Bott periodicity and stable quantum classes

We use Bott periodicity to relate previously defined quantum classes to certain "exotic Chern classes" on $BU$. This provides an interesting computational and theoretical framework for some Gromov-Witten invariants connected with cohomological field theories. This framework has applications to study of higher dimensional, Hamiltonian rigidity aspects of Hofer geometry of $\mathbb{CP} ^{n}$, one of which we discuss here.Comment: prepublication versio

Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

‘Gobbling drops’: the jetting–dripping transition in flows of polymer solutions

This paper discusses the breakup of capillary jets of dilute polymer solutions and the dynamics associated with the transition from dripping to jetting. High-speed digital video imaging reveals a new scenario of transition and breakup via periodic growth and detachment of large terminal drops. The underlying mechanism is discussed and a basic theory for the mechanism of breakup is also presented. The dynamics of the terminal drop growth and trajectory prove to be governed primarily by mass and momentum balances involving capillary, gravity and inertial forces, whilst the drop detachment event is controlled by the kinetics of the thinning process in the viscoelastic ligaments that connect the drops. This thinning process of the ligaments that are subjected to a constant axial force is driven by surface tension and resisted by the viscoelasticity of the dissolved polymeric molecules. Analysis of this transition provides a new experimental method to probe the rheological properties of solutions when minute concentrations of macromolecules have been added.Schlumberger FoundationMIT Class of 1951 Fellowship Fun

Quantum unsharpness and symplectic rigidity

We discuss a link between "hard" symplectic topology and an unsharpness principle for generalized quantum observables (positive operator valued measures). The link is provided by the Berezin-Toeplitz quantization.Comment: 26 pages, more preliminaries added, changes in the expositio

Interface dynamics in Hele-Shaw flows with centrifugal forces. Preventing cusp singularities with rotation

A class of exact solutions of Hele-Shaw flows without surface tension in a rotating cell is reported. We show that the interplay between injection and rotation modifies drastically the scenario of formation of finite-time cusp singularities. For a subclass of solutions, we show that, for any given initial condition, there exists a critical rotation rate above which cusp formation is prevented. We also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions. This condition admits a simple interpretation related to the linear stability problem.Comment: 4 pages, 2 figure

Symplectic geometry of quantum noise

We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.Comment: Revised version, 57 pages, 3 figures. Incorporates arXiv:1203.234

Morphogenesis of growing soft tissues

Recently, much attention has been given to a noteworthy property of some soft tissues: their ability to grow. Many attempts have been made to model this behaviour in biology, chemistry and physics. Using the theory of finite elasticity, Rodriguez has postulated a multiplicative decomposition of the geometric deformation gradient into a growth-induced part and an elastic one needed to ensure compatibility of the body. In order to fully explore the consequences of this hypothesis, the equations describing thin elastic objects under finite growth are derived. Under appropriate scaling assumptions for the growth rates, the proposed model is of the Foppl-von Karman type. As an illustration, the circumferential growth of a free hyperelastic disk is studied.Comment: 4 pages, 3 figure

A viscoelastic deadly fluid in carnivorous pitcher plants

Background : The carnivorous plants of the genus Nepenthes, widely distributed in the Asian tropics, rely mostly on nutrients derived from arthropods trapped in their pitcher-shaped leaves and digested by their enzymatic fluid. The genus exhibits a great diversity of prey and pitcher forms and its mechanism of trapping has long intrigued scientists. The slippery inner surfaces of the pitchers, which can be waxy or highly wettable, have so far been considered as the key trapping devices. However, the occurrence of species lacking such epidermal specializations but still effective at trapping insects suggests the possible implication of other mechanisms. Methodology/Principal Findings : Using a combination of insect bioassays, high-speed video and rheological measurements, we show that the digestive fluid of Nepenthes rafflesiana is highly viscoelastic and that this physical property is crucial for the retention of insects in its traps. Trapping efficiency is shown to remain strong even when the fluid is highly diluted by water, as long as the elastic relaxation time of the fluid is higher than the typical time scale of insect movements. Conclusions/Significance : This finding challenges the common classification of Nepenthes pitchers as simple passive traps and is of great adaptive significance for these tropical plants, which are often submitted to high rainfalls and variations in fluid concentration. The viscoelastic trap constitutes a cryptic but potentially widespread adaptation of Nepenthes species and could be a homologous trait shared through common ancestry with the sundew (Drosera) flypaper plants. Such large production of a highly viscoelastic biopolymer fluid in permanent pools is nevertheless unique in the plant kingdom and suggests novel applications for pest control