Asymptotically-good, multigroup decodable space-time block codes

For a family of Space-Time Block Codes (STBCs) C-1, C-2,..., with increasing number of transmit antennas N-i, with rates R-i complex symbols per channel use, i = 1, 2,..., we introduce the notion of asymptotic normalized rate which we define as lim(i ->infinity) R-i/N-i, and we say that a family of STBCs is asymptotically-good if its asymptotic normalized rate is non-zero, i. e., when the rate scales as a non-zero fraction of the number of transmit antennas. An STBC C is said to be g-group decodable, g >= 2, if the information symbols encoded by it can be partitioned into g groups, such that each group of symbols can be ML decoded independently of the others. In this paper we construct full-diversity g-group decodable codes with rates greater than one complex symbol per channel use for all g >= 2. Specifically, we construct delay-optimal, g-group decodable codes for number of transmit antennas N-t that are a multiple of g2left perpendicular(g-1/2)right perpendicular with rate N-t/g2(g-1) + g(2)-g/2N(t). Using these new codes as building blocks, we then construct non-delay-optimal g-group decodable codes with rate roughly g times that of the delay-optimal codes, for number of antennas N-t that are a multiple of 2left perpendicular(g-1/2)right perpendicular, with delay gN(t) and rate Nt/2(g-1) + g-1/2N(t). For each g >= 2, the new delay-optimal and non-delay- optimal families of STBCs are both asymptotically-good, with the latter family having the largest asymptotic normalized rates among all known families of multigroup decodable codes with delay T <= gN(t). Also, for g >= 3, these are the first instances of g-group decodable codes with rates greater than 1 reported in the literature.

Optimum steering input determination and path-tracking of all-wheel steer vehicles on uneven terrains based on constrained optimization

In this paper we propose a framework for optimum steering input determination of all-wheel steer vehicles (AWSV) on rough terrains. The framework computes the steering input which minimizes the tracking error for a given trajectory. Unlike previous methodologies of computing steering inputs of car-like vehicles, the proposed methodology depends explicitly on the vehicle dynamics and can be extended to vehicle having arbitrary number of steering inputs. A fully generic framework has been used to derive the vehicle dynamics and a non-linear programming based constrained optimization approach has been used to compute the steering input considering the instantaneous vehicle dynamics, no-slip and contact constraints of the vehicle. All Wheel steer Vehicles have a special parallel steering ability where the instantaneous centre of rotation (ICR) is at infinity. The proposed framework automatically enables the vehicle to choose between parallel steer and normal operation depending on the error with respect to the desired trajectory. The efficacy of the proposed framework is proved by extensive uneven terrain simulations, for trajectories with continuous or discontinuous velocity profile.

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference Delta U across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L x +/-infinity. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a ``vortex gas'' comprising N point vortices of the same strength (gamma = L Delta U/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32 000 vortices, with ensemble averages evaluated over up to 10(3) realizations and integration over 10(4)L/Delta U. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/Delta U. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via ``violent'' and ``slow'' relaxations P.-H. Chavanis, Physica A 391, 3657 (2012)], over flow time scales of order 10(2) and 10(4)L/Delta U, respectively (for N = 400). The final state involves a single structure that stochastically samples the domain, possibly constituting a ``relative equilibrium.'' The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter lambda close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166 +/- 0.0002 times Delta U. Regime II, extensively studied in the turbulent shear flow literature as a self-similar ``equilibrium'' state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term ``explosive'' as it lasts less than one L/Delta U. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers J. Sommeria, C. Staquet, and R. Robert, J. Fluid Mech. 233, 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

Globally coupled stochastic two-state oscillators: Fluctuations due to finite numbers

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number N of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a 1/N expansion and the Langevin equation via standard methods of Ito calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits N -> infinity and t -> infinity(t is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

A FUNCTIONAL MODEL FOR PURE Gamma-CONTRACTIONS

A pair of commuting operators (S,P) defined on a Hilbert space H for which the closed symmetrized bidisc Gamma = {(z(1) + z(2), z(1)z(2)) : vertical bar z(1)vertical bar <= 1, vertical bar z(2)vertical bar <= 1} subset of C-2 is a spectral set is called a Gamma-contraction in the literature. A Gamma-contraction (S, P) is said to be pure if P is a pure contraction, i.e., P*(n) -> 0 strongly as n -> infinity Here we construct a functional model and produce a set of unitary invariants for a pure Gamma-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation S - S*P = DpXDp, where X is an element of B(D-p), and is called the fundamental operator of the Gamma-contraction (S, P). We also discuss some important properties of the fundamental operator.

Universal power law in crossover from integrability to quantum chaos

We study models of interacting fermions in one dimension to investigate the crossover from integrability to nonintegrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L -> infinity nonintegrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law similar to L-eta when the integrable system is gapless. The exponent eta approximate to 3 appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the nonintegrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.

