Exact Lagrangian immersions with one double point revisited

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling of K obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image f(K) of the immersion. This leads to a new and simplified proof of some of the main results of arXiv:1111.5932, which treated Lagrangian immersions in the case n=2k by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.Comment: 39 pages, 2 figures. Version 2: A lengthy appendix now contains a detailed and largely self-contained proof of the existence of a C1-smooth structure on a certain compactified moduli space of Floer disks. The rest of the text is accordingly somewhat re-organised. Version 3: minor further change

Vortex dynamos

We investigate the kinematic dynamo properties of interacting vortex tubes. These flows are of great importance in geophysical and astrophysical fluid dynamics: for a large range of systems, turbulence is dominated by such coherent structures. We obtain a dynamically consistent 2(2)-(1)-dimensional velocity field of the form (u(x, y, t), upsilon(x, y, t), w(x, y, t)) by solving the z-independent Navier-Stokes equations in the presence of helical forcing. This system naturally forms vortex tubes via an inverse cascade. It has chaotic Lagrangian properties and is therefore a candidate for fast dynamo action. The kinematic dynamo properties of the flow are calculated by determining the growth rate of a small-scale seed field. The growth rate is found to have a complicated dependence on Reynolds number Re and magnetic Reynolds number Rm, but the flow continues to act as a dynamo for large Re and Rm. Moreover the dynamo is still efficient even in the limit Re much greater than Rm, providing Rm is large enough, because of the formation of coherent structures

The economic impact of demographic structure in OECD countries

We examine the impact of demographic structure, the proportion of the population in each age group, on growth, savings, investment, hours, interest rates and inflation using a panel VAR estimated from data for 20 OECD economies, mainly for the period 1970-2007. This flexible dynamic structure with interactions among the main macroeconomic variables allows us to estimate long-run effects of demographic structure on the individual countries. Our estimates confirm the importance of these effects

Constructing exact Lagrangian immersions with few double points

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space \R^6_\st with exactly one transverse double point. Our construction also yields a Lagrangian embedding S^1\times S^2\to\R^6_\st with vanishing Maslov class.Comment: In the new version corrected some misprints, added clarifications and filled a small gap in the proof of Lemma 3.

Exact Lagrangian immersions with a single double point

We show that if a closed orientable 2k-manifold K, k > 2, with Euler characteristic χ(K) ≠ -2 admits an exact Lagrangian immersion into C2k with one transverse double point and no other self intersections, then K is diffeomorphic to the sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K

The self-excitation damping ratio: A chatter criterion for time-domain milling simulations

Regenerative chatter is known to be a key factor that limits the productivity of high speed machining. Consequently, a great deal of research has focused on developing predictive models of milling dynamics, to aid engineers involved in both research and manufacturing practice. Time-domain models suffer from being computationally intensive, particularly when they are used to predict the boundary of chatter stability, when a large number of simulation runs are required under different milling conditions. Furthermore, to identify the boundary of stability each simulation must run for sufficient time for the chatter effect to manifest itself in the numerical data, and this is a major contributor to the inefficiency of the chatter prediction process. In the present article, a new chatter criterion is proposed for time-domain milling simulations, that aims to overcome this draw-back by considering the transient response of the modeled behavior, rather than the steady-state response. Using a series of numerical investigations, it is shown that in many cases the new criterion can enable the numerical prediction to be computed more than five times faster than was previously possible. In addition, the analysis yields greater detail concerning the nature of the chatter vibrations, and the degree of stability that is observed

Nearby Lagrangian fibers and Whitney sphere links

Let n>3, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ outside a compact set. Assume that $L$ is disjoint from the cotangent fiber at the origin. The projection of $L$ to the base extends to a map of the $n$-sphere $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.</jats:p