For<i>nn</i> systems of conservation laws in one dimension without source terms, the existence of global weak solutions was proved by Glimm [1]. Glimm constructed approximate solutions using a difference scheme by solving a class of Riemann problems.

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L1norm of potentials is known. We establish the minimization character-ization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough.

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 $p \geqslant1$.

Redirected walking (RDW) aims to reduce the collisions in the physical space for VR applications. However, most of the previous RDW methods do not consider future possibilities of collisions after imperceptibly redirecting users. In this paper, we combine the subtle RDW methods and reset strategy in our method design and propose a novel solution for RDW that can make better use of physical space and trigger fewer resets. The key idea of our method is to discretize the representation of possible user positions and orientations by a series of standard poses and rate them based on the possibilities of hitting obstacles of their reachable poses. A transfer path algorithm is proposed to measure the accessibility among standard poses and is used to support the calculation of the scores of standard poses. Using our method, the user can be redirected imperceptibly to the optimal pose with the best score among all the reachable poses from the user’s current pose during walking. Experiments demonstrate that our method outperforms state-of-the-art methods in various environment sizes and obstacle layouts.

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture [33, 38, 3] and its refinement [34], we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with b+ = 1 in terms of modular forms. This formula was proved earlier in [19] more generally for simply connected 4-manifolds with b+ = 1, as- suming the Kotschick-Morgan conjecture, and it was also derived by physical arguments in [31].