Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods

The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple

In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [2] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $\Gamma$ in a Hilbert space $H$. We prove the existence and uniqueness of a strong solution of this problem when $\Gamma$ is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when $\Gamma=K_\alpha$, where $K_\alpha={f\in L^2 (0,1)|f\geq -\alpha},\alpha\geq0$

Dynamic Set Intersection

Consider the problem of maintaining a family $F$ of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given $S,S'\in F$, report every member of $S\cap S'$ in any order. We show that in the word RAM model, where $w$ is the word size, given a cap $d$ on the maximum size of any set, we can support set intersection queries in $O(\frac{d}{w/\log^2 w})$ expected time, and updates in $O(\log w)$ expected time. Using this algorithm we can list all $t$ triangles of a graph $G=(V,E)$ in $O(m+\frac{m\alpha}{w/\log^2 w} +t)$ expected time, where $m=|E|$ and $\alpha$ is the arboricity of $G$. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in $O(m \alpha)$ time. We provide an incremental data structure on $F$ that supports intersection {\em witness} queries, where we only need to find {\em one} $e\in S\cap S'$. Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected time, where $N=\sum_{S\in F} |S|$. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using $M$ words of space, each update costs $O(\sqrt {M \log N})$ expected time, each reporting query costs $O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1})$ expected time where $op$ is the size of the output, and each witness query costs $O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N)$ expected time.Comment: Accepted to WADS 201

Spatial learning and memory in the tortoise (Geochelone carbonaria)

A single tortoise (Geochelone carbonaria) was trained in an eight-arm radial maze, with the apparatus and general procedures modeled on those used to demonstrate spatial learning in rats. The tortoise learned to perform reliably above chance, preferentially choosing baited arms, rather than returning to arms previously visited on a trial. Test sessions that examined control by olfactory cues revealed that they did not affect performance. No systematic, stereotyped response patterns were evident. In spite of differences in brain structure, the tortoise showed spatial learning abilities comparable to those observed in mammals

Emergence of Gapped Bulk and Metallic Side Walls in the Zeroth Landau level in Dirac and Weyl semimetals

Recent transport experiments have revealed the activation of longitudinal magnetoresistance of Weyl semimetals in the quantum limit, suggesting the breakdown of chiral anomaly in a strong magnetic field. Here we provide a general mechanism for gapping the zeroth chiral Landau levels applicable for both Dirac and Weyl semimetals. Our result shows that the zeroth Landau levels anticross when the magnetic axis is perpendicular to the Dirac/Weyl node separation and when the inverse magnetic length $l_B^{-1}$ is comparable to the node separation scale $\Delta k$. The induced bulk gap increases rapidly beyond a threshold field in Weyl semimetals, but has no threshold and is non-monotonic in Dirac systems due to the crossover between $l_B^{-1}>\Delta k$ and $l_B^{-1}<\Delta k$ regions. We also find that the Dirac and possibly Weyl systems host counterpropagating edge states between the zeroth Landau levels, leading to a state with metallic side walls and zero Hall conductance.Comment: 8 pages, 4 figure