Proof ��(i)It is obvious that ||��|| �� 0 for all �� and ||��|| = 0 if �� = 0. Conversely, if �̡�0, from Lemma 5, it follows that the midpoint function ��M �� 0M. selleck kinase inhibitor Thus we have||?��?||=inf?�͡�?��?||��||=inf?�͡�?��?(sup?0��a��1max?��L(a))��inf?�͡�?��?(sup?0��a��1|��L(a)+��R(a)|2)=sup?0��a��1|��M(a)|>0.(23)(ii)For all ��/ and b , we have||b?��?||=||?b��?||=inf?�͡�?b��?||��||=inf?b�͡��?b��?||b�͡�||=inf?�͡��?��?||b�͡�||=|b|||?��?||.(24)(iii)For all ��, ��/, we have that||?��?+?��?||=||?��+��?||=inf?�ء�?��+��?||��||��inf?�̡��?��?,�͡��?��?||��+��||��inf?�̡��?��?,�͡��?��?(||��||+||��||)��inf?�̡��?��?||��||+inf?�͡��?��?||��||=||?��?||+||?��?||.(25)We conclude that ||?|| is a norm on /.Now we show that the function �� : / �� / �� induced by ||?|| as ��(��, ��) = ||�� ? ��|| is exactly a metric on /.
Theorem 10 ��The function �� is a metric on /.Proof ��(i)By the definition of ��, it is obvious that ��(��, ��) �� 0, for any ��, ��/. If ��(��, ��) = 0, then ||�� ? ��|| = 0. Thus by Theorem 9, we get �� = ��. In addition, by Lemma 2 we have ��(��, ��) = ||�� ? ��|| = 0.(ii)For all ��, ��/, we have that��(?��?,?��?)=||?��?��?||=||(?1)?��?��?||=||��?��||=��(?��?,?��?).(26)(iii)For any ��, ��, ��/, we have��(?��?,?��?)=||?��?��?||=||?��?��+��?��?||?��||?��?��?||+||?��?��?||=��(?��?,?��?)+��(?��?,?��?).(27)We conclude that �� is a metric on /.4. ConclusionsIn this present investigation, we studied the norm induced by the supremum metric d�� on the space of fuzzy numbers. And then we proposed a method for constructing a norm on the quotient space of fuzzy numbers.
This norm is very natural and works well with the induced metric on /. The works in this paper enable us to study the fuzzy numbers in the new environment. We hope that our results in this paper may lead to significant, new, and innovative results in other related fields.AcknowledgmentsThis work was supported by the Mathematical Tianyuan Foundation of China (Grant no. 11126087), the National Natural Science Foundation of China (Grant no. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).
The synthesis and study of nanoscale materials have attracted much attention in recent years. One-dimensional nanostructures, including nanowires, nanorods, and nanotubes, have many amazing properties such as high density, high aspect ratio, and low threshold voltage in field emission.
On the other hand, the application of the giant magnetoresistance (GMR) effect found in 2-D metallic multilayers [1] has also been rigorously investigated for applications in the magnetic industry such as information storage and magnetic Entinostat sensors [2, 3]. The development of high-density perpendicular magnetic recording encourages the trend to investigate new types of magnetic structures as the medium.