Capacities from the Chiu-Tamarkin complex

In this paper, we construct a sequence $(c_k)_{k\in\mathbb{N}}$ of symplectic capacities based on the Chiu-Tamarkin complex $C_{T,\ell}$, a $\mathbb{Z}/\ell$-equivariant invariant coming from the microlocal theory of sheaves. We compute $(c_k)_{k\in\mathbb{N}}$ for convex toric domains, which are the same as the Gutt-Hutchings capacities. Our method also works for the prequantized contact manifold $T^*X\times S^1$. We define a sequence of "contact capacities" $([c]_k)_{k\in\mathbb{N}}$ on the prequantized contact manifold $T^*X\times S^1$, and we compute them for prequantized convex toric domains.Comment: Major revision. We add a discussion on the $S^1$-action at the end of Subsection 0.3. More details are provided in Subsection 1.4, Section 2 and Section 4. We rewrite Section 3. Comments are welcome

A Note on Homoclinic Orbits for Second Order Hamiltonian Systems

In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the existence of homoclinc orbits

Forced Oscillations of the Korteweg-de Vries Equation on a Bounded Domain and their Stability

It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Kortewg-de Vries (KdV) equation and the damped Benjamin-Bona-Mahony (BBM) equation. In this paper we intend to show the same results hold for the pure KdV equation (without the damping terms) posed on a bounded domain. Consideration is given to the initial-boundary-value problem uuxuxxx 0 \u3c x \u3c 1, t \u3e 0, (*) It is shown that if the boundary forcing is periodic with small amplitude, then the small amplitude solution of (*) becomes eventually time-periodic. Viewing (*) (without the initial condition ) as an infinite-dimensional dynamical system in the Hilbert space , we also demonstrate that for a given periodic boundary forcing with small amplitude, the system (*) admits a (locally) unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations

A hepatitis E outbreak by genotype 4 virus in Shandong province, China

AbstractHepatitis E vaccine was available in China in 2012, but the priority population for immunization is not clear. In 2013, a hepatitis E outbreak occurred in a company of Shandong province, China where most employees moved from other provinces and dined at the company’s cafeteria. A total of fourteen (19%, 14/73) case-patients were identified, and three of them had symptomatic infection with one death. The proportion of symptomatic infection was much higher among those aged ⩾50years than those aged <50years (2/2 vs. 1/12, P=0.03), and higher in males than females (3/8 vs. 0/6, P=0.21). Food in the company’s cafeteria might be the possible source of the outbreak. The findings from this outbreak investigation indicate that individuals aged ⩾50years, particularly males, might be the population of top priority for hepatitis E vaccination in China

Contact non-squeezing at large scale via generating functions

Using SFT techniques, Eliashberg, Kim and Polterovich (2006) proved that if $\pi R_2^2 \leq K \leq \pi R_1^2$ for some integer $K$ then there is no contact squeezing in $\mathbb{R}^{2n} \times S^1$ of the prequantization of the ball of radius $R_1$ into the prequantization of the ball of radius $R_2$. This result was extended to the case of balls of radius $R_1$ and $R_2$ with $1 \leq \pi R_2^2 \leq \pi R_1^2$ by Chiu (2017) and the first author (2016), using respectively microlocal sheaves and SFT. In the present article we recover this general contact non-squeezing theorem using generating functions, a classical method based on finite dimensional Morse theory. More precisely, we develop an equivariant version, with respect to a certain action of a finite cyclic group, of the generating function homology for domains of $\mathbb{R}^{2n} \times S^1$ defined by the second author (2011). A key role in the construction is played by translated chains of contactomorphisms, a generalization of translated points.Comment: 32 page