Hofer geometry of a subset of a symplectic manifold

To every closed subset X of a symplectic manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham(X, ω). We equip this group with a semi-norm · X,ω, generalizing the Hofer norm.We discuss Ham(X, ω) and · X,ω if X is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in R2n this diameter is bounded below by π2 , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in R2n, such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π/ k(n, d), where k(n, d) is an explicitly defined integer

Spectral and Hodge theory of `Witt' incomplete cusp edge spaces

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L^2$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$

Time-optimal torus theorem and control of spin systems

Given a compact, connected Lie group G with Lie algebra $$\mathfrak{g}$$ . We discuss time-optimal control of bilinear systems of the form (I) $$\dot U(t) = \left( {H_d + \sum\limits_{j = 1}^m {v_j (t)H_j } } \right)U(t),$$ where H d , H j ∈ $$\mathfrak{g}$$ , U ∈ G, and the v j act as control variables. The case G = SU(2 n ) has found interesting applications to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n-particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the system from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote by the Lie algebra spanned by H 1, ..., H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G/K we discuss in detail time-optimal control strategies for system (I) in the case where G/K carries the structure of a Riemannian symmetric spac

Internet Surveys by Direct Mailing: An Innovative Way of Collecting Data

This article describes a new method of collecting data by direct mailing via the Internet. Feasibility and capacities were evaluated through a worldwide opinion poll on global future risks of mankind and potential solutions. Within 1 day, a structured questionnaire was sent to 8,859 randomly selected e-mail addresses. One thousand seven hundred and thirteen were remailed properly completed, 90 within 4 days. Most respondents were residents of North America (64) and Europe (21 ), male (87), and 30 years old on average. Environmental destruction (52) was mentioned as the primary problem, followed by violence (45) and unemployment (45). Education (71 ) was the most frequently proposed solution to future problems. It is obvious that Internet surveys at this time are not repre sentative of the total population. However, they open new dimensions in the interrogation of experts and opinion leaders, especially considering their efficiency and potential for automation

Phase Measurement of Resonant Two-Photon Ionization in Helium

We study resonant two-color two-photon ionization of Helium via the 1s3p 1P1 state. The first color is the 15th harmonic of a tunable titanium sapphire laser, while the second color is the fundamental laser radiation. Our method uses phase-locked high-order harmonics to determine the {\it phase} of the two-photon process by interferometry. The measurement of the two-photon ionization phase variation as a function of detuning from the resonance and intensity of the dressing field allows us to determine the intensity dependence of the transition energy.Comment: 4 pages, 5 figures, under consideratio

Cardiac magnetic resonance in patients with cardiac resynchronization therapy: is it time to scan with resynchronization on?

Cardiac resynchronization therapy (CRT) is recommended in international guidelines for patients with heart failure due to important left ventricular systolic dysfunction (or heart failure with reduced ejection fraction) and ventricular conduction tissue disease. Cardiac magnetic resonance (CMR) represents the most powerful imaging tool for dynamic assessment of the volumes and function of cardiac chambers but is rarely utilized in patients with CRT due to limitations on the device, programming and scanning. In this review, we explore the known utility of CMR in this cohort with discussion of the risks and potential benefits of scanning whilst CRT is active, including a practical strategy for conducting high quality scans safely. Our contention is that imaging in patients with CRT could be improved further by keeping resynchronization therapy active with resultant benefits on research and also patient outcomes