Narrow Results

We show how resorting to dependable computer calculations makes it possible to compute all integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras and their equivalence classes under the automorphism group of the Lie algebra. We also prove that the set comprised of all integrable complex structures on such a Lie algebra is a smooth submanifold of ℝ³⁶.

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G₀ associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G₀, J) associated to G₀ and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spin^c-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.

Magnetic field effects have been instrumental to unveil several exotic phenomena in two-dimensional (2D) materials. Here, we show that graphene exhibits self-similar transport once the material is nanostructured with a magnetic field in complex fashion. In specific, when magnetic barriers are arranged according to the Cantor set rules. In this study, the charge carriers are described as quantum relativistic particles through an effective low-energy Hamiltonian. The transmission, transport and thermoelectric properties are computed with the transfer matrix method, the Landauer–Büttiker formalism and the Cutler–Mott formula, respectively. Self-similarity is reflected in the conductance spectra and Seebeck coefficient for different structural parameters such as generation number, the intensity of the magnetic field, the height of the barrier and the total length of the system. Moreover, well-defined scaling rules, which described fairly good the scalability between self-similar patterns, are obtained. We also compare the self-similar patterns of magnetic complex structures with the corresponding ones to magnetic-electric complex structures, finding better scalability for the former. It is worth mentioning that as far as we have corroborated the breaking symmetry associated to the magnetic field is paramount for the self-similar transport. So, magnetic complex structures constitute an excellent option to corroborate the peculiar phenomenon of self-similar transport.

A new approach called the ''Variational Theory of Complex Rays'' (VTCR) is being developed in order to calculate the vibrations of slightly damped elastic structures in the medium-frequency range. Here, the emphasis is put on the extension of this theory to analysis across a range of frequencies. Numerical examples show the capability of the VTCR to predict the vibrational response of a structure in a frequency range.

The objective of this paper is to determine the finite-dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name $i{D}_{\infty }$.

In this paper, we introduce the notion of a product structure on a 3-Bihom-Lie algebra, which is a Nijenhuis operator with some conditions. We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras. Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition. Similarly, we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures. Finally, we establish the relation between a complex structure and a product structure.