• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.853-868
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• 2015

Let G be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible G-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e., closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e., each lattice point appears precisely once in the formula, with coefficient +1. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.503-517
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• 2016

For positive integers d and n, let $[n]^d$ be the set of all vectors ($a_1,a_2,{\cdots},a_d$), where ai is an integer with $0{\leq}a_i{\leq}n-1$. A subset S of $[n]^d$ is called a Sidon set if all sums of two (not necessarily distinct) vectors in S are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $[n]^d$. First, let $\mathcal{Z}_{n,d}$ be the number of all Sidon sets in $[n]^d$. We show that ${\log}(\mathcal{Z}_{n,d})={\Theta}(n^{d/2})$, where the constants of ${\Theta}$ depend only on d. Next, we estimate the maximum size of Sidon sets contained in a random set $[n]^d_p$, where $[n]^d_p$ denotes a random set obtained from $[n]^d$ by choosing each element independently with probability p.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.677-693
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• 2018

In this work, we generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. The study of the local convergence based on Lipschitz constants is important because it provides the degree of difficulty for choosing initial points. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to verify the theoretical results and to show the convergence behavior.

• Proceedings of the Korean Geotechical Society Conference
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• 2004.03b
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• pp.442-449
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• 2004

This paper describes a new concept of finite element analysis, which is based on neural network based material models (NNCMs) without invoking any pre-chosen mathematical framework. NNCMs have several advantages over conventional constitutive models (CCMs) and once plugged in a finite element (FE) engine, can be used for FE analysis in a manner similar to CCMs. The paper demonstrates a FE framework in which NNCMs are incorporated and also proposes a strategy for data enhancement by invoking the assumption of isotropy of the material. It is shown through some illustrative examples that this provides a better training environment for a generalized NNCM in which stress and strain components are used as effects and cause. Form this study, it appears that there is a prima facia case for developing NNCMs for materials for which mathematical theories become too complex and a large number of material parameters and constants have to be identified or determined.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.12 no.2
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• pp.265-274
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• 1992

A mathematical model was proposed to predict the fate of a priority organic pollutant, anthracene, accidently spilled into a stream. The model consists of 6 differential equations with 5 input variables and 9 rate constants. Volatilization, biodegradation, adsorption/desorption, photodegradation as well as the convective inputs and outputs are included in the model. As a result of a series of dynamic simulations and sensitivity analyses under the given conditions, the concentrations of the organic chemical could be predicted within a detection limit in the stream. It was also suggested that the rate constant for diffusion/transport and adsorption rate constant are the most influential ones for predicting the chemical conentrations in dissolved and particulate phase. The model proposed appears to be a useful tool for assessing chemical spills.

• Journal of the Korean Mathematical Society
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• v.8 no.1
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• pp.25-30
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• 1971

Put ($$^*$$) $$G[x,y]={\sum}\limits^{p+q=n}_{p,q=0}[-n]_{p+q}c_{p,q}x^py^q$$, where $[{\lambda}]_m$ is the Pocbhammer symbol and the $c_{p,q}$ are arbitrary constants. Making use of the specialized forms of some of his earlier results (see [8] and [9] the author derives here bilateral generating functions of the type ($$^{**}$$) $${\sum}\limits^{\infty}_{n=0}{\frac{[\lambda]_n}{n!}}_2F_1[\array{{\rho}-n,\;{\alpha};\\{\lambda}+{\rho};}x]\;G[y,z]t^n$$ where ${\alpha}$, ${\rho}$ and ${\lambda}$ are arbitrary complex numbers. In particular, it is shown that when G[y, z] is a double hypergeometric polynomial, the right-band member of ($^{**}$) belongs to a class of general triple hypergeometric functions introduced by the author [7]. An interesting special case of ($^{**}$) when ${\rho}=-m,\;m$ being a nonnegative integer, yields a class of bilateral generating functions for the Jacobi polynomials $\{P_n{^{{\alpha},{\beta}}}(x)\}$ in the form ($$^{***}$$) $${\sum\limits^{\infty}_{n=0}}\(\array{m+n\\n}\)P{^{({\alpha}-n,{\beta}-n)}_{m+n}(x)\;G[y,z]{\frac{t^n}{n!}}$$, which provides a unification of several known results. Further extensions of ($^{**}$) and ($^{***}$) with G[y, z] replaced by an analogous multiple sum $H\[y_1,{\cdots},y_m\]$ are also discussed.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.117-126
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• 2003

