Representations of Rational Cherednik algebras with minimal support and torus knots

We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SL_m. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution by Berest-Etingof-Ginzburg and Gordon, and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras H_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry has a natural interpretation in terms of invariants of torus knots.Comment: 45 pages, latex; the new version contains a new subsection 3.4 on the Cohen-Macaulay property of subspace arrangements and a strengthened version of Theorem 1.

Phonon counting thermometry of an ultracoherent membrane resonator near its motional ground state

Generation of non-Gaussian quantum states of macroscopic mechanical objects is key to a number of challenges in quantum information science, ranging from fundamental tests of decoherence to quantum communication and sensing. Heralded generation of single-phonon states of mechanical motion is an attractive way towards this goal, as it is, in principle, not limited by the object size. Here we demonstrate a technique which allows for generation and detection of a quantum state of motion by phonon counting measurements near the ground state of a 1.5 MHz micromechanical oscillator. We detect scattered photons from a membrane-in-the-middle optomechanical system using an ultra-narrowband optical filter, and perform Raman-ratio thermometry and second-order intensity interferometry near the motional ground state ($\bar{n}=0.23\pm0.02$ phonons). With an effective mass in the nanogram range, our system lends itself for studies of long-lived non-Gaussian motional states with some of the heaviest objects to date.Comment: 11 pages, 10 figure

Equilibration of quantum Hall edge states by an Ohmic contact

Ohmic contacts are crucial elements of electron optics that have not received a clear theoretical description yet. We propose a model of an Ohmic contact as a piece of metal of the finite capacitance $C$ attached to a quantum Hall edge. It is shown that charged quantum Hall edge states may have weak coupling to neutral excitations in an Ohmic contact. Consequently, despite being a reservoir of neutral excitations, an Ohmic contact is not able to efficiently equilibrate edge states if its temperature is smaller than $\hbar\Omega_c$, where $\Omega_c$ is the inverse RC time of the contact. This energy scale for a floating contact may become as large as the single-electron charging energy $e^2/ C$.Comment: 5 pages, 4 figures; revised versio

Touchalytics: On the Applicability of Touchscreen Input as a Behavioral Biometric for Continuous Authentication

We investigate whether a classifier can continuously authenticate users based on the way they interact with the touchscreen of a smart phone. We propose a set of 30 behavioral touch features that can be extracted from raw touchscreen logs and demonstrate that different users populate distinct subspaces of this feature space. In a systematic experiment designed to test how this behavioral pattern exhibits consistency over time, we collected touch data from users interacting with a smart phone using basic navigation maneuvers, i.e., up-down and left-right scrolling. We propose a classification framework that learns the touch behavior of a user during an enrollment phase and is able to accept or reject the current user by monitoring interaction with the touch screen. The classifier achieves a median equal error rate of 0% for intra-session authentication, 2%-3% for inter-session authentication and below 4% when the authentication test was carried out one week after the enrollment phase. While our experimental findings disqualify this method as a standalone authentication mechanism for long-term authentication, it could be implemented as a means to extend screen-lock time or as a part of a multi-modal biometric authentication system.Comment: to appear at IEEE Transactions on Information Forensics & Security; Download data from http://www.mariofrank.net/touchalytics

Complexity and hierarchical game of life

Hierarchical structure is an essential part of complexity, important notion relevant for a wide range of applications ranging from biological population dynamics through robotics to social sciences. In this paper we propose a simple cellular-automata tool for study of hierarchical population dynamics

Modelling landslide debris flow with entrainment: development and validation

The volume and mobility of a debris flow could increase with distance travelled as it has the potential to entrain a substantial amount of channel-bed material along its travel path. This entrainment effect renders the debris flow more devastating to downslope populations and facilities. Over the past two decades, the Geotechnical Engineering Office (GEO) of Hong Kong has expended considerable effort to develop debris mobility modelling tools for use in routine engineering practice for forward prediction purposes. Recently, GEO has completed a study to enhance an in-house debris mobility code. Physical parameters which can be estimated from the field by engineers or geologists are incorporated in the code to predict entrainment effects in a simple and rational manner. This allows the modelling of varying entrainment potential along a debris flow path. The code has been checked against simplified analytical solutions and validated against field observations in a major historical landslide event involving highmobility debris flows in Hong Kong. The numerical modelling results indicated that simulated entrainment volume and mobility characteristics are broadly consistent with geological field mapping records