Computing Cup Products in \mathbb {Z}_2-Cohomology of 3D Polyhedral Complexes

Let \(I=(\mathbb {Z}^3,26,6,B)\) be a three-dimensional (3D) digital image, let \(Q(I)\) be an associated cubical complex, and let \(\partial Q(I)\) be a subcomplex of \(Q(I)\) whose maximal cells are the quadrangles of \(Q(I)\) shared by a voxel of \(B\) in the foreground—the object under study—and by a voxel of \(\mathbb {Z}^3\backslash B\) in the background—the ambient space. We show how to simplify the combinatorial structure of \(\partial Q(I)\) and obtain a 3D polyhedral complex \(P(I)\) homeomorphic to \(\partial Q(I)\) but with fewer cells. We introduce an algorithm that computes cup products in \(H^*(P(I);\mathbb {Z}_2)\) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in \(\mathbb {R}^3\) .     

Eigenvalues and eigenfunctions of the anharmonic oscillator V(x,y) = x 2 y 2

We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential V(x, y) = x ² y ² by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it     

Lasing action assisted by long‐range surface plasmons

It is shown that lasing action at subwavelength scales can be achieved in realistic plasmonic systems supporting long‐range surface plasmons (LRSPPs). To this end, a general numerical framework has been developed that is able to accurately account for the full spatio‐temporal lasing dynamics and the vastly different length‐ and time‐scales featured by this class of systems. Starting from a loss compensation regime for propagating LRSPPs, it is shown how the introduction of an optical feedback mechanism induces the formation of a self‐sustained laser oscillation at moderate pump intensities. The simplicity of the proposed subwavelength scale laser offers significant potential as a novel class of planar light sources in complex plasmonic circuits.     

Multiobjective shortest path problems with lexicographic goal-based preferences

Multiobjective shortest path problems are computationally harder than single objective ones. In particular, execution time is an important limiting factor in exact multiobjective search algorithms. This paper explores the possibility of improving search performance in those cases where the interesting portion of the Pareto front can be initially bounded. We introduce a new exact label-setting algorithm that returns the subset of Pareto optimal paths that satisfy a set of lexicographic goals, or the subset that minimizes deviation from goals if these cannot be fully satisfied. Formal proofs on the correctness of the algorithm are provided. We also show that the algorithm always explores a subset of the labels explored by a full Pareto search. The algorithm is evaluated over a set of problems with three objectives, showing a performance improvement of up to several orders of magnitude as goals become more restrictive.     

The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

Let \(f\) and \(g\) be holomorphic function-germs vanishing at the origin of complex analytic germs of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ \(f\bar{g}\) has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ \(f\bar{g}\) to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs of the form \(f\bar{g}\) defined in a complex surface germ, and we prove an A’Campo-type formula for the zeta function of its monodromy.     

Mixed norm spaces and rearrangement invariant estimates

Our main goal in this work is to further improve the mixed norm estimates due to Fournier [13], and also Algervik and Kolyada [1], to more general rearrangement invariant (r.i.) spaces. In particular we find the optimal domains and the optimal ranges for these embeddings between mixed norm spaces and r.i. spaces.     

Lipschitz factorization through subsets of Hilbert space

The Euclidean distortion of a metric space, a measure of how well the metric space can be embedded into a Hilbert space, is currently an active interdisciplinary research topic. We study the corresponding notion for mappings instead of spaces, which is that of Lipschitz factorization through subsets of Hilbert space. The main theorems are two characterizations of when a mapping admits such a factorization, both of them inspired by results dealing with linear factorizations through Hilbert space. The first is a nonlinear version of a classical theorem of Kwapień in terms of “dominated” sequences of vectors, whereas the second is a duality result by means of a tensor-product approach.     

Derived length of solvable groups of local diffeomorphisms

Let $G$ be a solvable subgroup of the group ${\mathrm{Diff}{{\,}_{}(\mathbb{C }^{n},0)}}$ of local complex analytic diffeomorphisms. Analogously as for groups of matrices we bound the solvable length of $G$ by a function of $n$ . Moreover we provide the best possible bounds for connected, unipotent and nilpotent groups.     

Bitumen-in-Water Emulsions: An Overview on Formation,Stability, and Rheological Properties

This paper describes the most relevant issues associated with the development of a technology; the formation of highly concentrated bitumen-in-water emulsions. Viscosity values for bitumen-in-water emulsions, containing between 70 and 85% (v/v) of bitumen, have been found to be several order of magnitude lower than the viscosity of the hydrocarbon itself. Thus, these emulsions, have potential applications in the processes of production, transportation, handling and commercialization of these extremely highly viscous hydrocarbons. The emulsions, the properties of which are discussed in this paper, were stabilized with mixtures of nonionic and natural surfactants (1,2) and formed using the HIPR (high internal phase ratio) technique (3). Information on the conditions required to produce emulsions with very narrow droplet diameter distributions is given. Results indicate that the mean droplet diameter, the droplet diameter distribution, and the bitumen volume fraction, significantly modify the rheological behavior. Emulsion stability was measured by following changes in the mean droplet diameter and in the rheological parameters with storage time.