We show that, for all choices of integers k > 2 and m, there are simple 3-connected k-crossing-critical graphs containing more than m vertices of each even degree ≤ 2k − 2. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least 5 in crossing-critical graphs remains open. Furthermore, our constructed graphs...

The genome must be highly compacted to fit within eukaryotic nuclei but must be accessible to the transcriptional machinery to allow appropriate expression of genes in different cell types and throughout developmental pathways. A growing body of work has shown that the genome, analogously to proteins, forms an ordered, hierarchical structure that closely correlates and may even be causally link...

Conclusive comparisons of approximate inference algorithms are difficult due a number of handicaps. The main problem is that most algorithms were developed independently for different domains and consequently have different input and output representations and formulations, for example, Bayesian networks vs Markov random fields (MRFs) [13]. Thus, researchers usually only compare novel approxima...

We introduce a new deenition of the (Lusternik-Schnirelmann) co-category of a CW complex X. This is accomplished by producing a dual of the fat wedge called the thin product. One then looks at fac-torizations of the fold map through the thin product analogously to the Whitehead deenition of category. We investigate the relationship between this deenition and the rational cocategory introduced b...

In this work a classical linear harmonic oscillator, evolving during a small time interval (so that simple non-linear, second order Taylor approximation of the dynamics is satisfied) and restarting (by a mechanism) in a strictly chosen series of the time moments, is considered. It is shown that given linear oscillator behaves dynamically analogously to discrete logistic equation, which, as an e...

We present a novel algorithm (Principal Sensitivity Analysis; PSA) to analyze the knowledge of the classifier obtained from supervised machine learning techniques. In particular, we define principal sensitivity map (PSM) as the direction on the input space to which the trained classifier is most sensitive, and use analogously defined k-th PSM to define a basis for the input space. We train neur...

In all-optical processors, enhancement of image details is the result of high-pass filtering. We describe an optoelectronic processor in which detail enhancement results from the digitally calculated difference between an original input image and its low-pass filtered version. The low-pass filtering is realized through the rank-order median and the morphological opening and closing operations c...

A continuous nested sequence of similar triangles converging to the Brocard point of a given triangle is investigated. All these triangles have the same Brocard point. For polygons, the Brocard point need not exist, but there is always a limit object for an analogously defined nested sequence of inner polygons. This limit object is a Brocard point if and only if the inner polygons are all simil...

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems canno...

In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In particular, we show that any sum set must exhibit higher-order regularity and that an abelian sum set is necessarily a reversible difference set. We next develop ...