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We consider the orientational order that arises from the alignment of anisotropic biological cells confined in bounded domains. Especially, it is known that topological defects, where the orientation angles are not well-defined, play an important role in the global pattern formation of cell alignment. Hence, it is important to investigate the relationship between the shape of the region and the location of the defects. The orientational order of cells is theoretically modelled as a harmonic function achieving the minimum elastic energy state of the nematic liquid crystal, and an analytical formula in simply connected domains is given by Miyazako & Nara (Miyazako, Nara 2022 R. Soc. Open Sci. 9, 211663 (doi:10.1098/rsos.211663)). In this paper, we extend this study to construct an analytic formula of cell angle in the doubly connected domains, where topological defects and boundary shapes are represented as the positions of logarithmic singularities and as conformal maps from a given region to a standard annular region, respectively. In addition, using this analytical formula, we propose a numerical algorithm to minimize elastic energy, from which we investigate cell alignment in doubly connected domains with anisotropic quadrilateral boundaries. Our numerical simulations suggest that pairs of topological defects tend to be located around corners of the boundaries, which will be a design principle of the boundary geometry for controlling the defect positions and cell alignment. [ABSTRACT FROM AUTHOR]

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem. [ABSTRACT FROM AUTHOR]

This study proposes an algorithm detecting atmospheric blocking by extracting topological features of geopotential height data at 500 hPa. The algorithm uses topological flow data analysis (TFDA) providing a unique symbolic representation, named the partially cyclically ordered rooted tree (COT) representation, and a discrete graph structure, called a Reeb graph, to each structurally stable Hamiltonian vector field based on the mathematical theory of topological classifications for streamline patterns. It recognizes blocking events more simply and effectively using fewer meteorological parameters than conventional algorithms. Furthermore, the algorithm can determine morphological types of blocking events, an Omega shape or a dipole pattern, whereas no effective algorithm has been available so far. The identified blocking events and their morphological types are consistent with synopticians' subjective judgments. [ABSTRACT FROM AUTHOR]

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyse equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite aspect ratio limit of the stability results known for Kirchhoff's ellipse (Love, Proc. Lond. Math. Soc., vol. s1–25, 1893, pp. 18–43; Mitchell & Rossi, Phys. Fluids, vol. 20, 2008, p. 054103) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite aspect ratio limits of the analogous solutions known for vortex patches. [ABSTRACT FROM AUTHOR]