continuous dynamical systems and mappings

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1: 32.16 Physical Applications

§32.16 Physical Applications

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2: About Color Map

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Continuous Phase Mapping

►For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. … ►

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Figure 3: Continuous phase mapping

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3: Bibliography

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Conformal Invariants, Inequalities, and Quasiconformal Maps. John Wiley & Sons Inc., New York.

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Notes:

With PC diskette, a Wiley-Interscience Publication

External Links:

ISBN 0-471-59486-5, MathReview (H. Renelt), ZentralBlatt

Referenced by:

§15.17(ii), §19.9(i).

Encodings:

BibTeX

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The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System. In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.), London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.17.

Encodings:

BibTeX

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Continuous Hahn polynomials. J. Phys. A 18 (16), pp. L1017–L1019.

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External Links:

ISSN 0305-4470, FullText, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.19, §18.20(ii), §18.21(ii).

Encodings:

BibTeX

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Spinning Tops: A Course on Integrable Systems. Cambridge Studies in Advanced Mathematics, Vol. 51, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-56129-9; 0-521-77919-7, MathReview (Emma Previato), ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

BibTeX

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Center for Algorithms and Interactive Scientific Software.

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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4: 18.39 Applications in the Physical Sciences

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Introduction and One-Dimensional (1D) Systems

… ►As in classical dynamics this sum is the total energy of the one particle system. … ►

1D Quantum Systems with Analytically Known Stationary States

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§18.39(v) Other Applications

►For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983). …

5: 1.5 Calculus of Two or More Variables

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§1.5(i) Partial Derivatives

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§1.5(ii) Coordinate Systems

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Polar Coordinates

… ►Again the mapping is one-to-one except perhaps for a set of points of volume zero. …

6: 23.20 Mathematical Applications

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§23.20(i) Conformal Mappings

… ►The interior of $R$ is mapped one-to-one onto the lower half-plane. … ►For examples of conformal mappings of the function $\mathrm{\wp }\left(z\right)$, see Abramowitz and Stegun (1964, pp. 642–648, 654–655, and 659–60). ►For conformal mappings via modular functions see Apostol (1990, §2.7). …

7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►When $\alpha$ is absolutely continuous, i. … ►

§1.18(vi) Continuous Spectra and Eigenfunction Expansions: Simple Cases

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§1.18(vii) Continuous Spectra: More General Cases

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8: 1.9 Calculus of a Complex Variable

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Continuity

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§1.9(iv) Conformal Mapping

… ►We then say that the mapping $w=f(z)$ is conformal (angle-preserving) at $z_0$. … ► …

9: 37.19 Other Orthogonal Polynomials of $d$ Variables

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§37.19(vi) OPs Associated with Root Systems

►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials of several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. In one variable they are essentially ultraspherical, Jacobi, continuous $q$-ultraspherical, or Askey–Wilson polynomials. …For general $q$ they occur as Macdonald polynomials for root system $A_n$ , as Macdonald polynomials for general root systems, and as Macdonald–Koornwinder polynomials; see Macdonald (1995, Chapter VI), Macdonald (2000, 2003), Koornwinder (1992). See §37.8 and §37.9 for Jacobi polynomials associated with root systems $B{C}_2$ and $A_2$, respectively.

10: 20.12 Mathematical Applications

… ►For an application of a generalization in affine root systems see Macdonald (1972). … ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …