On a new one-parameter generalization of dual-complex Jacobsthal numbers

In this paper we define dual-complex numbers with generalized Jacobsthal coefficients. We introduce one-parameter generalization of dual-complex Jacobsthal numbers - dual-complex r-Jacobsthal numbers. We investigate some algebraic properties of introduced numbers, among others Binet type formula, Catalan, Cassini, d’Ocagne and Honsberger type identities. Moreover, we present the generating function, summation formula and matrix generator for these numbers. The results are generalization of the properties for the dual-complex Jacobsthal numbers.

Dual-complex quaternion representation of gravitoelectromagnetism

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

Heterotic complex structure moduli stabilization for elliptically fibered Calabi-Yau manifolds

Holomorphicity of vector bundles can stabilize complex structure moduli of a Calabi-Yau threefold in N = 1 supersymmetric heterotic compactifications. In principle, the Atiyah class determines the stabilized moduli. In this paper, we study how this mechanism works in the context of elliptically fibered Calabi-Yau manifolds where the complex structure moduli space contains two kinds of moduli, those from the base and those from the fibration. Defining the bundle with spectral data, we find three types of situations when bundles’ holomorphicity depends on algebraic cycles exist only for special loci in the complex structure moduli, which allows us to stabilize both of these two moduli. We present concrete examples for each type and develop practical tools to analyze the stabilized moduli. Finally, by checking the holomorphicity of the four-flux and/or local Higgs bundle data in F-theory, we briefly study the dual complex structure moduli stabilization scenarios.

The Dual Complex of a Semi-log Canonical Surface

Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta $ of a log canonical pair $(X,\Delta )$ and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,\Delta ),$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $\Delta $.

Topology of tropical moduli of weighted stable curves

The moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.