Low elements in dominant Shi regions

This note is a complement of a recent paper about low elements in affine Coxeter groups. We explain in terms of ad-nilpotent ideals of a Borel subalgebra why the minimal elements of dominant Shi regions are low. We also give a survey of the bijections involved in the study of dominant Shi regions in affine Weyl groups.Comment: 8 page

A symmetric group action on the irreducible components of the Shi variety associated to W(A~n)W(\widetilde{A}_n)

Let $W_a$ be an affine Weyl group with corresponding finite root system $\Phi$. In \cite{JYS1} Jian-Yi Shi characterized each element $w \in W_a$ by a $\Phi^+$-tuple of integers $(k(w,\alpha))_{\alpha \in \Phi^+}$ subject to certain conditions. In \cite{NC1} a new interpretation of the coefficients $k(w,\alpha)$ is given. This description led us to define an affine variety $\widehat{X}_{W_a}$, called the Shi variety of $W_a$, whose integral points are in bijection with $W_a$. It turns out that this variety has more than one irreducible component, and the set of these components, denoted $H^0(\widehat{X}_{W_a})$, admits many interesting properties. In particular the group $W_a$ acts on it. In this article we show that the set of irreducible components of $\widehat{X}_{W(\widetilde{A}_n)}$ is in bijection with the conjugacy class of $(1~2~\cdots~n+1) \in W(A_n) = S_{n+1}$. We also compute the action of $W(A_n)$ on $H^0(\widehat{X}_{W(\widetilde{A}_n)})$.Comment: 18 pages, 5figures, 1 tabl

Tunneling Spectroscopy and Vortex Imaging in Boron-Doped Diamond

We present the first scanning tunneling spectroscopy study of single-crystalline boron doped diamond. The measurements were performed below 100 mK with a low temperature scanning tunneling microscope. The tunneling density of states displays a clear superconducting gap. The temperature evolution of the order parameter follows the weak coupling BCS law with $\Delta(0)/k_B T_c \simeq 1.74$. Vortex imaging at low magnetic field also reveals localized states inside the vortex core that are unexpected for such a dirty superconductor.Comment: 4 pages, 4 figures, replaced with revised versio

Ballistic effects in a proximity induced superconducting diffusive metal

Using a Scanning Tunneling Microscope (STM), we investigate the Local Density of States (LDOS) of artificially fabricated normal metal nano-structures in contact with a superconductor. Very low temperature local spectroscopic measurements (100 mK) reveal the presence of well defined subgap peaks at energy |E|<Delta in the LDOS at various positions of the STM tip. Although no clear correlations between the LDOS and the shape of the samples have emerged, some of the peak features suggest they originate from quasi-particle bound states within the normal metal structures (De Gennes St James states). Refocusing of electronic trajectories induced by the granular srtucture of the samples can explain the observation of spatially uncorrelated interference effects in a non-ballistic medium.Comment: 4 pages, 4 figure

Universal classification of twisted, strained and sheared graphene moir\'e superlattices

Moir\'e superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene's electronic properties. Yet, the exact crystallographic structure on which their band structure depends remains highly debated. In this scanning tunneling microscopy and density functional theory study, we have analysed graphene samples grown on multilayer graphene prepared onto SiC and on the close-packed surfaces of Re and Ir with ultra-high precision. We resolve small-angle twists and shears in graphene, and identify large unit cells comprising more than 1,000 carbon atoms and exhibiting non-trivial nanopatterns for moir\'e superlattices, which are commensurate to the graphene lattice. Finally, a general formalism applicable to any hexagonal moir\'e is presented to classify all reported structures.Comment: 14 pages, 6 figure

Synthesis of geopolymer emulsions

The understanding of emulsion geopolymer synthesis is a major issue for several industrial applications such as the formation of hierarchically porous material for filtration, lightweight materials for civil engineering or even the conditioning of radioactive mineral oil. Emulsion stability (irreversible coarsening, creaming…) are mainly controlled by the interfacial properties (surface tension and nature of the surfactant) and the viscosity ratio between the dispersed (hd) and the continuous phase (hc). The aim of this paper is thus to study model emulsions (composed of hexadecane (C16) as dispersed phase and metakaolin based geopolymer as continuous phase) with the highest volume fraction of C16 as possible. Surfactant was added to the mixture to stabilize the C16 droplets and geopolymer emulsion was synthesis under shear stirring. The influence of the viscosity of the geopolymer paste controlled by the water content was studied and results show that emulsions are unstable for a viscosity ratio hd/hc lower than 0.01. Up to 70% in volume of C16 was incorporated within the geopolymer and hierarchical porous network was thus obtained. Indeed after removal the C16 phase, the porous network was characterized and a specific surface area of 90 m²/g, a mean mesopore diameter of 19 nm, a macropore size distribution ranged between 10-200 µm (fig 1.) and a compression strength of around 0.5 MPa were obtained. Please click Additional Files below to see the full abstract