Balanced Islands in Two Colored Point Sets in the Plane

Let $S$ be a set of $n$ points in general position in the plane, $r$ of which are red and $b$ of which are blue. In this paper we prove that there exist: for every $\alpha \in \left [ 0,\frac{1}{2} \right ]$, a convex set containing exactly $\lceil \alpha r\rceil$ red points and exactly $\lceil \alpha b \rceil$ blue points of $S$; a convex set containing exactly $\left \lceil \frac{r+1}{2}\right \rceil$ red points and exactly $\left \lceil \frac{b+1}{2}\right \rceil$ blue points of $S$. Furthermore, we present polynomial time algorithms to find these convex sets. In the first case we provide an $O(n^4)$ time algorithm and an $O(n^2\log n)$ time algorithm in the second case. Finally, if $\lceil \alpha r\rceil+\lceil \alpha b\rceil$ is small, that is, not much larger than $\frac{1}{3}n$, we improve the running time to $O(n \log n)$

Area Law Violations and Quantum Phase Transitions in Modified Motzkin Walk Spin Chains

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here we consider a Motzkin walk with a different Hilbert space on each step of the walk spanned by elements of a {\it Symmetric Inverse Semigroup} with the direction of each step governed by its algebraic structure. This change alters the number of paths allowed in the Motzkin walk and introduces a ground state degeneracy sensitive to boundary perturbations. We study the frustration-free spin chains based on three symmetric inverse semigroups, \cS^3_1, \cS^3_2 and \cS^2_1. The system based on \cS^3_1 and \cS^3_2 provide examples of quantum phase transitions in one dimensions with the former exhibiting a transition between the area law and a logarithmic violation of the area law and the latter providing an example of transition from logarithmic scaling to a square root scaling in the system size, mimicking a colored \cS^3_1 system. The system with \cS^2_1 is much simpler and produces states that continue to obey the area law.Comment: 40 pages, 14 figures, A condensed version of this paper has been submitted to the Proceedings of the 2017 Granada Seminar on Computational Physics, Contains minor revisions and is closer to the Journal version. v3 includes an addendum that modifies the final Hamiltonian but does not change the main results of the pape

How metal films de-wet substrates - identifying the kinetic pathways and energetic driving forces

We study how single-crystal chromium films of uniform thickness on W(110) substrates are converted to arrays of three-dimensional (3D) Cr islands during annealing. We use low-energy electron microscopy (LEEM) to directly observe a kinetic pathway that produces trenches that expose the wetting layer. Adjacent film steps move simultaneously uphill and downhill relative to the staircase of atomic steps on the substrate. This step motion thickens the film regions where steps advance. Where film steps retract, the film thins, eventually exposing the stable wetting layer. Since our analysis shows that thick Cr films have a lattice constant close to bulk Cr, we propose that surface and interface stress provide a possible driving force for the observed morphological instability. Atomistic simulations and analytic elastic models show that surface and interface stress can cause a dependence of film energy on thickness that leads to an instability to simultaneous thinning and thickening. We observe that de-wetting is also initiated at bunches of substrate steps in two other systems, Ag/W(110) and Ag/Ru(0001). We additionally describe how Cr films are converted into patterns of unidirectional stripes as the trenches that expose the wetting layer lengthen along the W[001] direction. Finally, we observe how 3D Cr islands form directly during film growth at elevated temperature. The Cr mesas (wedges) form as Cr film steps advance down the staircase of substrate steps, another example of the critical role that substrate steps play in 3D island formation

A Decade of Short-Period Earthquake Rupture Histories From Multi-Array Back-Projection

Teleseismic back-projection imaging has emerged as a powerful tool for understanding the rupture propagation of large earthquakes. However, its application often suffers from artifacts related to the receiver array geometry. We developed a teleseismic back-projection technique that can accommodate data from multiple arrays. Combined processing of P and pP waveforms may further improve the resolution. The method is suitable for defining arrays ad-hoc to achieve a good azimuthal distribution for most earthquakes. We present a catalog of short-period rupture histories (0.5–2.0 Hz) for all earthquakes from 2010 to 2022 with MW ≥ 7.5 and depth less than 200 km (56 events). The method provides automatic estimates of rupture length, directivity, speed, and aspect ratio, a proxy for rupture complexity. We obtained short-period rupture length scaling relations that are in good agreement with previously published relations based on estimates of total slip. Rupture speeds were consistently in the sub-Rayleigh regime for thrust and normal earthquakes, whereas a tenth of strike-slip events propagated at supershear speeds. Many rupture histories exhibited complex behaviors, for example, rupture on conjugate faults, bilateral propagation, and dynamic triggering by a P wave. For megathrust earthquakes, ruptures encircling asperities were frequently observed, with downdip, updip, and balanced patterns. Although there is a preference for short-period emissions to emanate from central and downdip parts of the megathrust, emissions updip of the main asperity are more frequent than suggested by earlier results

Global correlations between maximum magnitudes of subduction zone interface thrust earthquakes and physical parameters of subduction zones

Peer reviewedPublisher PD

Canonical bases and higher representation theory

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case.Comment: 55 pages; DVI may not compile correctly, PDF is preferred. v2: added section on dual canonical bases. v3: improved exposition in line with new version of 1309.3796. v4: final version, to appear in Compositio Mathematica. v5: corrected references for proof of Theorem 4.