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  • 1
    ISSN: 1432-0940
    Keywords: 41A05 ; 41A63 ; Conditionally negative definite ; Completely monotone ; Radial ; Interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Motivated by the problem of multivariate scattered data interpolation, much interest has centered on interpolation by functions of the form $$f(x) = \sum\limits_{j = 1}^N {a_j g(\parallel x - x_j \parallel ),x \in R^s }$$ whereg:R + →R is some prescribed function. For a wide range of functionsg, it is known that the interpolation matricesA=g(∥x i −x j ∥) i,j=1 N are invertible for given distinct data pointsx 1,x 2,...,x N. More recently, progress has been made in quantifying these interpolation methods, in the sense of estimating the (l 2) norms of the inverses of these interpolation matrices as well as their condition numbers. In particular, given a suitable functiong:R + →R, and data inR s having minimal separationq, there exists a functionh s:R + →R +, which depends only ong ands, and a constantC s , which depends only ons, such that the inverse of the associated interpolation matrixA satisfies the estimate ‖A −1‖≤C s h s (q). The present paper seeks “converse” results to the inequality given above. That is, given a suitable functiong, a spatial dimensions, and a parameterq〉0 (which is usually assumed to be small), it is shown that there exists a data set inR s having minimal separationq, a constant $$\tilde C_s$$ depending only ons, and a functionk s (q), such that the inverse of the interpolation matrixA associated with this data set satisfies $$\parallel A^{ - 1} \parallel \geqslant \tilde C_s k_s (q)$$ . In some cases, it is seen thath s(q)=k s (q), so the bounds are optimal up to constants. In certain others,k s (q) is less thanh s (q), but nevertheless exhibits a behavior comparable to that ofh s (q). That is, even in these cases, the bounds are close to being optimal.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1432-0940
    Keywords: 41A05 ; 41A63 ; Positive definite ; Radial functions ; Polya frequency functions ; Toeplitz matrices ; Multiquadrics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A radial basis function approximation has the form whereϕ:R d →R is some given (usually radially symmetric) function, (y j ) 1 n are real coefficients, and the centers (x j ) 1 n are points inR d . For a wide class of functions ϕ, it is known that the interpolation matrixA=(ϕ(x j −x k )) j,k=1 n is invertible. Further, several recent papers have provided upper bounds on ||A −1||2, where the points (x j ) 1 n satisfy the condition ||x j −x k ||2≥δ,j≠k, for some positive constant δ. In this paper we calculate similar upper bounds on ||A −1||2 forp≥1 which apply when ϕ decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixA n = (ϕ(j−k)) j,k=1 n when ϕ(x)=(x 2+c 2)1/2, the Hardy multiquadric. In particular, we show that sup n ||A n −1 ||∞ is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||E −1|| p = ||E −1||2 for everyp≥1 when ϕ is a Gaussian. Indeed, we also show that this property persists for any function ϕ which is a tensor product of even, absolutely integrable Pólya frequency functions.
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  • 3
    ISSN: 0014-4827
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Biology , Medicine
    Type of Medium: Electronic Resource
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  • 4
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Product state distributions are reported for the vibrational predissociation of the NeCl2, B state, v'=6 through v'=13 levels. For the lower vibrational levels, Δv=−1 dissociation produces a bimodal Cl2 product rotational state distribution with the first maximum at j=4 and a secondary maximum at j=20. Surprisingly, the positions of these maxima are the same for v'=6, 7, 8, 9, and 10. For higher vibrational levels the limited available phase space constricts the observed rotational distribution allowing the Ne–Cl2 bond energy D0 to be determined. D0 is 54±2 cm−1 for the B electronic state, and 60±2 cm−1 for the ground electronic state. Δv=−2 dissociation produces a rotational distribution which, although not bimodal, is otherwise quite similar to that of the Δv=−1 channel, even though significantly more energy is released to product translation for Δv=−2. This behavior is quite different from what would be predicted by an impulsive half-collision model for the dynamics. Three dimensional quantum calculations on a simple atom–atom potential energy surface were able to reproduce most of the essential features of the experimental results. We conclude that the anisotropy of the initial wave function and that of the coupling between the covalent and van der Waals modes is more important in determining the product rotational distribution than is the kinematics of the dissociation trajectory. Since the rotational distribution produced by the Δv=−2 channel is similar to that of the Δv=−1 channel, the Δv=−2 dynamics probably occurs by a direct coupling between the quasibound state and the continuum rather than by a sequential mechanism with two Δv=−1 steps.
