For each positive rational number ϵ, we define K-theoretic ϵ-stable quasimaps to certain GIT quotients $W\sslash G$. For ϵ>1, this recovers the K-theoretic Gromov-Witten theory of $W\sslash G$ introduced in more general context by Givental and Y.-P. Lee. For arbitrary ϵ1 and ϵ2 in different stability chambers, these K-theoretic quasimap invariants are expected to be related by wall-crossing formulas. We prove wall-crossing formulas for genus zero K-theoretic quasimap theory when the target $W\sslash G$ admits a torus action with isolated fixed points and isolated one-dimensional orbits.

In this addendum note we fill in the gap left in the description of 2D homogeneous solutions to the stationary Euler system initiated in the previous publication. This completes the classification of all homogeneous stationary solutions. The note includes updated classification tables.

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb{Z}^d$, and deduce estimates on the group $e^{i t \Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q (\mathbb{Z}^d)$-weights for suitable $q \geqslant 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d \geqslant 3$ with potentials from $\ell^p (\mathbb{Z}^d)$ with suitable $p \geqslant1$.

By using Yau's Schwarz lemma and the Quillen determinant line bundles, several results about fibered algebraic surfaces and the moduli spaces of curves are improved and reproved.

We study the initial value problem of the thermal-diffusive combustion systemu 1, t= u 1, x, x u 1 u 2 2, u 2, t= du 2, xx+ u 1 u 2 2, x R 1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u 1, u 2) converges to a self-similar solution of the reduced systemu 1, t= u 1, xx u 1 u 2 2, u 2, t= du 2, xx, in the large time limit. In particular, u 1 decays to zero like O (t 1/2 ), where> 0 is an anomalous exponent depending on the initial data, andu 2 decays to zero with normal rate O (t 1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit.