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  • Author or Editor: S. Mukhopadhyay x
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Abstract  

If f: ℝ → ℝ is integrable in a right neighbourhood of x ∈ ℝ and if there are real numbers α 0, α 1, ..., α n−1 such that the limit lim

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim }\limits_{s \to \infty } s^{n + 1} \int_0^\delta {e^{ - st} } \left[ {f(x + t) - \sum\limits_{i = 0}^{n - 1} {\frac{{t^i }} {{i!}}\alpha _i } } \right]dt$$ \end{document}
exists, then this limit is called the right-hand Laplace derivative of f at x of order n and is denoted by LDn + f(x). There is a corresponding definition for the left-hand derivative and if they are equal the common value is the Laplace derivative LDn f(x). In this paper, it is shown that the basic properties of the Peano derivatives are also possessed by this derivative (cf. [5]).

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More than 5000 high precision seismic phases of 560 selected aftershocks ( M ≥ 2.0) of the January 26, 2001 Bhuj earthquake ( M W 7.7) in western India are used for joint determination of the hypocentral parameters and for 3D inversion of P-wave velocity and V p / V s structures in the source area. The aftershocks are located with an average rms of 0.19 s, and average error estimates of latitude, longitude and depth are 1.2, 1.1 and 2.3 km respectively. Most of the aftershocks occurred in an area 70 × 35 sq km; the intense activity was observed at a depth range 12–37 km. A bimodal distribution of aftershocks indicates that the main shock rupture propagated in upward and downward directions. Further, the best located aftershocks show two trends, one in northeast, parallel to Anjar Rapar Lineament, and the other in northwest parallel to the Bhachau Lineament. Fault-plane solutions of the northeast trending aftershocks indicate reverse faulting with left-lateral strike-slip component. These solutions are comparable with the main shock solution. The northwest trending aftershocks, on the other hand, show reverse faulting with right-lateral strike-slip motion. The estimated velocity structure indicates that the source zone of the Bhuj earthquake has a number of blocks showing lateral heterogeneities in P-and S-wave velocities. A block having higher P-and S-wave velocities appears to have uplift relative to its surroundings. The mainshock occurred at the boundary between the high V p , high V s uplifted block and the adjacent low V p , low V s block. Gravity observations support our 3D inversion results. This high velocity block is surrounded by rocks of higher V p / V s or lower rigidity, which possibly acted as a barrier zone.

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Abstract  

An integral more general than the SCP-integral is introduced, and among other properties, it is proved that iff is integrable in this sense with its second primitive ACG* and ifG is the indefinite integral of a functiong of bounded variation, thenfG is also integrable in this sense and the integration by parts formula holds. The integration by parts formula for the SCP-integral is also proved wheng is of bounded variation. This advances the existing result when, in addition,g is also continuous.

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