PT-symmetric couplings of dual pairs. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (24 Seiten). - (Preprint ; M23,03)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200049
Convexity and continuity of specific set-valued maps and their extremal value functions. - In: Journal of applied and numerical optimization, ISSN 2562-5535, Bd. 5 (2023), 1, S. 71-92
In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals over the image sets of the set-valued maps. Such extremal value functions play an important role for instance for derivative concepts for set-valued maps or for algorithmic approaches in set-valued optimization. We formulate conditions under which the set-valued maps and their extremal value functions inherit properties like (Lipschitz-)continuity and convexity.
https://doi.org/10.23952/jano.5.2023.1.05
Spectral inclusion property for a class of block operator matrices. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (12 Seiten). - (Preprint ; M23,02)
The numerical range and the quadratic numerical range is used to study the spectrum of a class of block operator matrices. We show that the approximate point spectrum is contained in the closure of the quadratic numerical range. In particular, the spectral enclosures yield a spectral gap. It is shown that these spectral bounds are tighter than classical numerical range bounds.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200026
Lower bounds for self-adjoint Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (11 Seiten). - (Preprint ; M23,01)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200011
Globally balancing spanning trees. - In: European journal of combinatorics, Bd. 109 (2023), 103644
https://doi.org/10.1016/j.ejc.2022.103644
Hamiltonicity of graphs perturbed by a random regular graph. - In: Random structures & algorithms, ISSN 1098-2418, Bd. 62 (2023), 4, S. 857-886
https://doi.org/10.1002/rsa.21122
Hamiltonicity of graphs perturbed by a random geometric graph. - In: Journal of graph theory, ISSN 1097-0118, Bd. 103 (2023), 1, S. 12-22
We study Hamiltonicity in graphs obtained as the union of a deterministic n-vertex graph H with linear degrees and a d-dimensional random geometric graph G d (n, r) for any d ≥ 1. We obtain an asymptotically optimal bound on the minimum r for which a.a.s. H ∪ G d (n, r) is Hamiltonian. Our proof provides a linear time algorithm to find a Hamilton cycle in such graphs.
https://doi.org/10.1002/jgt.22901
Relatively bounded perturbations of J-non-negative operators. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 17 (2023), 1, 14, insges. 30 S.
We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with Lp-potentials. Known bounds on the non-real eigenvalues of such operators are improved.
https://doi.org/10.1007/s11785-022-01263-2
On the complexity of finding well-balanced orientations with upper bounds on the out-degrees. - In: Journal of combinatorial optimization, ISSN 1573-2886, Bd. 45 (2023), 1, 30, S. 1-14
https://doi.org/10.1007/s10878-022-00962-y
A note on the invertibility of the Gabor frame operator on certain modulation spaces. - In: The journal of Fourier analysis and applications, ISSN 1531-5851, Bd. 29 (2023), 1, 3, S. 1-20
We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$, the weighted $$L^2$$-space $$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$, and the space $$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $$L^2$$spaces. In all cases, we prove that the space of Bessel vectors in $$V_j$$is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
https://doi.org/10.1007/s00041-022-09980-0