On fully supported eigenfunctions of quantum graphs

We prove that every metric graph which is a tree has an orthonormal sequence of generic Laplace-eigenfunctions, that are eigenfunctions corresponding to eigenvalues of multiplicity one and which have full support. This implies that the number of nodal domains $$\nu _n$$ ν n of the n-th eigenfunction of the Laplacian with standard conditions satisfies $$\nu _n/n \rightarrow 1$$ ν n / n → 1 along a subsequence and has previously only been known in special cases such as mutually rationally dependent or rationally independent side lengths. It shows in particular that the Pleijel nodal domain asymptotics from two- or higher dimensional domains cannot occur on these graphs: Despite their more complicated topology, they still behave as in the one-dimensional case. We prove an analogous result on general metric graphs under the condition that they have at least one Dirichlet vertex. Furthermore, we generalize our results to Delta vertex conditions and to edgewise constant potentials. The main technical contribution is a new expression for a secular function in which modifications to the graph, to vertex conditions, and to the potential are particularly easy to understand.

Nonlinear Elastic Deformation of Mindlin Torus

The nonlinear deformation and stress analysis of a circular torus is a difficult undertaking due to its complicated topology and the variation of the Gauss curvature. A nonlinear deformation (only one term in strain is omitted) of Mindlin torus was formulated in terms of the generalized displacement, and a general Maple code was written for numerical simulations. Numerical investigations show that the results obtained by nonlinear Mindlin, linear Mindlin, nonlinear Kirchhoff-Love, and linear Kirchhoff-Love models are close to each other. The study further reveals that the linear Kirchhoff-Love modeling of the circular torus gives good accuracy and provides assurance that the nonlinear deformation and stress analysis (not dynamics) of a Mindlin torus can be replaced by a simpler formulation, such as a linear Kirchhoff-Love theory of the torus, which has not been reported in the literature.

Continuous, Topologically Guided Protein Crystallization Controls Bacterial Surface Layer Self-Assembly

Bacteria assemble the cell envelope using localized enzymes to account for growth and division of a topologically complicated surface1–3. However, a regulatory pathway has not been identified for assembly and maintenance of the surface layer (S-layer), a 2D crystalline protein coat surrounding the curved 3D surface of a variety of bacteria4,5. By specifically labeling, imaging, and tracking native and purified RsaA, the S-layer protein (SLP) fromC. crescentus, we show that protein self-assembly alone is sufficient to assemble and maintain the S-layerin vivo. By monitoring the location of newly produced S-layer on the surface of living bacteria, we find that S-layer assembly occurs independently of the site of RsaA secretion and that localized production of new cell wall surface area alone is insufficient to explain S-layer assembly patterns. When the cell surface is devoid of a pre-existing S-layer, the location of S-layer assembly depends on the nucleation characteristics of SLP crystals, which grow by capturing RsaA molecules freely diffusing on the outer bacterial surface. Based on these observations, we propose a model of S-layer assembly whereby RsaA monomers are secreted randomly and diffuse on the lipopolysaccharide (LPS) outer membrane until incorporated into growing 2D S-layer crystals. The complicated topology of the cell surface enables formation of defects, gaps, and grain boundaries within the S-layer lattice, thereby guiding the location of S-layer assembly without enzymatic assistance. This unsupervised mechanism poses unique challenges and advantages for designing treatments targeting cell surface structures or utilizing S-layers as self-assembling macromolecular nanomaterials. As an evolutionary driver, 2D protein self-assembly rationalizes the exceptional S-layer subunit sequence and species diversity6.

Selected topics on the topology of ideal fluid flows

This is a survey of certain geometric aspects of inviscid and incompressible fluid flows, which are described by the solutions to the Euler equations. We will review Arnold’s theorem on the topological structure of stationary fluids in compact manifolds, and Moffatt’s theorem on the topological interpretation of helicity in terms of knot invariants. The recent realization theorem by Enciso and Peralta-Salas of vortex lines of arbitrarily complicated topology for stationary solutions to the Euler equations will also be introduced. The aim of this paper is not to provide detailed proofs of all the stated results but to introduce the main ideas and methods behind certain selected topics of the subject known as Topological Fluid Mechanics. This is the set of lecture notes, the author gave at the XXIV International Fall Workshop on Geometry and Physics held in Zaragoza (Spain) during September 2015.

Accuracy analysis of a multi-closed-loop deployable mechanism

The multi-closed-loop deployable mechanism generates distortion which is caused by the deviation of each rod after being assembled as a result of manufacturing error, which leads to the degradation of the antenna performance or excess of the given envelope. The multi-closed-loop deployable mechanisms consist of many closed loops and have complicated topology structure and loop constraints coupling. The positioning accuracy analysis of such mechanism is more difficult than the open-chain mechanisms or the single-loop mechanisms. In order to solve this problem, first, based on the minimum of elastic deformation energy, the distortion analysis model is constructed. Second, the sensitivity of the mechanism distortion to the individual rod deviation is investigated by examining the Lagrange multipliers. By means of this model, the allowable tolerance of multi-closed-loop deployable mechanism is discussed and calculated. Last but not least, the above method and model are applied to the four-closed-loop deployable mechanism to solve the problems of its distortion, sensitivity, and tolerance in both fully deployed and folded configurations. It is demonstrated in this paper that the model and method proposed are well accommodated to the accuracy analysis of such mechanisms. The achievement of the research will help reduce the difficulties of the assembling and manufacturing of the multi-closed-loop deployable mechanisms.

Classical spin dynamics of anisotropic Heisenberg dimmers

In the present work we study the deterministic spin dynamics of two interacting anisotropic magnetic particles in the presence of an external magnetic field using the Landau-Lifshitz equation. The interaction between particles is through the exchange energy. We study both conservative and dissipative cases. In the first one, we characterize the dynamical behavior of the system by monitoring the Lyapunov exponents and bifurcation diagrams. In particular, we explore the dependence of the largest Lyapunov exponent respect to the magnitude of applied magnetic field and exchange constant. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated topology of intricately intermingled chaotic and regular regions. In the dissipative case, we calculate the final saturation states as a function of the magnitude of the applied magnetic field, exchange constant as well as the anisotropy constants.