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We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.BL. For CBL we also need the following translation for the conflation connective:The following proposition is immediate.Proposition 5.1. For every formula A of BL (resp. CBL), the sequents t 1 (A) ⊢ t 1 (A) and t 2 (A) ⊢ t 2 (A) are derivable in D.BL (resp. D.CBL).Proof. By induction on the complexity of the formula A. If A is an atomic formula, the translation of t i (A) ⊢ t i (A) with i ∈ {1, 2} is A i ⊢ A i , hence it is derivable using (Id) in L 1 and L 2 , respectively. If A = A 1 ⊗ A 2 , then t i (A 1 ⊗ A 2 ) = t i (A 1 ) ⊓ 1 t i (A 2 ) and if A = A 1 ⊕ A 2 , then t i (A 1 ⊕ A 2 ) = t i (A 1 ) ⊔ 1 t i (A 2 ). By induction hypothesis, t i (A i ) ⊢ t i (A i ). So, it is enough to show that:

Abstract. In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term |ut| q−2 ut and a source term of the type |u| p−2 u.We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for q ≥ p. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for p > q.

In the present paper, we endow the logics of topological quasi Boolean algebras, topological quasi Boolean algebras 5, intermediate algebras of types 1-3, and pre-rough algebras with proper multi-type display calculi which are sound, complete, conservative, and enjoy cut elimination and subformula property. Our proposal builds on an algebraic analysis and applies the principles of the multi-type methodology in the design of display calculi.

In this paper, we consider the system of nonlinear viscoelastic equationswith initial and Dirichlet boundary conditions. We prove that, under suitable assumptions on the functions g i , f i (i = 1, 2) and certain initial data in the stable set, the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, there are solutions with positive initial energy that blow up in finite time.

In this paper, a nonlinear stochastic viscoelastic wave equation with linear damping is considered. By an appropriate energy inequality and estimations, we show that the local solution of the stochastic equations will blow up with positive probability or explosive in L2 sense under some sufficient conditions. Moreover, the upper bound of the blow-up time is given.

Most chemical processes, such as distillation, absorption, extraction, and catalytic reactions, are extremely complex processes affected by multiple factors. As a result, the relationships between their input and output variables are non-linear, and it is not easy to optimize or control them using traditional methods. Artificial neural network is a systematic structure composed of multiple neuron models. By simulating many basic functions of the nervous system of living organisms, nonlinear control can be realized without relying on mathematical models, and it is especially suitable for more complex control objects. This article will introduce artificial neural networks' basic principles and development history, and review its application research progress in chemical process control, fault diagnosis, and process optimization.

We investigate the asymptotic behavior of solutions of the non-autonomous Navier-Stokes equation with nonlinear damping in three-dimensional bounded domain. When 3 < β ≤ 5, the existence of pullback attractors is proved in V and H 2 (), respectively.