Fuzzy Modeling for Uncertain Nonlinear Systems Using Fuzzy Equations and Z-Numbers

In this paper, the uncertainty property is represented by Z-number as the coefficients and variables of the fuzzy equation. This modification for the fuzzy equation is suitable for nonlinear system modeling with uncertain parameters. Here, we use fuzzy equations as the models for the uncertain nonlinear systems. The modeling of the uncertain nonlinear systems is to find the coefficients of the fuzzy equation. However, it is very difficult to obtain Z-number coefficients of the fuzzy equations. Taking into consideration the modeling case at par with uncertain nonlinear systems, the implementation of neural network technique is contributed in the complex way of dealing the appropriate coefficients of the fuzzy equations. We use the neural network method to approximate Z-number coefficients of the fuzzy equations

Sparse approximation algorithms for high dimensional parametric initial value problems

We consider the efficient numerical approximation for parametric nonlinear systems of initial value Ordinary Differential Equations (ODEs) on Banach state spaces S over R or C. We assume the right hand side depends analytically on a vector y=(yj)j≥1 of possibly countably many parameters, normalized such that | y j  | ≤ 1. Such affine parameter dependence of the ODE arises, among others, in mass action models in computational biology and in stoichiometry with uncertain reaction rate constants. We review results by the authors on N-term approximation rates for the parametric solutions, i.e. summability theorems for coefficient sequences of generalized polynomial chaos (gpc) expansions of the parametric solutions {X(⋅ ; y)} y ∈ U with respect to tensorized polynomial bases of L 2(U). We give sufficient conditions on the ODEs for N-term truncations of these expansions to converge on the entire parameter space with efficiency (i.e. accuracy versus complexity) being independent of the number of parameters viz. the dimension of the parameter space U. We investigate a heuristic adaptive approach for computing sparse, approximate representations of the {X(t;y):0≤t≤T}⊂S. We increase efficiency by relating the accuracy of the adaptive initial value ODE solver to the estimated detail operator in the Smolyak formula. We also report tests which indicate that the proposed algorithms and the analyticity results hold for more general, nonaffine analytic dependence on parameters

Dose-Dependent Induction of Murine Th1/Th2 Responses to Sheep Red Blood Cells Occurs in Two Steps: Antigen Presentation during Second Encounter Is Decisive

The differentiation of CD4 T cells into Th1 and Th2 cells in vivo is difficult to analyze since it is influenced by many factors such as genetic background of the mice, nature of antigen, and adjuvant. In this study, we used a well-established model, which allows inducing Th1 or Th2 cells simply by low (LD, 105) or high dose (HD, 109) injection of sheep red blood cells (SRBC) into C57BL/6 mice. Signature cytokine mRNA expression was determined in specific splenic compartments after isolation by laser-microdissection. LD immunization with SRBC induced T cell proliferation in the splenic T cell zone but no Th1 differentiation. A second administration of SRBC into the skin rapidly generated Th1 cells. In contrast, HD immunization with SRBC induced both T cell proliferation and immediate Th2 differentiation. In addition, splenic marginal zone and B cell zone were activated indicating B cells as antigen presenting cells. Interestingly, disruption of the splenic architecture, in particular of the marginal zone, abolished Th2 differentiation and led to the generation of Th1 cells, confirming that antigen presentation by B cells directs Th2 polarization. Only in its absence Th1 cells develop. Therefore, B cells might be promising targets in order to therapeutically modulate the T cell response