Nowadays one of the most studied issues in economic or finance field is to get the best possible return with the minimum risk. Therefore, the objective of the paper is to select the optimal investment portfolio from SP500 stock market and CBOE Interest Rate 10-Year Bond to obtain the minimum risk in the financial market. For this purpose, the paper consists of: 1) the marginal density distribution of the two financial assets is described with kernel density estimation to get the ”high-picky and fat-tail” shape; 2) the relation structure of assets is studied with copula function to describe the correlation of financial assets in a nonlinear condition; 3) value at Risk (VaR) is computed through the combination of Copula method and Monte Carlo simulation to measure the possible maximum loss better. Therefore, through the above three steps methodology, the risk of the portifolio is described more accuratly than the conventional method, which always underestimates the risk in the finicial market. So it is necessary to pay attention to the happening of extreme cases like ”Black Friday 2008” and appropriate investment allocation is a wise strategy to make diversification and spread risks in financial market. uzzy regression model, fuzzy random variable, expected value, variance, confidence interval.

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Aggregation is the process of fusing a large data in one representative value. This is done in different ways, through what may be called ‘operators’, every operator having special characteristics. Expanding study of vague phenomena, through hesitant fuzzy information of hesitant fuzzy set (HFS) theory and their applications has attracted useful aggregation techniques. Paper explores the geometric Heronian mean (GHM) under hesitant fuzzy environment and defines some new geometric Heronian mean operators such as the hesitant fuzzy generalized geometric Heronian mean (HFGGHM) operator and the weighted hesitant fuzzy generalized geometric Heronian mean (WHFGGHM) operator. Further, we give definition of hesitant fuzzy geometric Heronian element (HFGHE), which is a basic calculation unit in HFGGHM and reflects the conjunction between two aggregated arguments. Properties of the new aggregation operators are reported and their special cases are considered. Furthermore, based on the WHFGGHM operator, an approach to deal with multi-criteria decision-making problems under hesitant fuzzy environment is developed. Finally, a practical example is provided to illustrate the multi-criteria decisionmaking process.

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This paper proposes homological analysis of statistical dependency graph. If a dependency graph model satisfy the condition of a chain complex, homological algebra can be applied. Especially, the degree of freedom can be viewed as a dual space of an original complex.

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The purpose of the present paper is to carry out reliability and profit analysis of a single-unit system considering the concept of inspection under warranty. Within warranty, failures are rectified by the manufacturer at no cost to the users provided warranty does not apply to product failure due to user-induced damage such as cracked screen, accident, misuse, physical damage, damage due to liquid and unauthorized modifications, etc. The cost to rectify failures beyond the warranty is borne by the users. After failure, unit goes under inspection within warranty. There is single repairman, which is always available with the system to do repair, inspection and replacement of the unit. Repairman inspects the failed unit to see the feasibility of its repair or replacement. If repair of the unit is not feasible, it is replaced by new one.The time to failure of the system follows negative exponential distribution while inspection and repair time distributions are taken as arbitrary. By using supplementary variable technique, various measures of system performance such as reliability, mean time to system failure (MTSF), availability of the system and profit function have been determined. The numerical results for reliability and profit function are also obtained in the form of tables for particular values of various parameters and repair cost.

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The authors have developed a method of calculating the non-linear elastic force that arises in the clearance seal of a finite length. The authors have analyzed the force impact on the dynamics of the rotor of a centrifugal machine. In this work authors studied impact of nonlinear hydrostatic force in an arbitrary long seal, opposite to the previous literature where this force assumed to be linear, and seals are short (e.g. the ratio of the length of the clearance seal to the seal’s radius is less than 0.5).

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Exponential fuzzy entropy with a single fuzzy set has been considered. Here we consider exponential information theoretic measures with pair of fuzzy sets. This leads to results that parallel the results as Shannon’s joint, conditional and mutual information measures between two fuzzy sets. Mathematics Subject Classification (2000): 94 A17

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