Let {$X_{nk}$\mid$1\;{\leq}\;k\;{\leq}\;n,\;n\;{\geq}\;1$} be an array of row negatively associated (NA) random variables which satisfy $P($\mid$X_{nk}$\mid$\;>\;x)\;{\leq}\;P($\mid$X$\mid$\;>\;x)$. For weighed sums ${{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}$ indexed by random variables {$T_n$\mid$n\;{\geq}$1$}, we establish a general weak law of large numbers (WLLN) of the form $({{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}\;-\;v_{[nk]})\;/b_{[an]}$ under some suitable conditions, where $\{a_n$\mid$n\;\geq\;1\},\; \{b_n$\mid$n\;\geq\;1\}$ are sequences of constants with $a_n\;>\;0,\;0\;<\;b_n\;\rightarrow \;\infty,\;n\;{\geq}\;1$, and {$v_{an}$\mid$n\;{\geq}\;1$} is an array of random variables, and the symbol [x] denotes the greatest integer in x.

• Composites Research
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• v.16 no.6
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• pp.10-15
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• 2003

In RTM process, the content of microvoids can be critical due to the fact that the presence of microvoids degrades mechanical properties on the fabricated composite parts. The present paper proposes an experimental method of observation in void formation and transport. VARTM processes are performed under observation with a digital video camera and then the microvoid formation in the flow front and transport are videotaped and observed both in channels and tows. The obtained data are used in the mathematical model in order to determine the model constants. Experimental results and expected results from the mathematical model show a good agreement with each other.

• KEPCO Journal on Electric Power and Energy
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• v.2 no.1
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• pp.89-96
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• 2016

The mechanism for laminar dust flame propagation can only be elucidated from a comprehensive mathematical model which incorporates conduction and radiation, as well as the chemical kinetics of particle devolatilization and gas phase and char reaction. The mathematical model for a flat, laminar, premixed coal-air flame is applied to the atmospheric coal-air mixtures studied by Smoot and co-workers, and comparisons are made with their measurements and predictions. Here the principal parameter for comparison is the laminar burning velocity. The studies of Smoot and co-workers are first reviewed and compared with those predicted by the present model. The effects of inlet temperature and devolatilization rate constants on the burning velocities are studied with the present model, and compared with their measurements and predictions. Their measured burning velocities are approximately predicted with the present model at relatively high coal concentrations, with a somewhat increased inlet temperature. From the comparisons, their model might over-estimate particle temperature and rates of devolatilization. This would enable coal-air mixtures to be burned without any form of preheat and would tend to increase their computed values of burning velocity.

• The Transactions of the Korean Institute of Power Electronics
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• v.19 no.1
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• pp.51-56
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• 2014

PV cell modeling is necessary both for software and hardware simulators in analyzing and testing the performance of PV generation systems. Unique I-V curve of a PV cell identifies its own characteristics by electrical equivalent model that is composed of diode constants ($I_o$, $v_t$), photo-generated current ($I_{ph}$), series resistance ($R_s$), and shunt resistance ($R_{sh}$). Photo-generated current can be easily estimated since it is proportional to irradiation level. However, other electrical parameters should be solved from the manufacturer's data sheet that is consisted with three remarkable operating points such as open circuit voltage ($V_{oc}$), short circuit current ($I_{sc}$), and maximum power voltage/current ($V_{MPP}/I_{MPP}$). This paper explains and analyzes mathematical process of a novel PV cell modeling algorithm that was proposed by the authors with the name of "K-algorithm".