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  • Articles  (93)
  • 1985-1989  (93)
  • Mathematics  (93)
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  • Articles  (93)
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  • 1
    Electronic Resource
    Electronic Resource
    R-separation of variables for the time-dependent Hamilton–Jacobi and Schrödinger equations (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1005-1015 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The theory of R-separation of variables is developed for the time-dependent Hamilton–Jacobi and Schrödinger equations on a Riemannian manifold V n where time-dependent vector and scalar potentials are permitted. As an application it is shown how to obtain all R-separable coordinates for the n-sphere and Euclidean n-space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527592
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  • 2
    Electronic Resource
    Electronic Resource
    Teukolsky–Starobinsky identities for arbitrary spin (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 2925-2929 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Teukolsky–Starobinsky identities are proven for arbitrary spin s. A pair of covariant equations are given that admit solutions in terms of Teukolsky functions for general s. The method of proof is shown to extend to the general class of space-times considered by Torres del Castillo [J. Math. Phys. 29, 2078 (1988)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528479
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  • 3
    Electronic Resource
    Electronic Resource
    Generalized Stäckel matrices (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 2168-2173 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Stäckel and differential-Stäckel matrices are generalized so that the matrix elements may be functions of the derivatives of the dependent variable as well as the independent variable. The inverses of these matrices are characterized and it is shown that for significant classes of linear and nonlinear partial differential equations, variable separation is accomplished via this generalized Stäckel mechanism.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526840
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  • 4
    Electronic Resource
    Electronic Resource
    Matrix operator symmetries of the Dirac equation and separation of variables (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1893-1900 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The set of all matrix-valued first-order differential operators that commute with the Dirac equation in n-dimensional complex Euclidean space is computed. In four dimensions it is shown that all matrix-valued second-order differential operators that commute with the Dirac operator in four dimensions are obtained as products of first-order operators that commute with the Dirac operator. Finally some additional coordinate systems for which the Dirac equation in Minkowski space can be solved by separation of variables are presented. These new systems are comparable to the separation in oblate spheroidal coordinates discussed by Chandrasekhar [S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford U.P., Oxford, 1983)].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527395
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  • 5
    Electronic Resource
    Electronic Resource
    Differential-Stäckel matrices (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 1560-1565 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We show that additive separation of variables for linear homogeneous equations of all orders is characterized by differential-Stäckel matrices, generalizations of the classical Stäckel matrices used for multiplicative separation of (second-order) Schrödinger equations and additive separation of Hamilton–Jacobi equations. We work out the principal properties of these matrices and demonstrate that even for second-order Laplace equations additive separation may occur when multiplicative separation does not.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526917
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  • 6
    Electronic Resource
    Electronic Resource
    Killing–Yano tensors and variable separation in Kerr geometry (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 2360-2365 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A complete analysis of the free-field massless spin-s equations (s=0, (1)/(2) ,1) in Kerr geometry is given. It is shown that in each case the separation constants occurring in the solutions obtained from a potential function can be characterized in an invariant way. This invariant characterization is given in terms of the Killing–Yano tensor admitted by Kerr geometry.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528565
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  • 7
    Electronic Resource
    Electronic Resource
    Separation of variables on n-dimensional Riemannian manifolds. I. The n-sphere Sn and Euclidean n-space Rn (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1721-1736 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The following problem is solved: What are all the "different'' separable coordinate systems for the Laplace–Beltrami eigenvalue equation on the n-sphere Sn and Euclidean n-space Rn and how are they constructed? This is achieved through a combination of differential geometric and group theoretic methods. A graphical procedure for construction of these systems is developed that generalizes Vilenkin's construction of polyspherical coordinates. The significance of these results for exactly soluble dynamical systems on these manifolds is pointed out. The results are also of importance for the analysis of the special functions appearing in the separable solutions of the Laplace–Beltrami eigenvalue equation on these manifolds.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527088
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  • 8
    Electronic Resource
    Electronic Resource
    Sequence comparison with concave weighting functions (1988)
    Springer
    Bulletin of mathematical biology 50 (1988), S. 97-120 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We consider efficient methods for computing a difference metric between two sequences of symbols, where the cost of an operation to insert or delete a block of symbols is a concave function of the block's length. Alternatively, sequences can be optimally aligned when gap penalties are a concave function of the gap length. Two algorithms based on the ‘candidate list paradigm’ first used by Waterman (1984) are presented. The first computes significantly more parsimonious candidate lists than Waterman's method. The second method refines the first to the point of guaranteeingO(N 2 lgN) worst-case time complexity, and under certain conditionsO(N 2). Experimental data show how various properties of the comparison problem affect the methods' relative performance. A number of extensions are discussed, among them a technique for constructing optimal alignments inO(N) space in expectation. This variation gives a practical method for comparing long amino sequences on a small computer.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02459948
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  • 9
    Electronic Resource
    Electronic Resource
    Approximate matching of regular expressions (1989)
    Springer
    Bulletin of mathematical biology 51 (1989), S. 5-37 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Given a sequenceA and regular expressionR, theapproximate regular expression matching problem is to find a sequence matchingR whose optimal alignment withA is the highest scoring of all such sequences. This paper develops an algorithm to solve the problem in timeO(MN), whereM andN are the lengths ofA andR. Thus, the time requirement is asymptotically no worse than for the simpler problem of aligning two fixed sequences. Our method is superior to an earlier algorithm by Wagner and Seiferas in several ways. First, it treats real-valued costs, in addition to integer costs, with no loss of asymptotic efficiency. Second, it requires onlyO(N) space to deliver just the score of the best alignment. Finally, its structure permits implementation techniques that make it extremely fast in practice. We extend the method to accommodate gap penalties, as required for typical applications in molecular biology, and further refine it to search for substrings ofA that strongly align with a sequence inR, as required for typical data base searches. We also show how to deliver an optimal alignment betweenA andR in onlyO(N+logM) space usingO(MN logM) time. Finally, anO(MN(M+N)+N 2logN) time algorithm is presented for alignment scoring schemes where the cost of a gap is an arbitrary increasing function of its length.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02458834
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  • 10
    Electronic Resource
    Electronic Resource
    A compact representation of proofs (1987)
    Springer
    Studia logica 46 (1987), S. 347-370 
    Details
    ISSN: 1572-8730
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Philosophy
    Notes: Abstract A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00370646
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