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In this article, we introduce two multivalued contractive mappings within the framework of Hausdorff controlled metric spaces, employing concepts of admissibility and C$$ C $$‐class functions. The first mapping is a (β∗,ζ)$$ \left({\beta}_{\ast },\zeta \right) $$‐generalized contractive multivalued mapping, and the second is a (β,W(ζ,ξ))$$ \left({\beta}_{,}\mathcal{W}\left(\zeta, \xi \right)\right) $$‐multivalued mapping. We establish conditions guaranteeing the existence of fixed points for these mappings. To support our theoretical findings, we provide a numerical example, demonstrating the independence of the contractive conditions for the mappings. Furthermore, we apply our results to a specific problem in polynomial equations. These findings complement several existing results in the literature. [ABSTRACT FROM AUTHOR]

We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi = c, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form xi = c. We clarify and formalize the relation between the two approaches. In particular, we show that ACR implies zero sensitivity and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR. [ABSTRACT FROM AUTHOR]

Two multi‐parametric iterative algorithms are developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete‐time linear periodic systems. An important feature of one of the proposed algorithms is that the information in the current and the last steps is used to update the iterative sequence. Necessary and sufficient conditions for these two algorithms to be convergent are provided in terms of the roots of a set of polynomial equations. Based on these conditions, a two‐dimensional section method is established to search suboptimal tuning parameters for these two algorithms. In addition, convergence properties are also analysed for some special cases of the obtained algorithms. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithms. It can be found that the presented algorithms have better convergence performance than some existing iterative algorithms by choosing proper tuning parameters. [ABSTRACT FROM AUTHOR]

Techniques involving choosing process combinations for optimisation without due consideration for relevant experimental designs is scientifically unreliable and irreproducible. Mathematical modelling, of which response surface methodology (RSM) is one, provides a precise map leading to successful optimisation. This paper identified key process variables, building the model and searching the solution through multivariate regression analysis, interpretation of resulting polynomial equations and response surface/contour plots as basic steps in adapting the central composite design to achieve process optimisation. It also gave information on appropriate RSM software packages and choice of order in RSM model and data economy in reducing the factorial experiments from large number parameter combinations to a far less number without losing any information including quadratic and interaction (if present) effects. It is expected that this paper will afford many food scientists and researchers the opportunity for adapting RSM as a mathematical model for achieving bioprocess optimisation in food systems. [ABSTRACT FROM AUTHOR]