• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1017-1024
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• 2023

A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of n with an even number of even parts minus the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over q-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.985-1001
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• 2023

Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S⁺⁺_p,q (a, b) and S^+-_p,q (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.671-697
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• 2024

In this paper, we formally introduce the notion of a general parametric digamma function Ψ(−s; A, a) and we find the Laurent expansion of Ψ(−s; A, a) at the integers and poles. Considering the contour integrations involving Ψ(−s; A, a), we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g. trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.25 no.3
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• pp.138-146
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• 2013

Tetrapod, one of the famous armor blocks for rubble mound breakwaters, has been widely used in the world. In order to evaluate the required weight of Tetrapod, many researchers have proposed various stability formulas. Since the stability formulas were proposed by curve-fitting the experimental data, some uncertainties are included in the formulas. The main uncertainties are associated with experimental data, derivation of the formula, and variability of the design variables. In this study, a new stability formula is developed by using M5' model tree, which reduces the uncertainty in the derivation of the formula. The index of agreement is used to evaluate the performance of the developed formula. The index of agreement for the new formula is higher by about 0.1 than the previous formula. The performance of the previous formula was not good when the predicted stability number is greater than about 3.0. However. the new formula is accurate regardless of the magnitude of stability number. As a result, the new formula performs better than the previous formula, while expressed in the form of a tree but still in an explicit form.

• The Pure and Applied Mathematics
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• v.20 no.1
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• pp.37-50
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• 2013

In the theory of hypergeometric functions of one or several variables, a remarkable amount of mathematicians's concern has been given to develop their transformation formulas and summation identities. Here we aim at presenting explicit expressions (in a single form) of the following weighted Appell's function $F_1$: $$(1+2x)^{-a}(1+2z)^{-b}F_1\;\(c,\;a,\;b;\;2c+j;\;\frac{4x}{1+2x},\;\frac{4z}{1+2z}\)\;(j=0,\;{\pm}1,\;{\ldots},\;{\pm}5)$$ in terms of Exton's triple hypergeometric $X_9$. The results are derived with the help of generalizations of Kummer's second theorem very recently provided by Kim et al. A large number of very interesting special cases including Exton's result are also given.

• Communications for Statistical Applications and Methods
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• v.24 no.4
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• pp.325-337
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• 2017

In this paper, we proposed a new long-term lifetime distribution with four parameters inserted in a risk competitive scenario with decreasing, increasing and unimodal hazard rate functions, namely the Weibull-Poisson long-term distribution. This new distribution arises from a scenario of competitive latent risk, in which the lifetime associated to the particular risk is not observable, and where only the minimum lifetime value among all risks is noticed in a long-term context. However, it can also be used in any other situation as long as it fits the data well. The Weibull-Poisson long-term distribution is presented as a particular case for the new exponential-Poisson long-term distribution and Weibull long-term distribution. The properties of the proposed distribution were discussed, including its probability density, survival and hazard functions and explicit algebraic formulas for its order statistics. Assuming censored data, we considered the maximum likelihood approach for parameter estimation. For different parameter settings, sample sizes, and censoring percentages various simulation studies were performed to study the mean square error of the maximum likelihood estimative, and compare the performance of the model proposed with the particular cases. The selection criteria Akaike information criterion, Bayesian information criterion, and likelihood ratio test were used for the model selection. The relevance of the approach was illustrated on two real datasets of where the new model was compared with its particular cases observing its potential and competitiveness.

• Journal of the Society of Naval Architects of Korea
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• v.29 no.1
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• pp.14-28
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• 1992

For a planar curve-sided paned with constant or linear density distributions of source or doublet in the singularity methods, Cantaloube and Rehbach(1986) show that the surface integral can be transformed into contour integral by using Stokes' formulas. As an extension of their formulations, this paper deals with a planar polygonal panel for which we derive the closed-forms of the potentials and the velocities induced by the singularity distributions. Test calculations show that the analytical evaluation of the closed-forms is superior to numerical integration(suggested by Cantaloube and Rehbach) of the contour integral. The compact and explicit expressions may produce accurate values of matrix elements of simultaneous linear equations in the singularity methods with much reduced computer tiome.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.423-445
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• 2021

In this paper, a new form of poly-Genocchi polynomials is defined by means of polylogarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which are used to express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials that possesses a double generating function is introduced. Finally, the type 2 Apostolpoly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.

• Journal of KIISE:Databases
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• v.27 no.2
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• pp.151-164
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• 2000

The Visual Object Query Language (VOQL) recently proposed for object databases has been successful in visualizing path expressions and set-related conditions, and providing formal semantics. However, VOQL has several problems. Due to unrealistic assumptions, only set-related conditions can be represented in VOQL. Due to the lack of explicit language construct for the notion of variables, queries are often awkard and less intuitive. In this paper, we propose VOQL*, which extends VOQL to remove these drawbacks. We introduce the notion of visual variables and refine the syntax and semantics of VOQL based on visual variables. We carefully design the language constructs of VOQL* to reflect the syntax of OOPC, so that the constructs such as visual variables, visual elements, VOQL* simple terms, VOQL* structured terms, VOQL* basic formulas, VOQL* formulas, and VOQL* query expressions are hierarchically and inductively constructed as those of OOPC. Most important, we formally define the semantics of each language construct of VOQL* by induction using OOPC. Because of the well-defined syntax and semantics, queries in VOQL* are clear, concise, and intuitive. We also provide an effective procedure to translate queries in VOQL* into those in OOPC. We believe that VOQL* is the first visual query language with the well-defined syntax reflecting the syntactic structure of logic and semantics formally defined by induction.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.28 no.5
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• pp.277-283
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• 2016

Tetrapod is one of the most widely used concrete armor units for rubble mound breakwaters. The calculation of the stability number of Tetrapods is necessary to determine the optimal weight of Tetrapods. Many empirical formulas have been developed to calculate the stability number of Tetrapods, from the Hudson formula in 1950s to the recent one developed by Suh and Kang. They were developed by using the regression analysis to determine the coefficients of an assumed formula using the experimental data. Recently, software engineering (or machine learning) methods are introduced as a large amount of experimental data becomes available, e.g. artificial neural network (ANN) models for rock armors. However, these methods are seldom used probably because they did not significantly improve the accuracy compared with the empirical formula and/or the engineers are not familiar with them. In this study, we propose an explicit method to calculate the stability number of Tetrapods using the weights and biases of an ANN model. This method can be used by an engineer who has basic knowledge of matrix operation without requiring knowledge of ANN, and it is more accurate than previous empirical formulas.