Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.

Entropy is commonly used as a way to measure the uncertainty of random variables. In uncertain set theory, a concept of entropy for uncertain sets has been defined by using logarithm. However, such an entropy fails to measure the uncertain degree of some uncertain sets. This paper aims at proposing a concept of elliptic entropy for uncertain sets and investigating its properties such as translation invariance and positive linearity. It also provides some formulas for calculating the elliptic entropy via inverse membership functions. Additionally, elliptic relative entropy for uncertain sets is presented as a measure of the difference between two membership functions, and some applications are considered in portfolio selection and clustering.

A random variable is a measurable function from an uncertainty space to the set of real numbers, which is used to model randomness. An uncertain variable is a measurable function from uncertainty space to the set of real numbers, which is used to describe uncertainty. However, randomness and uncertainty often simultaneously appear in a complex system. Uncertain random variable provides a useful tool to handle such a hybrid case. This concept integrates random variable and uncertain variable into a broader view. For uncertain random variables, a basic and important topic is to discuss the convergence of its sequence. Specifically, this paper focuses on studying the convergence in distribution for a sequence of uncertain random variables without a common chance distribution.

Entropy is a measure of the uncertainty associated with a variable whose value cannot be exactly predicted. Based on the notion of chance measure, a concept of uncertain random entropy is introduced and used to provide a quantitative measurement of the uncertainty associated with uncertain random variables and its properties are studied in this paper. Relative entropy is a measure of the difference between two distribution functions. In order to deal with the divergence of uncertain random variables via chance distributions, this paper proposes also the relative entropy for uncertain random variables, as well as it investigates some mathematical properties of this concept. As an application, a model is presented to formulate a minimum spanning tree problem with uncertain random edge weights involving a relative entropy chance distribution. Finally, a numerical example of an uncertain random network is put forward to illustrate the effectiveness of the proposed model.

The Liu process is a new tool to deal with the noise process based on uncertainty theory. In this paper, we view the foreign exchange rate as an uncertain processes, described by uncertain differential equations driven by the Liu process, and build an uncertain currency model. Then, the uncertain currency option problems are discussed. Moreover, European and American currency option pricing formulas are derived for the proposed uncertain currency model and some mathematical properties are studied. Finally, two numerical examples are documented.

This note presents the characterization of a fuzzy variable whose possibility distribution is a convex fuzzy set.

In this paper, several useful inequalities for uncertain variables are proved. A Borel-Cantelli lemma for uncertain measures is obtained and some convergence theorems for continuous uncertain measures are derived. Finally, these theorems are applied to compute the uncertainty distribution of Liu integral. We prove that the uncertain integral of a deterministic function with respect to a Liu process has a normal uncertainty distribution.

In many systems, randomness and uncertainty are present simultaneously. Uncertain random variables provide a tool to deal with these inexact phenomena. This paper proposes a concept of risk index to quantify the risk of an uncertain random system. In addition, a risk index theorem is proved in order to calculate the risk index, and is applied to series systems, parallel systems and standby systems. Finally, a concept of expected loss is suggested for an uncertain random system.

Train rescheduling is an important decision process in railway management. It aims to minimize the negative effects arising from the disturbances via real-time traffic management. Two main challenges are how to formulate the dynamic and complex rescheduling problem as an optimization model, and how to obtain a good solution within a short time limit. Focusing on the stochastic capacity recovery times of blocked tracks, we propose a new track-backup rescheduling (TBR) approach which optimally assigns each affected train a backup track, based on the estimation of recovery time, the original timetable, and track changing cost. Then, we formulate a mixed integer programming (MIP) model to obtain a conflict-free timetable which minimizes the delay cost and the expected track changing cost. A greedy algorithm is designed to reorder trains and reschedule the arrival and departure times, and then we use an MIP algorithm to solve the optimal track backup strategy. Based on the Beijing-Shanghai high-speed railway line, we conduct extensive experimental studies which show that the TBR approach can reduce the rescheduling cost by an average of 10.17% compared with traditional approaches. More important, the greedy-based algorithm is shown to be able to obtain good solutions (with an average error of only 2.85%) within 1.5 s, which implies the high potential of our approach in a real-time traffic management system where fast response is critical.

In this paper, a two-dimensional Hardy type inequality for fuzzy integrals is proved. Also, some illustrative examples are presented.

Liu's inference is a process of deriving consequences from uncertain knowledge or evidence via the tool of conditional uncertainty. Using membership functions, this paper derives some expressions of Liu's inference rule for uncertain systems. This paper also discusses Liu's inference rule with multiple antecedents and with multiple if-then rules.

We define a fuzzy subspace skeleton of data points and propose an algorithm for finding it. Such a skeleton is connected with data representation: if the data points (represented as columns of a given matrix X) belong exactly to this fuzzy skeleton, then under some mild conditions we can represent X of the form X = AS uniquely (up to scaling and permutation), where the matrices A and S with dimensions m times m and m times N respectively (often called mixing matrix or dictionary and source matrix) are such that S is r-sparse in sense that each column of S has at most m - r nonzero elements. In this paper we consider the case r ges 2 and develop a fuzzy algorithm for clustering over subspaces, which is essential for identification of the mixing matrix A. The idea of this clustering is the same as in the classical fuzzy clustering problem, but instead of balls, here we cluster over subspaces with co-dimension r. For identification of the source matrix, we apply a special source recovery algorithm. We illustrate our algorithms with examples. We note that our method is quite general, since the sparseness conditions could be obtained with some preprocessing methods and no independence conditions for the source signals are imposed (in contrast to independent component analysis).

In this paper, we discuss the defuzzification problem. We first propose a set defuzzification method, (from a fuzzy set to a crisp set) by using the Aumann integral. From the obtained set to a point, we have two methods of defuzzification. One of these uses the mean value method and the other uses a fuzzy measure. In the first case, we compare our mean value method with the method of the center of gravity. In the second case, we compare fuzzy measure method with the Choquet integral method. We also give there a sufficient condition so that the results in the last two methods are equivalent.

In this paper, comparisons of OWA operators and ordinal OWA operators are carried out by using lattice theoretic methods. It is proved first that in both cases, the set of all aggregation operators forms a lattice, then the concept of positive valuation is used to measure the "orness" of aggregation operators and the structures of all such possible "orness" measures are determined.