• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.477-492
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• 1994

For the eigenvalue problem of $Au = \lambda u$ where A is considered as a semi-bounded self-adjoint operator on a Hilbert space, we are used to apply two complentary methods finding upper bounds and lower bounds to the eigenvalues. The most popular method for finding upper bounds may be the Rayleigh-Ritz method which was developed in the 19th century while a method for computing lower bounds may be the method of intermediate eigenvalue problems which has been developed since 1950's. In the method of intermediate eigenvalue problems (IEP), we consider the original operator eigenvalue problem as a perturbation of a simpler, resolvable, self-adjoint eigenvalue problem, called a base problem, that gives rough lower bounds.

• Journal of the Korean Operations Research and Management Science Society
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• v.27 no.3
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• pp.145-156
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• 2002

Business managers are often required to use LP problems to deal with uncertainty inherent in decision making due to rapid changes in today＇s business environments. Uncertain parameters can be easily formulated in the two-stage stochastic LP problems. However, since solution methods are complex and time-consuming, a common approach has been to use modified formulations to provide upper and lower bounds on the two-stage stochastic LP problem. One approach is to use an expected value problem, which provides upper and lower bounds. Another approach is to use “walt-and-see” problem to provide upper and lower bounds. The objective of this paper is to propose a modified approach of “wait-and-see” problem to provide an upper bound and to compare the relative error of optimal value with various upper and lower bounds. A computing experiment is implemented to show the relative error of optimal value with various upper and lower bounds and computing times.

• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.111-121
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• 1992

A new method(EVF method) is applied to get lower bounds to frequencies of rotating uniform beams which are clamped or simply supported at one end and free at the other. For the upper bounds, the Rayleigh-Ritz method is employed. Numerical results are presented.

• The Journal of Korea Institute of Information, Electronics, and Communication Technology
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• v.13 no.5
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• pp.432-435
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• 2020

The elasticity tensor for water may be employed to model the fully saturated porous material. Mostly water is assumed to be incompressible with a bulk modulus, however, the upper and lower bounds of off-diagonal components of the elasticity tensor of porous materials filled with water are violated when the bulk modulus is relatively high. In many cases, the generalized Hill inequality describes the general bounds of Voigt and Reuss for eigenvalues, but the bounds for the component of elasticity tensor are more realistic because the principal axis of eigenvalues of two phases, matrix and water, are not coincident. Thus in this paper, for anisotropic material containing pores filled with water, the bounds for the component of elasticity tensor are expressed by the rule of mixture and the upper and lower bounds of fully saturated porous materials are violated for low porosity and high bulk modulus of water.

• Journal of Communications and Networks
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• v.18 no.2
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• pp.182-189
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• 2016

The focus in this paper is on obtaining tight, simple algebraic-form bounds and invertible expressions for the average symbol error probability (ASEP) of M-ary phase shift keying (MPSK) in a class of composite fading channels. We employ the mixture gamma (MG) distribution to approximate the signal-to-noise ratio (SNR) distributions of fading models, which include Nakagami-m, Generalized-K ($K_G$), and Nakagami-lognormal fading as specific examples. Our approach involves using the tight upper and lower bounds that we recently derived on the Gaussian Q-function, which can easily be averaged over the general MG distribution. First, algebraic-form upper bounds are derived on the ASEP of MPSK for M > 2, based on the union upper bound on the symbol error probability (SEP) of MPSK in additive white Gaussian noise (AWGN) given by a single Gaussian Q-function. By comparison with the exact ASEP results obtained by numerical integration, we show that these upper bounds are extremely tight for all SNR values of practical interest. These bounds can be employed as accurate approximations that are invertible for high SNR. For the special case of binary phase shift keying (BPSK) (M = 2), where the exact SEP in the AWGN channel is given as one Gaussian Q-function, upper and lower bounds on the exact ASEP are obtained. The bounds can be made arbitrarily tight by adjusting the parameters in our Gaussian bounds. The average of the upper and lower bounds gives a very accurate approximation of the exact ASEP. Moreover, the arbitrarily accurate approximations for all three of the fading models we consider become invertible for reasonably high SNR.

• Communications of the Korean Mathematical Society
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• v.12 no.2
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• pp.393-401
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• 1997

Let $A_1,A_2,\cdots,A_m$ and $B_1,B_2,\cdots,B_n$ be two sequences of events on a given probability space. Let $X_m$ and $Y_n$, respectively, be the number of those $A_i$ and $B_j$, which occur we establish new upper and lower bounds on the probability $P(X=r, Y=t)$ which improve upper bounds and classical lower bounds in terms of the bivariate binomial moment $S_{r,t},S_{r+1,t},S_{r,t+1}$ and $S_{r+1,t+1}$.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.461-469
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• 1995

Some bounds for anisotropic torsional rigidity with one plane of elastic symmetry perpendicular to the axis of the beam are derived by making use of the isoperimetric inequalities, complementary variational principles, and the maximum principle. Upper and lower bounds are obtained by applying the isoperimetric inequalities. While the upper bound investigated by the variational principles and maximum principle. The analysis is patterned after the work of Payne and Weinbeger [J. Math. Anal. Appl. 2(1961). pp. 210-216].

• Journal of Korean Society of Industrial and Systems Engineering
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• v.34 no.1
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• pp.67-73
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• 2011

Using the known result of the expected busy period for a controllable M/G/1 queueing model operating under the triadic Max (N, T, D) policy, its upper and lower bounds are derived to approximate its corresponding actual value. Both bounds are represented in terms of the expected busy periods for the dyadic Min (N, T), Min (N, D) and Min (T, D) and simple N, T and D operating policies. All three input variables N, T and D are equally contributed to construct such bounds for better estimation.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.33 no.2
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• pp.97-104
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• 2010

Using the known result of the expected busy period for the triadic Min (N, T, D) operating policy applied to a controllable M/G/1 queueing model, its upper and lower bounds are derived to approximate its corresponding actual value. Both bounds are represented in terms of the expected busy periods for the dyadic Min (N, T), Min (N, D) and Min (T, D) and simple N, T and D operating policies. All three input variables N, T and D are equally contributed to construct such bounds for better approximations.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.36 no.1
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• pp.58-63
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• 2013

Using the known result of the expected busy period for the triadic Med (N, T, D) operating policies applied to a controllable M/G/1 queueing model, its upper and lower bounds are derived to approximate its corresponding actual values. Both bounds are represented in terms of the expected busy periods for the dyadic Min (N, T), Min (N, D) and Min (T, D) or Max (N, T), Max (N, D) and Max (T, D) with the simple N, T and D operating policies without using any other types of triadic operating policies such as Min (N, T, D) and Max (N, T, D) policies. All three input variables N, T and D are equally contributed to construct such bounds for estimation of the expected busy period.