• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1631-1637
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• 2013

This paper generalizes the Aleksandrov problem and Mazur Ulam theorem to the case of $n$-normed spaces. For real $n$-normed spaces X and Y, we will prove that $f$ is an affine isometry when the mapping satisfies the weaker assumptions that preserves unit distance, $n$-colinear and 2-colinear on same-order.

• Bulletin of the Korean Chemical Society
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• v.14 no.1
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• pp.123-127
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• 1993

New lipophilic Crown-4 compounds of 16-membered rings containing furan (neutral carrier,I), tetrahydrofuran (neutral carrier,II) and lithium complex of the latter (neutral carrier,III) have been synthesized and tested as the active sensors for lithium ion in poly(vinyl chloride) (PVC) membrane electrode, in the presence and absence of an anion excluder, tetrakis(4-chloro-phenyl)borate (KTClPB), 2-nitrophenyl phenyl ether (NPPE), tris(2-ethylhexyl)phosphate (TEHP), o-nitrophenyl octyl ether (NPOE), dioctyl adipate (DOA), bis(2-ethylhexyl)adipate (BEHA), di-n-octylphenyl phosphonate (DOPP) were used as plasticizing solvent mediators. The electrode response function had a nearly Nernstian slope of 54-61 mV per decade (25$^{\circ}$C) within the concentration range of $10^{-1}-10^{-4}$ M LiCl and the detection limits for all electrodes were ca. $5{\times}10^{-4}$ M. The response time of the electrode was faster at the higher lithium concentration and the response of the electrode was stable for longer than 6 months. The sensor membranes exhibit improved response times and increased lifetimes as compared to the system described earlier.