Vol. 7 No. 1 (2015)

Analogues of the classical omitting types theorems of first-order logic are proved for propositional logic. For an infinite cardinal $\kappa$, a sufficient criterion is given for the omission of $\kappa$-many types in a propositional language with $\kappa$ propositional variables.

The paper generalizes the self-adjoint differential operator $\Box$ on hypersurfaces of a constant curvature manifold to submanifolds, introduced by Cheng-Yau. Using the series of such operators, a new way to prove Minkowski-Hsiung integral formula is given and some integral formulas for compact submanifolds is derived. An application to a hypersurface of a Riemannian manifold with harmonic Riemannian curvature is presented.

In this paper we treat the numerical approximation of the two-phase parabolic obstacle-like problem: \[\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,\] where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain. We introduce a certain variation form, which allows us to define a notion of viscosity solution. We use defined viscosity solutions framework to apply Barles-Souganidis theory. The numerical projected Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of the discretized scheme to the unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.

In the paper we consider an operator algebra generated by a family of partial isometries associated with a self-mapping on a countable set and by multipliers. An action of the unit circle on this algebra is specified that determines its $\mathbb{Z}$-grading. Under some conditions on the mapping the algebra is isomorphic to the crossed product of its fixed point subalgebra and the semigroup $\mathbb{N}$.

We consider the problem of the construction of the backward stochastic differential equation in the Markovian case. We suppose that the forward equation has a diffusion coefficient depending on some unknown parameter. We propose an estimator of this parameter constructed by the discrete time observations of the forward equation and then we use this estimator for approximation of the solution of the backward equation. The question of asymptotic optimality of this approximation is also discussed.