• Kyungpook Mathematical Journal
• /
• v.49 no.4
• /
• pp.731-742
• /
• 2009

In this paper, by introducing parameters ${\lambda}$, ${\alpha}$and two pairs of conjugate exponents (p, q), (r, s) and applying the improved Euler-Maclaurin's summation formula, we establish a reverse of the slightly sharper Hilbert-type inequality. As applications, the strengthened version and the equivalent form are given.

• Economic and Environmental Geology
• /
• v.55 no.3
• /
• pp.231-239
• /
• 2022

The irregular shape of the disturbances is a fundamental issue for mining engineers at the Sidi Chennane phosphate deposit in Morocco. A precise classification of disturbed areas is therefore necessary to understand their part in the overall volume of phosphate. In this paper, we investigate the theoretical and practical aspects of studying and measuring multifractal spectrums as a defining and representative parameter for distinguishing between the phosphate deposit of a low rate of disturbances and the deposit of a high rate. An empirical multifractal approach was used by analyzing the disturbed areas through the geoelectric images of an area located in the Sidi Chennane phosphate deposit. The Generalized fractal dimension, D(q), the Singularities of strength, α(q), the local dimension, f(α) and their conjugate parameter the mass exponent, τ(q) as well as f(α)-α spectrum were the common multifractal parameters used. The results reported show wide variations of the analyzed images, indicating that the multifractal analysis is an indicator for evaluate and characterize the disturbed areas within the phosphates deposits through the studied geoelectric images. This could be the starting point for future work aimed at improving phosphate exploration planning.

• Korean Journal of Mathematics
• /
• v.15 no.1
• /
• pp.13-26
• /
• 2007

We prove that every strong null sequence in a Banach space X lies inside the range of a vector measure of bounded variation if and only if the condition $\mathcal{N}_1(X,{\ell}_1)={\Pi}_1(X,{\ell}_1)$ holds. We also prove that for $1{\leq}p<{\infty}$ every strong ${\ell}_p$ sequence in a Banach space X lies inside the range of an X-valued measure of bounded variation if and only if the identity operator of the dual Banach space $X^*$ is ($p^{\prime}$,1)-summing, where $p^{\prime}$ is the conjugate exponent of $p$. Finally we prove that a Banach space X has the property that any sequence lying in the range of an X-valued measure actually lies in the range of a vector measure of bounded variation if and only if the condition ${\Pi}_1(X,{\ell}_1)={\Pi}_2(X,{\ell}_1)$ holds.