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One considers a group S of subjects and one uses the measured values of EEG waves and the levels of anxiety given by the psychologists for the members of S in order to construct a monotone measure µ. This measure µ is such that the levels of anxiety of the members of S are Choquet integrals with respect to µ. One infers that this is true for any other subject. This inference is validated by comparing the values obtained via the above mentioned method with the results given by the psychologists for another group of subjects S'. We think that this procedure is completely new for the computation of the anxiety degree. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature. [ABSTRACT FROM AUTHOR]

The purpose of this article is to present some new nonlinear retarded integral inequalities which can be utilized to study the existence, stability, boundedness, uniqueness, and asymptotic behavior of solutions of nonlinear retarded integro-differential equations, and these inequalities can be used in the symmetrical properties of functions. These inequalities also generalize some former famous inequalities in the literature. Two examples as applications will be provided to demonstrate the strength of our inequalities in estimating the boundedness and global existence of the solution to initial value problems for nonlinear integro-differential equations and differential equations which can be seen in graphs. This research work will ensure opening new opportunities for studying nonlinear dynamic inequalities on a time-scale structure of a varying nature.

The ocean environment is complex and changeable because of all kinds of noise interferences, such as salt cliffs, ships around and other electromagnetic interferences, so the measurement information is prone to be lost. It is difficult to describe the complex noise and the acquisition probability of measurement information. In this paper, a continuous discrete variational Bayesian filter (CD VBF) is proposed to solve the problems of the heavy tailed noises and measurement random loss for state estimation. The variational Bayesian (VB) approach can effectively estimate the state vector, scale matrices, degree of freedom (DOF) parameters, Bernoulli random variables and the acquisition probability of measurement information. The performances of the proposed algorithm and traditional algorithms are tested in simulations and underwater experiments. The simulation results illustrate that the CD VBF has better localization accuracy and robustness. The experimental results demonstrate that the localization accuracy is improved by CD VBF and an optimal solution of iteration number is acquired.

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Matrix algebra is an efficient way to represent the relationship between different concepts in the multiple criteria decision analysis theory. The main purpose of this paper is to provide a compact and systematic representation of the capacities and nonlinear integrals based decision theory and models from the perspective of matrices or vectors. We first investigate the forms and properties of the matrix representation of some equivalent transformations of capacities and introduce some new types of equivalent representations. Then we represent respectively the specified conditions corresponding to several particular families of capacities as well as the calculation formulas of three types of nonlinear integrals in terms of vectors and matrices without the process of presorting the integrand values. Finally, we discuss the applications of the matrix representations in the capacity identification models and illustrate them through a multiple criteria decision making problem. The use of matrix-vector formalism facilitates formulation of the relevant optimization problems and their solutions in capacity based decision analysis can be easily programmed and implemented by means of standard software tools and libraries in the environments such as R and Matlab.

One can find a subadditive measure μ and a sequence { f n } n ∈ N of nonnegative measurable functions converging in μ -measure to such a function f with the property that for any p > 1 , f n p no longer converges in μ -measure to f p. Therefore, in nonadditive measure theory an additional and intrinsic discussion is needed to prove integral convergence theorems for functions integrable to the p th power. The purpose of the paper is to propose a method for automatically deriving p th power versions of convergence in measure theorems of nonlinear integrals from already established ones. The proposed method is based on the robustness of some properties of integral functional under the operation f ↦ f p raising a nonnegative function f to the p th power. [ABSTRACT FROM AUTHOR]

The Vitali convergence in measure theorem for the abstract Lebesgue integral is fundamental in Lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications. In this paper, in a unified way using the perturbation method the Vitali convergence in measure theorem is established for nonlinear integrals such as the Choquet, Šipoš, Sugeno, and Shilkret integrals, and their symmetric and asymmetric extensions. It is derived from the Fatou and the reverse Fatou type lemmas for perturbative nonlinear integral functionals. [ABSTRACT FROM AUTHOR]

The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures.

We present a unified approach to the monotone convergence theorem for nonlinear integrals such as the Choquet, the Šipoš, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of monotone convergence theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ -additive. [ABSTRACT FROM AUTHOR]

In this study, a new classification model - Multiple Nonlinear Integral with Multiple projections is proposed, which includes Double Nonlinear Integral extending to new variants for adapting for complicated data distribution and enhancing classification accuracy. When the performance is not satisfying by projecting with classical Nonlinear Integral, the second projection is need to stretch data in one dimension space to two dimension space. The value of two projection forms the 2-dimensional coordinates. All data in two-dimensional space can be classified by a straight line easily. The rest may be deduced by analogy, if the result is still not good for decision, the Multiple Nonlinear Integral can repeat n times double projections, in which n will be an optimized value to balance the performance and the complexity. The repeating can help adjust the data distribution in 2-dimensional space until being classified easily. The classification model based on Multiple Nonlinear Integral is applied to two kinds of datasets. One kind comes from the classical database; another kind is the real data about the HBV (Hepatitis B Virus) collected from hospital. The experimental results show that the new model has better performance compared with the classical algorithm and the classical Nonlinear Integral. Especially to the HBV data, Multiple Nonlinear Integral presents the superior on diagnosis to the others. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a class of the second-order quasilinear equations with characteristic invariants equal to the corresponding characteristic roots. By using the classical method of characteristics, we construct general integrals in terms of characteristic invariants and investigate the initial Cauchy and the Goursat-type problems for a mixed hyperbolic-parabolic quasilinear equation from the class under consideration. [ABSTRACT FROM AUTHOR]

The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz's theorem remains valid for monotone measures. When considered measurable space (X , A) is S-compact, the null-continuity condition is also sufficient for Riesz's theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff's theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures. [ABSTRACT FROM AUTHOR]