On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl's theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl's theorem for f(T).     

On tropical Clifford’s Theorem

We prove an analogue of Clifford’s inequality for tropical curves. Next we focus on the hyperelliptic case and we characterize divisors attaining equality. Finally we speculate whether inequality in tropical Clifford’s Theorem does imply hyperellipticity, in analogy with the classical case.     

On Wiener’s Theorem for functions periodic at infinity

We consider the functions periodic at infinity with values in a complex Banach space. The notions are introduced of the canonical and generalized Fourier series of a function periodic at infinity. We prove an analog of Wiener’s Theorem on absolutely convergent Fourier series for functions periodic at infinity whose Fourier series are summable with weight. The two criteria are given: for the function periodic at infinity to be the sum of a purely periodic function and a function vanishing at infinity and for a function to be periodic at infinity. The results of the article base on substantially use on spectral theory of isometric representations.     

On Maingé’s Approach for Hierarchical Optimization Problems

This paper aims at investigating an iterative method for solving a system of variational inequalities with fixed-point set constraints. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé. Strong convergence results are established in the setting of Hilbert spaces. We propose an alternative analysis that allows us to relax some assumption imposed in his paper for convergence of the considered method. As a complementary result, we show how to adapt these processes to the case when the constraints involve operators belonging to the class of hemi-contractive mappings; this goes beyond the scope of Maingé’s result.     

On Analogues of Heilbronn’s Theorem

Mathematical Notes - Continued fractions with rational partial quotients arise in a natural way in the course of applying any $$k$$ -ary gcd algorithm to the ratio of natural numbers $$a$$ , $$b$$...     

On a Generalization of Zaslavsky’s Theorem for Hyperplane Arrangements

We define arrangements of codimension-1 submanifolds in a smooth manifold which generalize arrangements of hyperplanes. When these submanifolds are removed the manifold breaks up into regions, each of which is homeomorphic to an open disc. The aim of this paper is to derive formulas that count the number of regions formed by such an arrangement. We achieve this aim by generalizing Zaslavsky’s theorem to this setting. We show that this number is determined by the combinatorics of the intersections of these submanifolds.     

On Hawking’s Local Rigidity Theorem for Charged Black Holes

We show the existence of the Hawking vector field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth Einstein–Maxwell space–time without assuming the underlying space–time is analytic. This extends a result of Friedrich et al. (Commun Math Phys 204:691–707, 1971), which holds in the interior of the black hole region. Moreover, we also show, in the presence of an additional Killing vector field T which is tangent to the horizon and not vanishing on the bifurcate sphere, then space–time must be locally axially symmetric without the analyticity assumption. This axial symmetry plays a fundamental role in the classification theory of stationary black holes.     

On the Hochstadt–Lieberman Theorem for Discontinuous Boundary-valued Problems

In this paper, we discuss the half inverse problem for Sturm–Liouville equations with boundary conditions dependent on the spectral parameter and a finite number of discontinuities inside the interval and prove the Hochstadt–Liberman type theorem for the above boundary-valued problem.     

On a Generalization of Voronin’s Theorem

Mathematical Notes - Voronin’s theorem states that the Riemann zeta-function ζ(s) is universal in the sense that all analytic functions that are defined and have no zeros on the right...     

On Quillen?s Theorem A for posets

A theorem of McCord of 1966 and Quillen?s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen?s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.     

Pontryagin’s Maximum Principle for Multidimensional Control Problems in Image Processing

In the present paper, we prove a substantially improved version of the Pontryagin maximum principle for convex multidimensional control problems of Dieudonné-Rashevsky type. Although the range of the operator describing the first-order PDE system involved in this problem has infinite codimension, we obtain first-order necessary conditions in a completely analogous form as in the one-dimensional case. Furthermore, the adjoint variables are subjected to a Weyl decomposition. We reformulate two basic problems of mathematical image processing (determination of optical flow and shape from shading problem) within the framework of optimal control, which gives the possibility to incorporate hard constraints in the problems. In the convex case, we state the necessary optimality conditions for these problems.     

Hölder Continuity and Optimal Control for Nonsmooth Elliptic Problems

The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.