• Journal for History of Mathematics
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• v.18 no.3
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• pp.67-78
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• 2005

In this paper we deals with Leibniz's definition of infinite and infinitely small quantities, infinite quantities and theory of quantified indivisibles in comparison with Galileo's concept of indivisibles. Leibniz developed 'method of indivisible' in order to introduce the integrability of continuous functions. also we deals with this demonstration, with Leibniz's rules of arithmetic of infinitely small and infinite quantities.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2002.04a
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• pp.369-376
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• 2002

The stochastic analysis of semi-infinite domain is presented using the weighted integral method, which is improved to include the higher order terms in expanding the displacement vector. To improve the weighted integral method, the Lagrangian remainder is taken into account in the expansion of the status variable with respect to the mean value of the random variables. In the resulting formulae only the 'proportionality coefficients' are introduced in the resulting equation, therefore no additional computation time and memory requirement is needed. The equations are applied in analyzing the semi-infinite domain. The results obtained by the improved weighted integral method are reasonable and are in good agreement with those of the Monte Carlo simulation. To model the semi-infinite domain, the Bettess's infinite element is adopted, where the theoretical decomposition of the strain-displacement matrix to calculate the deviatoric stiffness of the semi-infinite domains is introduced. The calculated value of mean and the covariance of the displacement are revealed to be larger than those given by the finite domain assumptions which is thought to be rational and should be considered in the design of structures on semi-infinite domains.

• Geomechanics and Engineering
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• v.33 no.4
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• pp.369-380
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• 2023

The aim of this paper is to investigate stress analysis of semi-infinite soils consisting of two layers with twin rectangular tunnels under static loads. The region close to the ground surface and tunnel modelled within finite elements. In order to use a more realistic model, the far region is modelled within infinite elements. The material model of the layered soil is considered as elastic and isotropic. In the finite element solution of the problem, two dimensional (2D) plane solid elements are used with sixteen-nodes rectangular finite and eight-nodes infinite shapes. Finite and infinite elements are ordered to be suitable for the tunnel and the soils. The governing equations of the problem are obtained by using the virtual work principle. In the numerical process, the five-point Gauss rule is used for the calculation of the integrations. In order to validate using methods, comparison studies are performed. In the numerical results, the stress distributions of the two layered soils containing twin rectangular tunnels presented. In the presented results, effects of the location of the tunnels on the stress distributions along soil depth are obtained and discussed in detail. The obtained results show that the locations of the tunnels are very effective on the stress distribution on the soils.

• East Asian mathematical journal
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• v.17 no.1
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• pp.135-146
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• 2001

Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

• Kyungpook Mathematical Journal
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• v.60 no.4
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• pp.797-803
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• 2020

The Brück conjecture is still open for an entire function f with hyper order of no less than 1/2, which is not an integer. In this paper, it is proved that the hyper order of solutions of a linear complex differential equation that is related to the Brüuck Conjecture is infinite. The results show that the conjecture holds in a special case when the hyper order of f is 1/2.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.18 no.10
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• pp.1454-1460
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• 1993

The scattering pattern, due to an E-Polarized wave incident on M circular parallel dielectric cylinders, is computed. The multiply-scattered fields between the cylinders are considered. Modeling of infinite cylindrical scatterer of arbitrary cross sections by a number of circular cylinders is executed. By enforcing the boundary conditions on the surface of each cylinder, an infinite set of equations is obtained. The first order of scattering results from the excitation of each cylinder by only the incident wave. The second order results from the excitation of each cylinder by the first order of scattering from the remaining cylinders, and so no to an infinite order of scattering. Although the resulting equation is of infinite size, proper truncation yields very accurate numerical results.

• The Journal of the Acoustical Society of Korea
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• v.31 no.4
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• pp.214-224
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• 2012

The objective of this paper is to design multichannel spherical loudspeaker array by considering various positioning methods such as Gaussian grid, Lebedev grid and packing method. For the spatial sound manipulation, which is to make desired sound field by controling multiple sound sources, the Kirchhoff- Helmholtz integral states that sound fields can be reproduced in terms of infinite control sources on the integral surface. But since we cannot control infinite number of sources for the implementation, we have to allocate finite number of sound sources which can approximately act as infinite number of sources. To manipulate sound field inside of a sphere (which is typical example of three dimensional array) by controlling sound sources on the surface, three methods of allocating sound sources, which are Gaussian grid, Lebedev grid and packing method, are reviewed. For each geometry, the performances of manipulation rendered by time-reversal operator and higher-order ambisonics are compared.

• Journal of the Korean Data and Information Science Society
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• v.24 no.3
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• pp.637-642
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• 2013

In this paper, we consider the maximum entropy test in in nite order autoregressiv models. Its asymptotic distribution is derived under the null hypothesis. A bootstrap version of the test is discussed and its performance is evaluated through Monte Carlo simulations.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1495-1506
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• 2021

The growth of solutions of second order complex differential equations f" + A(z)f' + B(z)f = 0 with transcendental entire coefficients is considered. Assuming that A(z) has a finite deficient value and that B(z) has either Fabry gaps or a multiply connected Fatou component, it follows that all solutions are of infinite order of growth.

• Journal of the Korean Statistical Society
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• v.29 no.3
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• pp.351-360
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• 2000

This paper considers the problem of testing a variance change in infinite order autoregressive models. A cusum of squares test based on the residuals from an AR(q) model is constructed analogous to Inclan and Tiao (1994)'s test statistic, where q is a sequence of positive integers diverging to $\infty$. It is shown that under regularity conditions the limiting distribution of the test statistic is the sup of a standard Brownian bridge. Simulation results are given to illustrate the performance of the test.