In the early 60s, Harary and Hill conjectured H(n) := 1 4b2 cbn−1 2 cbn−2 2 cbn−3 2 c to be the minimum number of crossings among all drawings of the complete graph Kn. It has recently been shown that this conjecture holds for so-called shellable drawings of Kn. For n ≥ 11 odd, we construct a non-shellable family of drawings of Kn with exactly H(n) crossings. In particular, every edge in our dr...

Avoided crossings are common in the quasienergy spectra of strongly driven nonlinear quantum wells. In this paper we examine the sinusoidally driven particle in a square potential well to show that avoided crossings can alter the structure of Floquet states in this system. Two types of avoided crossings are identi ed: one type leads only to temporary changes (as a function of driving eld streng...

A smoothing theorem for normal crossings to Calabi-Yau manifolds was proved by Y. Kawamata and Y. Namikawa ([KaNa]). This paper is a study of the observation that the Picard groups and Chern classes of these Calabi-Yau manifolds are constructible from the normal crossings in such smoothings. We provide and prove the formula for the construction in its full generality and various applications ar...

|We present a neural network algorithm for minimizing edge crossings in drawings of nonplanar graphs. This is an important subproblem encountered in graph layout. The algorithm nds either the minimum number of crossings or an approximation thereof and also provides a linear embedding realizing the number of crossings found. The parallel time complexity of the algorithm is O(1) for a neural netw...

The multi–scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot properly cope with crossings or bifurcations since it only looks for elongated structures. Therefore, we disentangle crossings/bifurcations via (multiple scale) invertible orientation scores and apply vesselness filters in this domain. This new method via scale–orientation scores pe...

In contrast to bounded systems, quantum chaos in extended systems may be associated with fractal spectra, metal-insulator transitions due to avoided band crossings, and spreading wave packets. In this lecture we point out the role of avoided band crossings for spectral transitions in the example of the kicked Harper model. We explain the coexistence of localized and extended eigenfunctions oo t...

The spectral flow of the hermitian Dirac–Wilson operator H(m) has been used to construct a lattice version of the index of the Dirac operator. We clarify some aspects of this construction by showing the following (in 4D): When the curvature of the lattice gauge field satisfies an approximate smoothness condition, crossings of the origin by eigenvalues of H(m) can only happen when m is close to ...

The multi-scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot properly cope with crossings or bifurcations since it only looks for elongated structures. Therefore, we disentangle crossings/bifurcations via (multiple scale) invertible orientation scores and apply vesselness filters in this domain. This new method via scale-orientation scores pe...

In quantum chemistry, the dynamics of heavy molecules is described by Schrödinger equations with energy level crossings. These crossings generate energy transfers at leading order between the modes as was earlier noticed by Landau and Zener in the 30’s. The mathematical analysis of these transfers relies on the use of normal forms. Here, we analyze the implications of Y. Colin de Verdière’s rec...

A variety of phenomena in physical and biological sciences can be mathematically understood by considering the statistical properties of level crossings of random Gaussian processes. Notably, a growing number of these phenomena demand a consideration of correlated level crossings emerging from multiple correlated processes. While many theoretical results have been obtained in the last decades f...