Bender--Knuth Billiards in Coxeter Groups

Let $(W,S)$ be a Coxeter system, and write $S=\{s_i:i\in I\}$, where $I$ is a finite index set. Fix a nonempty convex subset $\mathscr{L}$ of $W$. If $W$ is of type $A$, then $\mathscr{L}$ is the set of linear extensions of a poset, and there are important Bender--Knuth involutions $\mathrm{BK}_i\colon\mathscr{L}\to\mathscr{L}$ indexed by elements of $I$. For arbitrary $W$ and for each $i\in I$, we introduce an operator $\tau_i\colon W\to W$ (depending on $\mathscr{L}$) that we call a noninvertible Bender--Knuth toggle; this operator restricts to an involution on $\mathscr{L}$ that coincides with $\mathrm{BK}_i$ in type $A$. Given a Coxeter element $c=s_{i_n}\cdots s_{i_1}$, we consider the operator $\mathrm{Pro}_c=\tau_{i_n}\cdots\tau_{i_1}$. We say $W$ is futuristic if for every nonempty finite convex set $\mathscr{L}$, every Coxeter element $c$, and every $u\in W$, there exists an integer $K\geq 0$ such that $\mathrm{Pro}_c^K(u)\in\mathscr{L}$. We prove that finite Coxeter groups, right-angled Coxeter groups, rank-3 Coxeter groups, affine Coxeter groups of types $\widetilde A$ and $\widetilde C$, and Coxeter groups whose Coxeter graphs are complete are all futuristic. When $W$ is finite, we actually prove that if $s_{i_N}\cdots s_{i_1}$ is a reduced expression for the long element of $W$, then $\tau_{i_N}\cdots\tau_{i_1}(W)=\mathscr{L}$; this allows us to determine the smallest integer $\mathrm{M}(c)$ such that $\mathrm{Pro}_c^{{\mathrm{M}}(c)}(W)=\mathscr{L}$ for all $\mathscr{L}$. We also exhibit infinitely many non-futuristic Coxeter groups, including all irreducible affine Coxeter groups that are not of type $\widetilde A$, $\widetilde C$, or $\widetilde G_2$.Comment: 51 pages, 12 figure

Los diez errores principales de los artistas marciales al defenderse de un arma blanca

Existen diez errores mortales que hacen que la mayoría de técnicas de artes marciales a manos desnudas sean ineficaces contra un cuchillo. El principal error se deriva del hecho de que pocos sistemas marciales actuales enseñan el uso de armas blancas. La premisa de este artículo es que para defenderse eficazmente de un cuchillo, usted necesita conocer el uso del mismo. Nuestro propósito es motivar a los instructores de artes marciales a analizar más críticamente sus técnicas de defensa sin armas frente a un arma blanca basándose en una clara comprensión del uso del cuchillo, y volver a despertar el interés por las artes de las armas blancas en la comunidad de artes marciales

Effects of Microbial Processes on Electrolytic and Interfacial Electrical Properties of Unconsolidated Sediments

The effect of microbial processes on electrical properties of unconsolidated sediments was investigated in a laboratory experiment consisting of biotic and abiotic sand columns. The biotic column (nutrient, diesel and bacteria) showed (a) temporal increase in the real, imaginary, and surface conductivity, and (b) temporal decrease in the formation factor. The abiotic columns (nutrient; and nutrient and diesel) showed no significant changes. Increase in microbial population numbers, decrease in organic carbon source, nitrate, and sulfate and increase in dissolved inorganic carbon and fluid conductivity were indicative of microbial activity in the biotic column. We also measure relative increase in the interfacial electrical properties that exceed relative increase in the electrolytic conductivity. Thus changes in the real and imaginary conductivity were induced by microbial processes. These results suggest that interpretation of geoelectrical data from near surface environments should consider effects of microbial processes

Autonomous error correction of a single logical qubit using two transmons

Large-scale quantum computers will inevitably need quantum error correction to protect information against decoherence. Traditional error correction typically requires many qubits, along with high-efficiency error syndrome measurement and real-time feedback. Autonomous quantum error correction (AQEC) instead uses steady-state bath engineering to perform the correction in a hardware-efficient manner. We realize an AQEC scheme, implemented with only two transmon qubits in a 2D scalable architecture, that actively corrects single-photon loss and passively suppresses low-frequency dephasing using six microwave drives. Compared to uncorrected encoding, factors of 2.0, 5.1, and 1.4 improvements are experimentally witnessed for the logical zero, one, and superposition states. Our results show the potential of implementing hardware-efficient AQEC to enhance the reliability of a transmon-based quantum information processor

Charge-Transport Anisotropy in a Uniaxially Aligned Diketopyrrolopyrrole-Based Copolymer.

Aligned films of a semiconducting DPP-based copolymer exhibit highly anisotropic charge transport with a band-like temperature dependence along the alignment direction and hole mobilities of up to 6.7 cm(2) V(-1) s(-1) . X-ray diffraction measurements reveal an exceptional degree of in-plane alignment, high crystallinity, and a dominant face-on orientation of the polymer backbones. The surprising charge-transport properties are interpreted in a tie-chain model consistent with anisotropic activation energies.The authors gratefully acknowledge funding from the Engineering and Physical Sciences Research Council (EPSRC) through a program grant (EP/G060738/1). C.R.M. acknowledges support from the Australia Research Council (FT100100275 and DP130102616). This work was performed in part at the SAXS/WAXS[59] and soft X-ray[60] beam lines at the Australian Synchrotron. C.R.M. and E.G. thank Nigel Kirby of the Australian Synchrotron for technical assistance.This is the final version of the article. It first appeared from Wiley via http://dx.doi.org/10.1002/adma.20150243