• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.267-271
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• 2011

Let 0 < s < 1. For $f^{\ast}$ representing the symmetric radial decreasing rearrangement of f, we build up a fractional version of Polya-$Szeg{\ddot{o}}$ inequality: $${\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f^{\ast}(x){\mid}^2dx{\leq}{\int}_{\mathbb{R}^n}{\mid}(-\Delta)^{s/2}f(x){\mid}^2dx$$.

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.149-158
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• 2024

A least squares problem without correspondences is expressed as the following optimization: Π∈P^min_{m, x∈ℝn} ║Ax-Πy║, where A ∈ ℝ^m×n and y ∈ ℝ^m are given. In general, solving such an optimization problem is highly challenging. In this paper we use the rearrangement inequalities to find the closed form of solutions for certain cases. Moreover, despite the stringent constraints, we successfully tackle the nonlinear least squares problem without correspondences by leveraging rearrangement inequalities.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.929-941
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• 2000

We study a parabolic system of chemotaxis introduced by E.F. Keler and L.A. Segel. First, norm behaviors of the blow-up solution are proven. Then some kind of symmetry breaking and the concentration toward the boundary follow when the L$^1$norm of the initial value is less than 8$\pi$. Meanwhile a method of rearrangement is porposed toprove an inequality of Trudinger-Moser's type.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.7A
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• pp.646-653
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• 2010

This paper provides a unified framework for the joint optimal subchannel and power allocation in multi-hop relay network, where each node in the network has multiple parallel subchannels such as in OFDM or MIMO system. When there are multiple parallel subchannels between nodes, the relay node decides how to match the subchannel at the first hop with the one at the second hop aside from determining the power allocation. Joint optimal design of subchannel matching and power allocation is, in general, known to be very difficult to solve due to the combinatorial nature involved in subchannel matching. Despite this difficulty, we use a simple rearrangement inequality and show that seemingly difficult problems can be efficiently solved. This includes several existing solution methods as special cases. We also provide various design examples to show the effectiveness of the proposed framework.