Efficient loop-check for KD 45 logic

We introduce a new sequent calculus for KD 45 logic. A loop-check technique is used to determine whether a sequent derivable or not. We concentrate ourselves on the efficiency of the loop-check technique used. The efficiency is obtained by making the loop-check to act locally (then we need to check only one or two current sequents), instead of a global loop-check used in known works as [3]. Moreover, the sequent calculus introduced uses a marked operator □ to integrate loop-check into an inference rule. Besides the efficient loop-check used, the sequent calculus introduced produces smaller derivation trees that reduce the derivation time. We also prove a lemma which determines the maximal number of modality rule applications in one branch of a tree. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 55–66, January–March, 2006.     

Equivalences of higher derived brackets

This note elaborates on Th. Voronov’s construction [Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (1-3) (2005) 133-153; Th. Voronov, Higher derived brackets for arbitrary derivations, Travaux Math. XVI (2005) 163-186] of L_∞-structures via higher derived brackets with a Maurer-Cartan element. It is shown that gauge equivalent Maurer-Cartan elements induce L_∞-isomorphic structures. Applications in symplectic, Poisson and Dirac geometry are discussed.     

Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Hyperspace dynamical system (E²,f²) induced by a given dynamical system (E,f) has been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on E² employed in these studies is non-metrizable when E is not compact metrizable, e.g., E=Rn. Consequently, metric related dynamical concepts of (E²,f²) such as sensitivity on initial conditions and metric-based entropy, could not even be defined. Moreover, a condition on (E²,f²) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on E² is again HLCSC, thus metrizable. In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (E,f), these three conditions on (E,f) are respectively equivalent to mixing, weak mixing and transitivity on (E²,f²) (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.     

Oscillatory and phase dimensions of solutions of some second-order differential equations

In order to measure fractal oscillatority of solutions at t=∞, we define oscillatory and phase dimensions of solutions of a class of second-order nonlinear differential equations. The relation between these two dimensions is found using formulas for box dimension of chirps and nonrectifiable spirals. Applications include the Liénard equation and weakly damped oscillators.     

A note on the Undirected Rural Postman Problem polytope

This note refers to the article by G. Ghiani and G. Laporte ``A branch-and-cut algorithm for the Undirected Rural Postman Problem', Math. Program. 87 (2000). We show that some conditions for the facet-defining property of the basic non-trivial inequalities are not sufficient and that the Rural Postman Problem polytope is more complex even when focusing on canonical inequalities only.     

An explicit formula for a Bennequin-type invariant of apparent contours

We derive an explicit formula for the computation of a Vassiliev-type invariant of an apparent contour introduced recently by Ohmoto and Aicardi using appropriate linking numbers of its Legendrian lift. Our formula does not require the construction of the Legendrian lift and only takes into account the nodes, the cusps, the extremal points and the orientation of the apparent contour. In this way the computation can be implemented into a computer program; we demonstrate its use with some examples.     

Instability of solitary waves for a generalized Benney–Luke equation

We study linear instability of solitary wave solutions of a one-dimensional generalized Benney–Luke equation, which is a formally valid approximation for describing two-way water wave propagation in the presence of surface tension. Further, we implement a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions of the model equation which is used to validate the theory presented.     

Plane brownian motion in a region with holes

We consider a fixed family of balls with decreasing radii in the plane. We establish a relationship between a Dirichlet problem in a region without the balls and the solution of a Schroedinger equation in the complete region. Then we find upper bounds for the probability that a brownian motion exits the region without touching these balls. This is used to study harmonic measure and entire functions.     

A Wiener-Hopf approach for the scattering by a disk

Dedicated to Professor Erhard Meister on the occasion of his 65^th birthday.     

White noise driven SPDEs with reflection

Summary We study reflected solutions of a nonlinear heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by space-time white noise. The nonlinearity appears both in the drift and in the diffusion coefficient. Roughly speaking, at any point (t, x) where the solutionu(t, x) is strictly positive it obeys the equation, and at a point (t, x) whereu(t, x) is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. Existence of a minimal solution is proved. The construction uses a penalization argument, a new existence theorem for SPDEs whose coefficients depend on the past of the solution, and a comparison theorem for solutions of white-noise driven SPDEs.Partially supported by DRET under contract 901636/A000/DRET/DS/SR