Symbolic calculus on the time-frequency half-plane

The study concerns a special symbolic calculus of interest for signal analysis. This calculus associates functions on the time-frequency half-plane f>0 with linear operators defined on the positive-frequency signals. Full attention is given to its construction which is entirely based on the study of the affine group in a simple and direct way. The correspondence rule is detailed and the associated Wigner function is given. Formulas expressing the basic operation (star-bracket) of the Lie algebra of symbols, which is isomorphic to that of the operators, are obtained. In addition, it is shown that the resulting calculus is covariant under a three-parameter group which contains the affine group as subgroup. This observation is the starting point of an investigation leading to a whole class of symbolic calculi which can be considered as modifications of the original one.Comment: 25 pages, Latex, minor changes and more references; to be published in the "Journal of Mathematical Physics" (special issue on "Wavelet and Time-Frequency Analysis"

Macroscopic Discontinuous Shear Thickening vs Local Shear Jamming in Cornstarch

We study the emergence of discontinuous shear-thickening (DST) in cornstarch, by combining macroscopic rheometry with local Magnetic Resonance Imaging (MRI) measurements. We bring evidence that macroscopic DST is observed only when the flow separates into a low-density flowing and a high-density jammed region. In the shear-thickened steady state, the local rheology in the flowing region, is not DST but, strikingly, is often shear-thinning. Our data thus show that the stress jump measured during DST, in cornstach, does not capture a secondary, high-viscosity branch of the local steady rheology, but results from the existence of a shear jamming limit at volume fractions quite significantly below random close packing.Comment: To be published in PR

Influence of electromagnetic interferences on the gravimetric sensitivity of surface acoustic waveguides

Surface acoustic waveguides are increasing in interest for (bio)chemical detection. The surface mass modification leads to measurable changes in the propagation properties of the waveguide. Among a wide variety of waveguides, Love mode has been investigated because of its high gravimetric sensitivity. The acoustic signal launched and detected in the waveguide by electrical transducers is accompanied by an electromagnetic wave; the interaction of the two signals, easily enhanced by the open structure of the sensor, creates interference patterns in the transfer function of the sensor. The influence of these interferences on the gravimetric sensitivity is presented, whereby the structure of the entire sensor is modelled. We show that electromagnetic interferences generate an error in the experimental value of the sensitivity. This error is different for the open and the closed loop configurations of the sensor. The theoretical approach is completed by the experimentation of an actual Love mode sensor operated under liquid in open loop configuration. The experiment indicates that the interaction depends on the frequency and the mass modifications.Comment: 28 pages, 8 figure

Absorption of water and lubricating oils into porous nylon

Oil and water absorption from air into sintered porous nylon can be described by infiltration into the pores of the material. This process can be modeled by a diffusion-like mechanism. For water absorption, we find a formal diffusion coefficient of 1.5 x 10(exp -4)sq cm/min when the nylon is initially dry. The diffusion coefficient is 4 x 10(exp -6)sq cm/min when the nylon is oil-impregnated prior to air exposure. In a 52% RH atmosphere, dry nylon absorbs 3% w/w water, and oil-impregnated nylon absorbs 0.6% w/w water. For oil absorption there are three steps: (1) surface absorption and infiltration into (2) larger and (3) smaller pores. Surface absorption is too fast to be measured in these experiments. The diffusion coefficient for the second step is 6 x 10(exp -4)sq cm/min for SRG-60 oil into dry nylon and 4 x 10(exp -4)sq cm/min for air-equilibrated nylon. The diffusion coefficient for the third step is about 1 x 10(exp -6)sq cm/min for both cases. The total amount of oil absorbed is 31% w/w. The interaction between water and nylon is not as strong as that between water and cotton-phenolic: oil can replace water, and only a small amount of water can enter previously oil-impregnated nylon

Probably Safe or Live

This paper presents a formal characterisation of safety and liveness properties \`a la Alpern and Schneider for fully probabilistic systems. As for the classical setting, it is established that any (probabilistic tree) property is equivalent to a conjunction of a safety and liveness property. A simple algorithm is provided to obtain such property decomposition for flat probabilistic CTL (PCTL). A safe fragment of PCTL is identified that provides a sound and complete characterisation of safety properties. For liveness properties, we provide two PCTL fragments, a sound and a complete one. We show that safety properties only have finite counterexamples, whereas liveness properties have none. We compare our characterisation for qualitative properties with the one for branching time properties by Manolios and Trefler, and present sound and complete PCTL fragments for characterising the notions of strong safety and absolute liveness coined by Sistla

Saddle-splay modulus of a particle-laden fluid interface

The scaled-particle theory equation of state for the two-dimensional hard-disk fluid on a curved surface is proposed and used to determine the saddle-splay modulus of a particle-laden fluid interface. The resulting contribution to saddle-splay modulus, which is caused by thermal motion of the adsorbed particles, is comparable in magnitude with the saddle-splay modulus of a simple fluid interface.Comment: 10 pages, 2 figure

On the transcendence degree of the differential field generated by Siegel modular forms

It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.Comment: 21 pages, AmSTeX, uses picture.sty for 1 LaTeX picture; submitted for publicatio

Duality and KPZ in Liouville Quantum Gravity

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma <2, and h_\epsilon(z) denotes the mean value on the circle of radius \epsilon centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case \gamma >2 is shown to be related to the quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality \gamma\gamma'=4.Comment: 4 pages, 1 figur

A Quantitative Measure of Interference

We introduce an interference measure which allows to quantify the amount of interference present in any physical process that maps an initial density matrix to a final density matrix. In particular, the interference measure enables one to monitor the amount of interference generated in each step of a quantum algorithm. We show that a Hadamard gate acting on a single qubit is a basic building block for interference generation and realizes one bit of interference, an ``i-bit''. We use the interference measure to quantify interference for various examples, including Grover's search algorithm and Shor's factorization algorithm. We distinguish between ``potentially available'' and ``actually used'' interference, and show that for both algorithms the potentially available interference is exponentially large. However, the amount of interference actually used in Grover's algorithm is only about 3 i-bits and asymptotically independent of the number of qubits, while Shor's algorithm indeed uses an exponential amount of interference.Comment: 13 pages of latex; research done at http://www.quantware.ups-tlse.fr