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1: 3.11 Approximation Techniques
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§3.11(v) Least Squares Approximations
… ► … ► … ► … ►For further information on least squares approximations, including examples, see Gautschi (1997a, Chapter 2) and Björck (1996, Chapters 1 and 2). …2: Bibliography Y
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-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian.
Phys. Rev. A 11 (4), pp. 1144–1156.
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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State.
American Journal of Physics 57 (1), pp. 85–86.
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3: Guide to Searching the DLMF
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Table 1: Query Examples
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Query | Matching records contain |
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Fourier or series |
at least one of the words “Fourier” or “series”. |
Fourier (transform or series) |
at least one of “Fourier transform” or “Fourier series”. |
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J_n@(x or z)= |
at least one of the math fragments or , emphasizing that is a function. |
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sin x and (J_nu(z) or I_nu(z)) |
both and at least one of the two functions or . |
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trigonometric^2 + trig$^2 |
any sum of the squares of two trigonometric functions such as . |
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4: 26.18 Counting Techniques
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►With the notation of §26.15, the number of placements of nonattacking rooks on an chessboard that avoid the squares in a specified subset is
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►The number of ways of placing labeled objects into labeled boxes so that at least one object is in each box is
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5: Bibliography P
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria.
Comput. J. 9 (4), pp. 404–407.
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6: 1.11 Zeros of Polynomials
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►Every monic (coefficient of highest power is one) polynomial of odd degree with real coefficients has at least one real zero with sign opposite to that of the constant term.
A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
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►The square roots are chosen so that
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►Resolvent cubic is with roots , , , and , , .
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7: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into parts with difference at least
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denotes the number of partitions of into parts with difference at least 3, except that multiples of 3 must differ by at least 6.
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8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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§1.18(ii) spaces on intervals in
… ►For a Lebesgue–Stieltjes measure on let be the space of all Lebesgue–Stieltjes measurable complex-valued functions on which are square integrable with respect to , … ►Eigenfunctions corresponding to the continuous spectrum are non- functions. … ►What then is the condition on to insure the existence of at least a single eigenvalue in the point spectrum? The discussions of §1.18(vi) imply that if then there is only a continuous spectrum. Surprisingly, if on any interval on the real line, even if positive elsewhere, as long as , see Simon (1976, Theorem 2.5), then there will be at least one eigenfunction with a negative eigenvalue, with corresponding eigenfunction. …9: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Numerical Methods for Least Squares Problems.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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