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31: Guide to Searching the DLMF

… ►If a query does not return any hits, the DLMF search system relaxes the query to match and retrieve what may be intended by the original query. …Therefore, if your query is Ai^2+Bi^2, the system modifies the query so it will find the equations containing the latter expressions. … ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. … ►The syntax of the special functions can be LaTeX-like or as employed in widely used computer algebra systems. …

32: 4.48 Software

… ►All scientific programming languages, libraries, and systems support computation of at least some of the elementary functions in standard floating-point arithmetic (§3.1(i)). …

33: 18.37 Classical OP’s in Two or More Variables

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§18.37(iii) OP’s Associated with Root Systems

►Orthogonal polynomials associated with root systems are certain systems of trigonometric polynomials in several variables, symmetric under a certain finite group (Weyl group), and orthogonal on a torus. …In several variables they occur, for $q=1$, as Jack polynomials and also as Jacobi polynomials associated with root systems; see Macdonald (1995, Chapter VI, §10), Stanley (1989), Kuznetsov and Sahi (2006, Part 1), Heckman (1991). 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).

34: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. …

35: 36.14 Other Physical Applications

… ►Diffraction catastrophes describe the “semiclassical” connections between classical orbits and quantum wavefunctions, for integrable (non-chaotic) systems. …

36: Barry I. Schneider

… ►Schneider has served as Chair and Co-Chair of the APS Division of Computational Physics and the Topical Group on Few-Body Systems and Multipartical Dynamics and has been the organizer of a number of conferences and invited sessions here and abroad. …

37: 3.2 Linear Algebra

… ►To solve the system … ►

§3.2(ii) Gaussian Elimination for a Tridiagonal Matrix

… ►For more information on solving tridiagonal systems see Golub and Van Loan (1996, pp. 152–160). ►

§3.2(iii) Condition of Linear Systems

… ►The larger the value $\kappa (𝐀)$, the more ill-conditioned the system. …

38: 18.38 Mathematical Applications

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Integrable Systems

►The Toda equation provides an important model of a completely integrable system. …While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations. … ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. …Eigenvalue equations involving Dunkl type operators have as eigenfunctions nonsymmetric analogues of multivariable special functions associated with root systems. …

39: Bibliography M

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Affine root systems and Dedekind’s $\eta$-function. Invent. Math. 15 (2), pp. 91–143.

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External Links:

Document, MathReview (D.-N. Verma), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

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Some conjectures for root systems. SIAM J. Math. Anal. 13 (6), pp. 988–1007.

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External Links:

ISSN 0036-1410, Document, MathReview (S. Milne), ZentralBlatt

Referenced by:

§17.13, §17.14.

Encodings:

BibTeX

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Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).

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External Links:

ISSN 1286-4889, FullText, MathReview, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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Notes:

System for the manipulation of symbolic and numerical expressions. Includes a collection of special functions. Utilizes exact fractions, arbitrary precision integers, and arbitrary precision floating point numbers.

External Links:

Home Page

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.

Encodings:

BibTeX

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SciFace Software, Paderborn, Germany.

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Notes:

Computer algebra system for doing symbolic and exact algebraic computations as well as numerical calculations with almost arbitrary accuracy

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

…

40: 20.12 Mathematical Applications

… ►For an application of a generalization in affine root systems see Macdonald (1972). …