zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Inequalities and mathematical properties of uncertain variables. (English) Zbl 1254.28020
Summary: Uncertain variables are measurable functions from uncertainty spaces to the set of real numbers. In this paper, some important inequalities of uncertain variables, for example, extension of Jensen’s inequality, Lyapunov’s inequality, and refined Markov inequalities are presented. In addition, some mathematical properties of uncertain variables are also given and proven.

28E10Fuzzy measure theory
60E15Inequalities in probability theory; stochastic orderings
Full Text: DOI
[1] Chen X., Liu B. (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making 9(1): 69--81 · Zbl 1196.34005 · doi:10.1007/s10700-010-9073-2
[2] Gao X. (2009) Some properties of continuous uncertain measure. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 17(3): 419--426 · Zbl 1180.28011 · doi:10.1142/S0218488509005954
[3] Gao X., Gao Y., Ralescu D. (2010) On Liu’s inference rule for uncertain systems. International Journal of Uncertainty. Fuzziness and Knowledge-Based Systems 18(1): 1--11 · Zbl 1207.68386 · doi:10.1142/S0218488510006349
[4] Li X., Liu B. (2006) A suficient and necessary condition for credibility measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 14(5): 527--535 · Zbl 1113.28014 · doi:10.1142/S0218488506004175
[5] Liu B. (2002) Random fuzzy dependent-chance programming and its hybrid intelligent algorithm. Information Sciences 141(3-4): 259--271 · Zbl 1175.90439 · doi:10.1016/S0020-0255(02)00176-7
[6] Liu B. (2003) Inequalities and convergence concepts of fuzzy and rough variables. Fuzzy Optimization and Decision Making 2(2): 87--100 · doi:10.1023/A:1023491000011
[7] Liu B. (2004) Uncertainty theory. Springer, Berlin · Zbl 1072.28012
[8] Liu B. (2007) Uncertainty theory. (2nd ed.). Springer, Berlin · Zbl 1141.28001
[9] Liu B. (2008) Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems 2(1): 3--16
[10] Liu B. (2009) Some research problems in uncertainty theory. Journal of Uncertain Systems 3(1): 3--10
[11] Liu B. (2010) Uncertainty theory: A branch of mathematics for modeling human uncertainty. Springer, Berlin
[12] Liu, B. (2011). Uncertainty theory (4th ed.). http://orsc.edu.cn/liu/ut.pdf .
[13] Liu B. (2010) Uncertain risk analysis and uncertain reliability analysis. Journal of Uncertain Systems 4(3): 163--170
[14] Liu B. (2010) Uncertain set theory and inference rule with application to uncertain control. Journal of Uncertain Systems 4(2): 83--98
[15] Liu Y., Ha M. (2010) Expected value of function of uncertain variables. Journal of Uncertain Systems 4(3): 181--186
[16] Liu B., Liu Y. (2002) Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10(4): 445--450 · doi:10.1109/TFUZZ.2002.800692
[17] You C. (2009) Some convergence theorems of uncertain sequences. Mathematical and Computer Modelling 49(3-4): 482--487 · Zbl 1165.28310 · doi:10.1016/j.mcm.2008.07.007
[18] Zadeh L. (1965) Fuzzy sets. Information and control 8(3): 338--353 · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
[19] Zadeh L. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1(1): 3--28 · Zbl 0377.04002 · doi:10.1016/0165-0114(78)90029-5
[20] Zhang Z. (2011) Some discussions on uncertain measure. Fuzzy Optimization and Decision Making, 10(1): 31--43 · Zbl 1305.28028 · doi:10.1007/s10700-010-9091-0
[21] Zhu Y., Liu B. (2005) Some inequalities of random fuzzy variables with application to moment convergence. Computers and Mathematics with Applications 50(5-6): 719--727 · Zbl 1084.60013 · doi:10.1016/j.camwa.2005.04.015