• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.657-668
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• 2016

Given a partition ${\lambda}=({\lambda}_1,{\lambda}_2,{\ldots},{\lambda}_l)$ of a positive integer n, let Tab(${\lambda}$, k) be the set of all tabloids of shape ${\lambda}$ whose weights range over the set of all k-compositions of n and ${\mathcal{OP}}^k_{\lambda}_{rev}$ the set of all ordered partitions into k blocks of the multiset $\{1^{{\lambda}_l}2^{{\lambda}_{l-1}}{\cdots}l^{{\lambda}_1}\}$. In [2], Butler introduced an inversion-like statistic on Tab(${\lambda}$, k) to show that the rank-selected $M{\ddot{o}}bius$ invariant arising from the subgroup lattice of a finite abelian p-group of type ${\lambda}$ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(${\lambda}$, k) and ${\mathcal{OP}}^k_{\hat{\lambda}}$. When k = 2, we also introduce a major-like statistic on Tab(${\lambda}$, 2) and study its connection to the inversion statistic due to Butler.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.123-132
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• 2021

Let K be a number field and L a finite abelian extension of K. Let E be an elliptic curve defined over K. The restriction of scalars ResKLE decomposes (up to isogeny) into abelian varieties over K $$Res^L_KE{\sim}{\bigoplus_{F{\in}S}}A_F,$$ where S is the set of cyclic extensions of K in L. It is known that if L is a quadratic extension, then AL is the quadratic twist of E. In this paper, we consider the case that K is a number field containing a primitive third root of unity, $L=K({\sqrt[3]{D}})$ is the cyclic cubic extension of K for some D ∈ K×/(K×)3, E = Ea : y2 = x3 + a is an elliptic curve with j-invariant 0 defined over K, and EaD : y2 = x3 + aD2 is the cubic twist of Ea. In this case, we prove AL is isogenous over K to $E_a^D{\times}E_a^{D^2}$ and a property of the Selmer rank of AL, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

• Proceedings of the KWS Conference
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• 2002.10a
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• pp.675-680
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• 2002

Cracking and bulging in welded and internally lined pressure vessels that work in thermal-mechanical cycling service have been well known problems in the petrochemical, power and nuclear industries. However, published literature and industry surveys show that similar problems have been occurring during the last 50 years. A better understanding of the causes of cracking and bulging causes is needed to improve the reliability of these pressure vessels. This study attempts to add information required for increasing the knowledge and fundamental understanding required. Typical examples of this problem are the coke drums in the delayed coking units refinery process. This case was selected for experimental work, field study and results comparison. Delayed coking units are among the refinery units that have higher economical yields. To shut down these units represents a high negative economical impact in refinery operations. Also, the maintenance costs associated with repairs are commonly very high. Cracking and bulging occurrences in the coke drums, most often at the weld areas, characterize the history of the operation of delayed coking units. To design and operate more robust coke drums with fewer problems, an improved metallurgical understanding of the cracking and bulging mechanisms is required. A methodology that is based field experience revision and metallurgical analyses for the screening of the most important variables, and subsequent finite element analyses to verify hypotheses and to rank the variables according to their impact on the coke drum lives has been developed. This indicated approach provides useful information for increasing coke drum reliability. The results of this work not only order the most important variables according to their impact in the life of the vessels, but also permit estimation of the life spans of coke drums. In conclusion, the current work shows that coke drums may fail as a combination of thermal fatigue and other degradation mechanisms such as: corrosion at high and low temperatures, detrimental metallurgical transformations and plastic deformation. It was also found that FEA is a very valuable tool for understanding cracking and bulging mechanisms in these services and for ranking the design, fabrication, operation and maintenance variables that affect coke drum reliability.

• Structural Engineering and Mechanics
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• v.45 no.5
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• pp.595-611
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• 2013

This paper aims to compare three collocation point methods associated with the odd order stochastic response surface method (SRSM) in a systematical and quantitative way. The SRSM with the Hermite polynomial chaos is briefly introduced first. Then, three collocation point methods, namely the point method, the root method and the without origin method underlying the odd order SRSMs are highlighted. Three examples are presented to demonstrate the accuracy and efficiency of the three methods. The results indicate that the condition that the Hermite polynomial information matrix evaluated at the collocation points has a full rank should be satisfied to yield reliability results with a sufficient accuracy. The point method and the without origin method are much more efficient than the root method, especially for the reliability problems involving a large number of random variables or requiring complex finite element analysis. The without origin method can also produce sufficiently accurate reliability results in comparison with the point and root methods. Therefore, the origin often used as a collocation point is not absolutely necessary. The odd order SRSMs with the point method and the without origin method are recommended for the reliability analysis due to their computational accuracy and efficiency. The order of SRSM has a significant influence on the results associated with the three collocation point methods. For normal random variables, the SRSM with an order equaling or exceeding the order of a performance function can produce reliability results with a sufficient accuracy. The order of SRSM should significantly exceed the order of the performance function involving strongly non-normal random variables.