• Journal of the Chungcheong Mathematical Society
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• v.36 no.2
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• pp.125-133
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• 2023

We establish a characterization theorem of elliptic paraboloids in the (n+1)-dimensional Euclidean space 𝔼ⁿ⁺¹ with extrinsic properties such as the (n+1)-dimensional volumes of regions enclosed by the hyperplanes and hypersurfaces, and the n-dimensional areas of projections of the sections of hypersurfaces cut off by hyperplanes.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.259-272
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• 2017

In this paper we study the mapping properties of the finite field restricted averaging operators to various algebraic varieties. We derive necessary conditions for the boundedness of the generalized restricted averaging operator related to arbitrary algebraic varieties. It is shown that the necessary conditions are in fact sufficient in the specific case when the Fourier transform on varieties has enough decay estimates. Our work extends the known optimal result on regular varieties such as paraboloids and spheres to certain lower dimensional varieties.

• Journal of Korean Association for Spatial Structures
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• v.4 no.4 s.14
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• pp.113-128
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• 2004

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloidal and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components $u_r,\;u_{\theta},\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in ${\theta}$, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.853-867
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• 2013

In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space $\mathbb{F}^d_q$ over a finite field $\mathbb{F}_q$ with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and ${\mid}E{\parallel}F{\mid}{\geq}Cq^d$ for some large C > 1, then ${\mid}{\Pi}(E,F){\mid}{\geq}cq$ for some 0 < c < 1. In particular, we find a connection between the size of the dot product set and the number of lines through both the origin and a nonzero point in the given set E. As an application of this observation, we obtain more sharpened results on the generalized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.

• Journal of Korean Association for Spatial Structures
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• v.2 no.3 s.5
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• pp.81-92
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• 2002

The objective of this paper is to help to make decision of the appropriate structural types in long span structured building due to range of span. For the intention, based on 7 forces of structural element, it is analized the relationships among 6 configurations of structural element(d/1), 25 structural types, 4 materials, and span-length known with 186 sample from 1850 to 1996. 1) bending forces: $club(1/100{\sim}1/10),\;plate(1/100{\sim}1/10),\;rahmen(steel,\;10{\sim}24m)\;simple\;beam(PC,\;10{\sim}35m)$ 2) shearing forces: $shell(1/100{\sim}1/1000)\;hyperbolic\;paraboloids(RC,25{\sim}97m)$ 3) shearing+bending forces: plate, folded $plate(RC21{\sim}59m)$ 4) compression axial forces: club, $arch(RC,\;32{\sim}65m)$ 5) compression+tension forces: shell, braced dome $shell(RC,\;40{\sim}201m),\;vault\;shell(RC,\;16{\sim}103m)$ 6) compression+tension axial forces: $rod(1/1000{\sim}1/100)$, cable(below 1/1000)+rod, coble+rod+membrane(below 1/1000), planar $truss(steel,\;31{\sim}134m),\;arch\;truss(31{\sim}135m),\;horizontal\;spaceframe(29{\sim}10\;8m),\;portal\;frame(39{\sim}55m),\;domical\;space\;truss(44{\sim}222m),\;framed\;\;membrane(45{\sim}110m),\;hybrid\;\;membrane\;(42{\sim}256m)$ 7) tension forces: cable, membrane, $suspension(60{\sim}150m),\;cable\;\;beam(40{\sim}130m),\;tensile\;membrane(42{\sim}136m),\;cable\;-slayed(25{\sim}90m),\;suspension\;membrane(24{\sim}97m),\;single\;layer\;pneumatic\;structure(45{\sim}231m),\;double\;layer\;pneumatic\;structures(30{\sim}44m)$