• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.625-633
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• 1995

Classical Fatous' theorem asserts that the non-tangential limit of the Poisson integral of an integrals function on $[\pi, \pi]$ exists a.e.

• The Pure and Applied Mathematics
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• v.4 no.1
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• pp.87-92
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• 1997

Some function of a complete finite measure space (for short, CFMS) into the duals and pre-duals of weakly compactly generated (for short, WCG) spaces are considered. We shall show that a universally weakly measurable function f of a CFMS into the dual of a WCG space has RS property and bounded weakly measurable functions of a CFMS into the pre-duals of WCG spaces are always Pettis integrable.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.1035-1058
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• 2021

Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.20 no.8
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• pp.2524-2539
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• 1996

A design process and an axisymmetric boundary integral equation method for precise structural analysis of the aircraft gas turbine rotor disk were developed. This axisymmetric boundary integral equation method for stress and steady-state thermal analysis was improved in solution accuracy by appling an implicit technique for Cauchy principal value evaluation, a subelement technique for weak singular integral evaluation and a double exponentical integral technoque for internal point solution near boundary surfaces. Stresses, temperatures, low cycle fatigue lifes and critical speeds for the turbine rotor disk of the thrust 1421 N class turbojet engine were analysed in a pratical calculation model problem.

• Interaction and multiscale mechanics
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• v.1 no.3
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• pp.329-337
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• 2008

This paper presents the derivation of the generalized governing equations in theory of elasticity, their weak forms and the some applications in the numerical analysis of structural mechanics. Unlike the differential equations in classical elasticity theory, the generalized equations of the equilibrium and compatibility equations presented here take the form of integral equations, and the generalized equilibrium equations contain the classical differential equations and the boundary conditions in a single equation. By using appropriate test functions, the weak forms of these generalized governing equations can be established. It can be shown that various variational principles in structural analysis are merely the special cases of these weak forms of generalized governing equations in elasticity. The present weak forms of elasticity equations extend greatly the choices of the trial functions for approximate solutions in the numerical analysis of various engineering problems. Therefore, the weak forms of generalized governing equations in elasticity provide a powerful modeling tool in the computational structural mechanics.

• Journal for History of Mathematics
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• v.21 no.3
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• pp.67-96
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• 2008

In the 19th century Fourier and Dirichlet studied the expansion of "arbitrary" functions into the trigonometric series and this led to the development of the Riemann's definition of the integral. Riemann's integral was considered to be of the highest generality and was discussed intensively. As a result, some weak points were found but, at least at the beginning, these were not considered as the criticism of the Riemann's integral. But after Jordan introduced the theory of content and measure-theoretic approach to the concept of the integral, and after Borel developed the Jordan's theory of content to a theory of measure, Lebesgue joined these two concepts together and obtained a new theory of integral, now known as the "Lebesgue integral".

• International Journal of Ocean Engineering and Technology Speciallssue:Selected Papers
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• v.3 no.1
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• pp.14-22
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• 2000

The improved Green integral equation for the calculation of time-harmonic potentials in the radiation diffraction problem about a freely floating body in the presence of moderate or weak current is presented. The forward-speed Green function presented by Brard is used. The correct free surface boundary conditions on the physical free surface are employed as well as an appropriate boundary conditions on the non-physical inner free surface. The default in the existing Green integral equation as well as in the source integral equation is discussed in detail.

• Structural Engineering and Mechanics
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• v.55 no.3
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• pp.655-674
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• 2015

In this paper, a new and simplified method is presented in which the natural frequencies of the uniform and non-uniform beams are calculated through simple mathematical relationships. The various vibration problems such as: Rayleigh beam under variable axial force, axial vibration of a bar with and without end discrete spring, torsional vibration of a bar with an attached mass moment of inertia, flexural vibration of the beam with laterally distributed elastic springs and also flexural vibration of the beam with effects of viscose damping are investigated. The governing differential equations are first obtained and then; according to a harmonic vibration, are converted into single variable equations in terms of location. Through repetitive integrations, the governing equations are converted into weak form integral equations. The mode shape functions of the vibration are approximated using a power series. Substitution of the power series into the integral equations results in a system of linear algebraic equations. The natural frequencies are determined by calculation of a non-trivial solution for system of equations. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained with those obtained from other published references and results of available finite element software.

• Structural Engineering and Mechanics
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• v.43 no.5
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• pp.561-582
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• 2012

Despite popularity of FEM in analysis of static and dynamic structural problems and the routine applicability of FE softwares, analytical methods based on simple mathematical relations is still largely sought by many researchers and practicing engineers around the world. Development of such analytical methods for analysis of free vibration of non-prismatic beams is also of primary concern. In this paper a new and simple method is proposed for determination of vibration frequencies of non-prismatic beams under variable axial forces. The governing differential equation is first obtained and, according to a harmonic vibration, is converted into a single variable equation in terms of location. Through repetitive integrations, integral equation for the weak form of governing equation is derived. The integration constants are determined using the boundary conditions applied to the problem. The mode shape functions are approximated by a power series. Substitution of the power series into the integral equation transforms it into a system of linear algebraic equations. Natural frequencies are determined using a non-trivial solution for system of equations. Presented method is formulated for beams having various end conditions and is extended for determination of the buckling load of non-prismatic beams. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained to those obtained using available finite element software.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.183-190
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• 2012

The main purpose of this paper is to introduce a new type of $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$): $\tilde{E}^{({\alpha},{\omega})}_{n,q}$ and $\tilde{E}^{({\alpha},{\omega})}_{n,q}(x)$, respectively. By using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_p$, we can obtain some results and derive some recurrence identities for the $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$).