel ◽  
C.P Dyt
Keyword(s):  
Error Growth ◽  
Volume Preserving

scholarly journals Third derivatives of the integrable part of an asteroid Hamiltonian

2007 ◽  
pp. 53-60 ◽  
Author(s):  
R. Pavlovic
Keyword(s):  
Third Order ◽  
Geometrical Condition ◽  
The Third ◽  
Derivatives Of ◽  
Quasi Convexity ◽  
Explicit Expressions

To apply the theorem of Nekhoroshev (1977) to asteroids, one first has to check whether a necessary geometrical condition is fulfilled: either convexity, or quasi-convexity, or only a 3-jet non-degeneracy. This requires computation of the derivatives of the integrable part of the corresponding Hamiltonian up to the third order over actions and a thorough analysis of their properties. In this paper we describe in detail the procedure of derivation and we give explicit expressions for the obtained derivatives. .

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