Geometric characterization of continuously defective elastic crystals

We show how some differential geometric structures associated with a concept of a homogeneous space appear naturally in a kinematic model of continuously distributed defects in an elastic crystal solid and discuss how one can use them to describe the defectiveness of such a continuum.

Observability and Geometric Approach of 2D Hybrid Systems

A connection is emphasized between two branches of the Systems Theory, namely the Geometric Approach and 2D Systems, with a special regard to the concept of observability. An algorithm is provided which determines the maximal subspace which is invariant with respect to two commutative matrices and which is included in a given subspace. Observability criteria are obtained for a class of 2D systems by using a suitable 2D observability Gramian and some such criteria are derived for LTI 2D systems, as well as the geometric characterization of the subspace of unobservable states. The presented algorithm is applied to determine this subspace.

Planar Lattice Subsets with Minimal Vertex Boundary

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

On curves with circles as their isoptics

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

A system of local/nonlocal p-Laplacians: The eigenvalue problem and its asymptotic limit as p → ∞

In this work, given p ∈ ( 1 , ∞ ), we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector ( u p , v p ), for the following local/nonlocal PDE system (0.1) − Δ p u + ( − Δ ) p r u = 2 α α + β λ | u | α − 2 | v | β u in Ω − Δ p v + ( − Δ ) p s v = 2 β α + β λ | u | α | v | β − 2 v in Ω u = 0 on R N ∖ Ω v = 0 on R N ∖ Ω , where Ω ⊂ R N is a bounded open domain, 0 < r , s < 1 and α ( p ) + β ( p ) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ ∞ , and the uniformly convergence of the pair ( u p , v p ) to the ∞-eigenvector ( u ∞ , v ∞ ). Finally, the triple ( u ∞ , v ∞ , λ ∞ ) verifies, in the viscosity sense, a limiting PDE system.