• Journal for History of Mathematics
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• v.37 no.4
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• pp.93-105
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• 2024

Research on the caustic, the envelope of light rays, began with the geometric optics studies of Huygens and others in the 17th century. One of the most important problems in optics in the 17th century was focusing the sun's rays. This was a problem that had to be solved in order to manufacture various practical optical instruments at the time. Beginning with research on the caustic during this period, the concept of envelope became generalized and expanded to various fields until the 19th century. This paper examines the mathematical history involved in the study of envelope curves. We compare several methods of defining the envelope and provide an example of calculating the envelope accordingly.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.409-420
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• 1998

In this paper as a univesal Banach space of the separable Banach spaces we investigate the complemented Banach subspaces of $_wL_{\hat {I}}$. Also, using Peck's theorem and the properties of the envelope norm of $_wL_{\hat {I}}$ we will find a canonical basis of $l_1^n, l_\infty^n$ for each n.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1305-1314
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• 2010

In this paper, we deal with a class of one-dimensional reflected backward stochastic differential equations driven by a Brownian motion and the martingales of Teugels associated with an independent L$\acute{e}$vy process having a stochastic Lipschitz coefficient. We derive the existence and uniqueness of solutions for these equations via Snell envelope and the fixed point theorem.

• 한국데이터정보과학회:학술대회논문집
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• 2002.06a
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• pp.51-73
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• 2002

Empirical likelihood ratio method is a new technique in nonparametric inference developed by A. Owen (1988, 2001). Sometimes empirical likelihood has difficulties to define itself. As such a case in point, we discuss the way to define a modified empirical likelihood for the location of symmetry using well-known points of symmetry as a side conditions. The side condition of symmetry is defined through a finite subset of the infinite set of constraints. The modified empirical likelihood under symmetry studied in this paper is to construct a constrained parameter space $\theta+$ of distributions imposing known symmetry as side information. We show that the usual asymptotic theory (Wilks theorem) still hold for the empirical likelihood ratio on the constrained parameter space and the asymptotic distribution of the empirical NPMLE of difference of two symmetric points is obtained.