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By creating some new concepts and methods: checking tree, long unit path, direct contradiction unit pair, indirect contradiction unit pair, additional contradiction unit pair, 2-unit layer and 3-unit layer, redundant units, and destroying parallel pairs , we successfully transform solving a 3SAT problem to solving 2SAT problems in polynomial time. Thus we proved that NP=P.
We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in \emph{affine logic} the translation schemes due to Kolmogorov and Gödel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and {\it vice versa}. By contrast, in the setting of \emph{Łukasiewicz logic}, we can prove a general result asserting that a wide class of formula translations including those of Kolmogorov, Gödel, Gentzen and Glivenko not only satisfy Troelstra's criteria with respect to a natural intuitionistic fragment of Łukasiewicz logic but are all equivalent.
In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with constant density, in the domain $\mathbb{T}\times \mathbb{R}$. We begin by directly investigating the Couette shear flow, where we characterize the linear growth of the compressible part of the fluid while proving time decay for the incompressible part (inviscid damping with slower rates). Then we extend the analysis to monotone shear flows near Couette, where we are able to give an upper bound, superlinear in time, for the compressible part of the fluid. The incompressible part enjoys an inviscid damping property, analogous to the Couette case. In the pure Couette case, we exploit the presence of an additional conservation law (which connects the vorticity and the density on the moving frame) in order to reduce the number of degrees of freedom of the system. The result then follows by using weighted energy estimates. In the general case, unfortunately, this conservation law no longer holds. Therefore we define a suitable weighted energy functional for the whole system, which can be used to estimate the irrotational component of the velocity but does not provide sharp bounds on the solenoidal component. However, even in the absence of the aforementioned additional conservation law, we are still able to show the existence of a functional relation which allows us to recover somehow the vorticity from the density, on the moving frame. By combining the weighted energy estimates with the functional relation we also recover the inviscid damping for the solenoidal component of the velocity.
We construct two Fréchet algebras admitting countably many mutually inequivalent Fréchet algebra topologies. The second example is a modification of the maiden (and first) example of a non-Banach Fréchet algebra with two inequivalent Fréchet algebra topologies, considered by Read to show that the famous Singer-Wermer conjecture does not hold in the Fréchet case. We discuss a couple of avatars of the Singer-Wermer conjecture in Fréchet algebras. In particular, this conjecture (and so, its weaker version, too) does hold by the second example which also admits countably many mutually equivalent Fréchet algebra topologies. However the stronger version of this conjecture (i.e., the Kaplansky's conjecture) does not hold by the second example.
The elastodynamic Peach-Koehler force is computed for a fully-regularized straight dislocation with isotropic core in continuum isotropic elastic elasticity, in compact forms involving partial mass or impulsion functions relative to shear and compressional waves. The force accounts for both dynamic radiation damping and inertia. The expressions are valid indifferently for subsonic or supersonic velocities. Results are compared with the case of a flat-core dislocation of the Peierls-Eshelby type, for a motion of jump from rest to constant velocity. In the steady-state limit, the Lagrangian function relevant to expressing the force in the flat-core case must be replaced by a related but different function for the regularized dislocation. However, by suitably defining the regularizing dislocation width, the steady-state limits of the force for the fully-regularized and flat-core dislocations can be matched exactly.
In 1952, Michael posed a question about functional continuity of commutative Fréchet algebras in his memoir, known as Michael's problem in the literature. We settle this in the affirmative for the non-commutative Fréchet algebras, along with its equivalent form. Indeed, we continue our previous works to directly attack these problems. We show that the existence of a discontinuous character on the non-commutative test case for this problem -- the Fréchet algebra ${\cal U}_{nc}$ of all entire functions on $\ell^\infty$ -- allows us to generate another Fréchet algebra topology on ${\cal U}_{nc}$, inequivalent to the usual Fréchet algebra topology. An important tool is a topological version of the tensor algebra over a Banach space. The elementary, but crucial, idea is to express the test algebra ${\cal U}_{nc}$ as a {\it weighted Fréchet tensor algebra} over the Banach space $\ell^1$. We give some important applications in automatic continuity theory. In particular, we provide affirmative answers to significant, long-standing problems from 1967 (extending Johnson's result for non-commutative, semi-simple Fréchet algebras; Dales' survey, Question 9), 1976 (extending Esterle's result, giving another approach for Michael's problem for more general $(F)$-algebras), and 1978 (a weaker version of an equivalent form of the non-commutative Michael's problem; Dales' survey, p. 143).
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