Second-order logic

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence \forall P,\forall x (Px \lor \neg Px) says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but

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Regularity of solutions for a fourth-order elliptic system via Conservation law

Changyu Guo

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.

2019

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Rhythmic Modulation of Theta Oscillations Supports Encoding of Spatial and Behavioral Information in the Rat Hippocampus

Oscillatory patterns of activity in various frequency ranges are ubiquitously expressed in cortical circuits. While recent studies in humans emphasized rhythmic modulations of neuronal oscillations ("second-order" rhythms), their potential involvement in information coding remains an open question. Here, we show that a rhythmic (similar to 0.7 Hz) modulation of hippocampal theta power, unraveled by second-order spectral analysis, supports encoding of spatial and behavioral information. The phase preference of neuronal discharge within this slow rhythm significantly increases the amount of information carried by action potentials in various motor/cognitive behaviors by (1) distinguishing between the spikes fired within versus outside the place field of hippocampal place cells, (2) disambiguating place firing of neurons having multiple place fields, and (3) predicting between alternative future spatial trajectories. This finding demonstrates the relevance of second-order spectral components of brain rhythms for decoding neuronal information.

Elsevier2012

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A second-order cut-cell method for the numerical simulation of 2D flows past obstacles

We present a new second-order method, based on the MAC scheme on cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. In this approach, the solid boundary is embedded in the cartesian computational mesh. Discretizations of the viscous and convective terms are formulated in the context of finite volume methods ensuring local conservation properties of the scheme. Classical second-order centered schemes are applied in mesh cells which are sufficiently far from the obstacle. In the mesh cells cut by the obstacle, first-order approximations are proposed. The resulting linear system is nonsymmetric but the stencil remains local as in the classical MAC scheme on cartesian grids. The linear systems are solved by a fast direct method based on the capacitance matrix method. The time integration is achieved with a second-order projection scheme. While in cut-cells the scheme is locally first-order, a global second-order accuracy is recovered. This property is assessed by computing analytical solutions for a Taylor-Couette problem. The efficiency and robustness of the method is supported by numerical simulations of 2D flows past a circular cylinder at Reynolds number up to 9500. Good agreement with experimental and published numerical results are obtained. (c) 2012 Elsevier Ltd. All rights reserved.

Pergamon-Elsevier Science Ltd2012

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