Concept# Compactness theorem

Summary

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.
The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.
The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Althoug

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MATH-381: Mathematical logic

Branche des mathématiques en lien avec le fondement des mathématiques et l'informatique théorique. Le cours est centré sur la logique du 1er ordre et l'articulation entre syntaxe et sémantique.

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In this paper, we obtain interior Holder continuity for solutions of the fourth-order elliptic system Delta(2)u = Delta(V center dot del u) + div(w del u) + W center dot del u formulated by Lamm and Riviere [Comm. Partial Differential Equations 33 (2008) 245-262]. Boundary continuity is also obtained under a standard Dirichlet or Navier boundary condition. We also use conservation law to establish a weak compactness result which generalizes a result of Riviere for the second-order problem.

2019A proof of existence is given for a stationary model of alloy solidification. The system is composed of heat equation, solute equation and Navier-Stokes equations. In rite latter Carman-Kozeny penalization of porous medium models the mushy zone. The problem is first regularized and a sequence of regularized solutions is built thanks to Leray-Schauder's fixed point Theorem. A solution is then extracted by compactness argument.

1995Morgan Almanza, Yoan René Cyrille Civet, Thomas Guillaume Martinez, Yves Perriard

Electroactive polymers show promising characteristic such as lightness, compactness, flexibility and large displacements making them a candidate for application in cardiac assist devices. This revives the need for quasi- square wave voltage supply switching between 0 and several kilo-Volts, that must be efficient, to limit the heat dissipation, and compact in order to be implanted. The high access resistance, associated to compliant electrodes, represents an additional difficulty. Here, a solid-state Marx modulator is adapted to cope with electroactive polymer characteristics, taking advantage of an efficient energy transfer over a sequential multistep charge/discharge process. To ensure compactness, efficiency as well as the needs of an implanted device, a wireless magnetic field based communication, and power transfer system has been implemented. This work demonstrates the benefit of this design through simulations, and experimental validation on a cardiac assist device. At a voltage of 7 kV, an efficiency of up to 88% has been achieved over a complete charge/discharge cycle.

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