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  • 5
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The structure and vibrational predissociation dynamics of HeCl2 are studied by pump–probe spectroscopy and by three-dimensional quantum mechanical calculations. Parity selected excitation spectroscopy is used to confirm the essential features of the previous analysis of the HeCl2 B←X laser excited fluorescence spectra. Product vibrational and rotational state distributions are measured for the v'=6, 8, 12, 20, and 24 levels of HeCl2 in the B state. For the v'=6 and 8 levels the dependence of the product state distribution on the initially excited rotational state is also measured. Although the dissociation dynamics are dominated by Δv=−1, V→T energy transfer, several interesting effects are revealed by monitoring the product rotational degrees of freedom. Due to the symmetry of the HeCl2 potential, the parity of the initially excited HeCl2 rotational state is conserved during the dissociation dynamics. Even when a single initial rotational state is excited, the observed product rotational state distribution is bimodal. The product rotational distribution is nearly independent of the amount of kinetic energy released to the product degrees of freedom. Three-dimensional quantum mechanical calculations using a simple potential energy surface are remarkably successful at reproducing the details of the experimental measurements. Only five parameters of the potential were adjusted to calculate the excitation spectrum, the vibrational predissociation product state distributions, and the lifetimes of the excited states. Analysis of the dissociation mechanism in terms of simple models, however, is not straightforward. In particular, the impulsive, quasiclassical half-collision model is not compatible with the observed independence of the product rotational state distribution from the amount of kinetic energy which is released. The close agreement between state-to-state experiment and quantum theory on the HeCl2 dynamics shows that the shortcomings of more approximate theories are fundamental and cannot be attributed to lack of knowledge of the true potential energy surface.
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  • 6
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Rotational–vibrational distributions are reported for Cl2 product states of the vibrational predissociation of the v=6 through v=12 levels of the ArCl2 molecule in the B electronic state. For v=6 and 7, the Δv=−1 dissociation channel dominates the predissociation dynamics, and the observed rotational distribution appears to be slightly inverted. This inversion does not appear to be due to "momentum gap'' constraints but may be due to kinematic symmetry selection rules. Above v=7 the molecule decays via the Δv=−2 channel. Unlike the rotational distributions observed for HeCl2 and NeCl2, those of the ArCl2 Δv=−2 channel are highly structured and have a strong dependence on the initially excited vibrational level. It is argued that the dependence of the observed rotational distribution on the initial vibrational state (for the dissociation of ArCl2) is due to intermediate resonances in a sequential, IVR mechanism, whereas the dissociation of HeCl2 and NeCl2 occurs by a direct mechanism. Although this conclusion cannot be proved at this time, experiments and calculations are suggested that would directly measure the participation of intermediate levels.
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  • 7
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: A technique has been developed for determining the angular correlation between a photofragment's angular momentum vector J, its recoil velocity vector v, and the transition dipole moment of the parent molecule μp . Doppler profile spectroscopy used in conjunction with laser-induced fluorescence probing by polarized light can be used to determine the correlations. The pairwise correlations between these vectors as well as their triple correlation are discussed for limiting cases using a classical approach as well as for the general case using a quantum approach based on density matrices. The current formulations differ in two ways from the recent approach of Dixon, who used a bipolar expansion of the correlated velocity and angular momentum distributions. The physical basis for the influence of the vector correlations on the Doppler profile is somewhat more transparent in the current formulations, and the direct connection between the measured correlations and the t-matrix elements occurring in the theory of Balint-Kurti and Shapiro for the photodissociation of a triatomic molecule is also demonstrated.
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  • 8
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 88 (1988), S. 3692-3708 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: Photodissociation of OCS in the region from 222–248 nm has been investigated by monitoring the CO and S(1D2) primary photoproducts; as well as the secondary production of S(3P2), S(3P1), and S(3P0) using fluorescence induced by a tunable vacuum ultraviolet laser source based on four-wave mixing in magnesium vapor. The quantum yield of S(3P) was found to be 0.00±0.02 at 222 nm. Thus, in contrast to our preliminary report, the present more detailed investigation shows that the sole sulfur product appears to be S(1D). The CO photofragment is produced almost exclusively in v=0 [CO(v=1)/ CO(v=0)≤0.02], but the rotational distribution is inverted and peaked at very high rotational levels. The peak shifts from J=56 for dissociation at 222 nm to J=31 at 248 nm. Doppler profiles of the CO rotational transitions reveal (1) that all observed levels are produced in coincidence with S(1D), (2) that for 222 nm photolysis the fragment recoil anisotropy shifts from a distribution characterized by β=1.9 at J=67 toward one characterized by β=0 near J=54, (3) that the CO velocity vector is aligned nearly perpendicular to its angular momentum vector, and (4) that the CO angular momentum vector is also aligned parallel to that component of the transition dipole which lies perpendicular to the recoil velocity. These results are interpreted in terms of a model for the dissociation in which excitation takes place to two surfaces of A' and A‘ symmetry derived from a bent 1Δ configuration. Dissociation of OCS clusters was also investigated and was found to produce a photochemistry completely different from that of the monomers. Rotationally cold CO as well as S2 in both the X3Σ−g and a1Δg states was observed.
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  • 9
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 86 (1987), S. 1636-1637 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
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  • 10
    ISSN: 0945-3245
    Keywords: 41A05 ; 41A63
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper investigates some aspects of discrete least squares approximation by translates of certain classes of radial functions. Its specific aims are (i) to provide conditions under which the associated least squares matrix is invertible and (ii) to give upper bounds for the Euclidean norms of the inverses of these matrices (when they exist).
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