Let Ω ⊆ Rn be a compact convex set and x be a point in the exterior of Ω. The aperture angle of x relative to Ω is defined as the maximal angle of the smallest closed convex cone that contains Ω − x. This note provides an explicit formula, based on eigenvalues of symmetric matrices, for the aperture angle of a point relative to an ellipsoid.

In this paper, the natural boundary element method (NBEM) for an anisotropic hyperbolic problem in an exterior elliptic domain is investigated. By the theory of the natural boundary reduction (NBR), the natural integral equation (NIE) and the Poisson integral formula of the problem considered are obtained, and the numerical method of the NIE is given. Finally, some numerical examples are presen...

Abstract: Generalized linear systems are considered, which contain in their state-space representation matrices with elements functions of bounded variation and controls in the space of regulated functions. The Perron-Stieltjes integral is used to obtain a variation-of-parameters formula. On this basis the formula of the state of the system as well as the input-output map are derived. The funda...

<p style='text-indent:20px;'>We use Perron method to prove the existence of ancient solutions exterior problem for parabolic k-Hessian equations <inline-formula><tex-math id="M1">\begin{document}$ -u_tS_k(D^2u) = 1 $\end{document}</tex-math></inline-formula> with prescribed asymptotic behavior at infinity.</p>

Exterior Dirichlet problems for two-dimensional (2D) lattice waves on the infinite triangular are considered. Namely, we study 2D discrete Helmholtz equation in a plane with hole. New results obtained existence and uniqueness of solution case real wave number [Formula: see text] without passing to complex number. Besides, Green’s representation formula is derived help difference potentials. To ...

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal densit...

in exterior domains in Rn where f : (0,∞) → (0,∞) is continuous function. We give lower bounds on the growth of f(s) at s = 0 and/or s = ∞ such that inequality (0.1) has no C positive solution in any exterior domain of Rn. Similar results were obtained by Armstrong and Sirakov [Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Eq...

We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase s(λ) for all λ > 0. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural C∞ topology. This followed earlier wor...

Analogy between Anderson [Formula: see text]-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: (1) A description MIQF dimension text] signature in terms ”lattices” text]; (2) construction exterior powers having text].

Let V be a four-dimensional real vector space with a fixed orientation. Then the wedge product can be viewed, up to a positive factor, as a canonical quadratic form of signature (3, 3) on the six-dimensional space ΛV . This gives a homomorphism from the identity component of the general linear group GL(V ) to the conformal group of the indefinite form, which is a local isomorphism. The signific...