Full blow-up range for co-rotaional wave maps to surfaces of revolution

We construct blow-up solutions of the energy critical wave map equation on $\mathbb{R}^{2+1}\to \mathcal N$ with polynomial blow-up rate ($t^{-1-\nu}$ for blow-up at $t=0$) in the case when $\mathcal{N}$ is a surface of revolution. Here we extend the blow-up range found by C\^arstea ($\nu>\frac 12$) based on the work by Krieger, Schlag and Tataru to $\nu>0$. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere

Optimal polynomial blow up range for critical wave maps

We prove that the critical Wave Maps equation with target $S^2$ and origin $\mathbb{R}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(\lambda(t)r)+\epsilon(t,r)$$where $Q: \mathbb{R}^2 \to S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu > 0$. This extends the work [13], where such solutions were constructed under the assumption $\nu > 1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates

A Universal Constraint on the Infrared Behavior of the Ghost Propagator in QCD

With proposing a unified description of the fields variation at the level of generating functional, we obtain a new identity for the quark-gluon interaction vertex based on gauge symmetry, which is similar to the Slavnov-Taylor Identities(STIs) based on the Becchi-Rouet-Stora-Tyutin transformation. With these identities, we find that in Landau gauge, the dressing function of the ghost propagator approaches to a constant as its momentum goes to zero, which provides a strong constraint on the infrared behaviour of ghost propagator.Comment: 4 pages, no figur

Volatility, valuation ratios, and bubbles: an empirical measure of market sentiment

We define a sentiment indicator based on option prices, valuation ratios, and interest rates. The indicator can be interpreted as a lower bound on the expected growth in fundamentals that a rational investor would have to perceive to be happy to hold the market. The bound was unusually high in the late 1990s, reflecting dividend growth expectations that in our view were unreasonably optimistic. Our approach exploits two key ingredients. First, we derive a new valuation ratio decomposition that is related to the Campbell–Shiller loglinearization but that resembles the Gordon growth model more closely and has certain other advantages. Second, we introduce a volatility index that provides a lower bound on the market's expected log return