Nonlocal problems for hyperbolic equations of the first order describe the dynamic of population [1]–[3]. During the last thirty years the existence and uniqueness of the solution of nonlocal problems for the system of hyperbolic equations have been considered in a number of papers [4]–[8]. The authors assumed that the nonlinear functions satisfy the Lipschitz condition with respect to the unkn...

We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez [13]) associated with matrix-valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result [39...

with a1, a2 > 0. (Indeed any rank 2 matrix can be written as RA with A as above, and R orthogonal. Hence any affine transformation can be written as h ◦ f with f as claimed and g a rigid motion.) Hence, by Caratheodory uniqueness theorem it is sufficient to show that λ2(ASu(α1, α2)) = |det(A)|λ2(Su(α1, α2)) for all u = (u1, u2) ∈ R, αi ∈ R+ (3) Su(α, β) ≡ { x = (x1, x2) ∈ R : x1 ∈ (u1, u1 + α1]...

We prove a general Borg-type result for reflectionless unitary CMV operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle. In the course of deriving the Borg-type result we also use exponential...

A requirement common to most dynamic vision applications is the ability to track objects in a sequence of frames. This problem has been extensively studied in the past few years, leading to several techniques, such as Unscented Particle Filter based trackers, that exploit a combination of the (assumed) target dynamics, empirically learned noise distributions and past position observations. Whil...

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract CaratheodoryTchakaloff (CATCH) submeshes for compressed Least Squares fitting. 2010 AMS subject classification: 41A10, 65D32.

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KAN̄ and Bruhat big cell NMAN̄ . Then the space G/MAN̄ may be (almost) identified with N and with K/M , and these identifications induce the (generalised) Cayley transform C : N → K/M . We show that C is a conformal map of Carnot–Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplicat...

This paper introduces a generalized Schur algorithm in the Krein space with an indefinite inner product. Concepts such as Caratheodory classes and Schur classes used in the classical Schur algorithm cannot be applied in the Krein space since the positivedefiniteness corresponds merely to the nonsingularity in the Krein space. We note also that these problems appear when fast algorithms for subo...

A set K of vertices in a connected graph is M-convex if and only if for every pair of vertices in K, all vertices of all chordless paths joining them also lie in K. Carathtodory, Helly and Radon type theorems are proved for M-convex sets. The Caratheodory number is 1 for complete graphs and 2 for other graphs, The Helly number equals the size of a maximum clique. The Radon number is one more th...