On Precoding for Constant K-User MIMO Gaussian Interference Channel With Finite Constellation Inputs

This paper considers linear precoding for the constant channel-coefficient K-user MIMO Gaussian interference channel (MIMO GIC) where each transmitter-i (Tx-i) requires the sending of d(i) independent complex symbols per channel use that take values from fixed finite constellations with uniform distribution to receiver-i (Rx-i) for i = 1, 2, ..., K. We define the maximum rate achieved by Tx-i using any linear precoder as the signal-to-noise ratio (SNR) tends to infinity when the interference channel coefficients are zero to be the constellation constrained saturation capacity (CCSC) for Tx-i. We derive a high-SNR approximation for the rate achieved by Tx-i when interference is treated as noise and this rate is given by the mutual information between Tx-i and Rx-i, denoted as I(X) under bar (i); (Y) under bar (i)]. A set of necessary and sufficient conditions on the precoders under which I(X) under bar (i); (Y) under bar (i)] tends to CCSC for Tx-i is derived. Interestingly, the precoders designed for interference alignment (IA) satisfy these necessary and sufficient conditions. Furthermore, we propose gradient-ascentbased algorithms to optimize the sum rate achieved by precoding with finite constellation inputs and treating interference as noise. A simulation study using the proposed algorithms for a three-user MIMO GIC with two antennas at each node with d(i) = 1 for all i and with BPSK and QPSK inputs shows more than 0.1-b/s/Hz gain in the ergodic sum rate over that yielded by precoders obtained from some known IA algorithms at moderate SNRs.

Unusual eigenvalue spectrum and relaxation in the Levy-Ornstein-Uhlenbeck process

We consider the rates of relaxation of a particle in a harmonic well, subject to Levy noise characterized by its Levy index mu. Using the propagator for this Levy-Ornstein-Uhlenbeck process (LOUP), we show that the eigenvalue spectrum of the associated Fokker-Planck operator has the form (n + m mu)nu where nu is the force constant characterizing the well, and n, m is an element of N. If mu is irrational, the eigenvalues are all nondegenerate, but rational mu can lead to degeneracy. The maximum degeneracy is shown to be 2. The left eigenfunctions of the fractional Fokker-Planck operator are very simple while the right eigenfunctions may be obtained from the lowest eigenfunction by a combination of two different step-up operators. Further, we find that the acceptable eigenfunctions should have the asymptotic behavior vertical bar x vertical bar(-n1-n2 mu) as vertical bar x vertical bar -> infinity, with n(1) and n(2) being positive integers, though this condition alone is not enough to identify them uniquely. We also assert that the rates of relaxation of LOUP are determined by the eigenvalues of the associated fractional Fokker-Planck operator and do not depend on the initial state if the moments of the initial distribution are all finite. If the initial distribution has fat tails, for which the higher moments diverge, one can have nonspectral relaxation, as pointed out by Toenjes et al. Phys. Rev. Lett. 110, 150602 (2013)].

DIMENSION FREE BOUNDEDNESS OF RIESZ TRANSFORMS FOR THE GRUSHIN OPERATOR

Let G = -Delta(xi) - vertical bar xi vertical bar(2) partial derivative(2)/partial derivative eta(2) be the Grushin operator on R-n x R. We prove that the Riesz transforms associated to this operator are bounded on L-p(Rn+1), 1 < p < infinity, and their norms are independent of dimension n.

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

The periodic 3D Navier-Stokes equations are analyzed in terms of dimensionless, scaled, L-2m-norms of vorticity D-m (1 <= m <= infinity). The first in this hierarchy, D-1, is the global enstrophy. Three regimes naturally occur in the D-1-D-m plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime 1]; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

BELIEF PROPAGATION FOR OPTIMAL EDGE COVER IN THE RANDOM COMPLETE GRAPH

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wastlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the (2) limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

Flexibility in flapping foil suppresses meandering of induced jet in absence of free stream

Thrust-generating flapping foils are known to produce jets inclined to the free stream at high Strouhal numbers St = fA/U-infinity, where f is the frequency and A is the amplitude of flapping and U-infinity is the free-stream velocity. Our experiments, in the limiting case of St —> infinity (zero free-stream speed), show that a purely oscillatory pitching motion of a chordwise flexible foil produces a coherent jet composed of a reverse Benard-Karman vortex street along the centreline, albeit over a specific range of effective flap stiffnesses. We obtain flexibility by attaching a thin flap to the trailing edge of a rigid NACA0015 foil; length of flap is 0.79 c where c is rigid foil chord length. It is the time-varying deflections of the flexible flap that suppress the meandering found in the jets produced by a pitching rigid foil for zero free-stream condition. Recent experiments (Marais et al., J. Fluid Mech., vol. 710, 2012, p. 659) have also shown that the flexibility increases the St at which non-deflected jets are obtained. Analysing the near-wake vortex dynamics from flow visualization and particle image velocimetry (PIV) measurements, we identify the mechanisms by which flexibility suppresses jet deflection and meandering. A convenient characterization of flap deformation, caused by fluid-flap interaction, is through a non-dimensional effective stiffness', EI* = 8 EI/(rho V-TEmax(2) s(f) c(f)(3)/2), representing the inverse of the flap deflection due to the fluid-dynamic loading; here, EI is the bending stiffness of flap, rho is fluid density, V-TEmax is the maximum velocity of rigid foil trailing edge, s(f) is span and c(f) is chord length of the flexible flap. By varying the amplitude and frequency of pitching, we obtain a variation in EI* over nearly two orders of magnitude and show that only moderate EI*. (0.1 less than or similar to EI * less than or similar to 1 generates a sustained, coherent, orderly jet. Relatively `stiff' flaps (EI* greater than or similar to 1), including the extreme case of no flap, produce meandering jets, whereas highly `flexible' flaps (EI* less than or similar to 0.1) produce spread-out jets. Obtained from the measured mean velocity fields, we present values of thrust coefficients for the cases for which orderly jets are observed.

Transition from dissipative to conservative dynamics in equations of hydrodynamics

We show, by using direct numerical simulations and theory, how, by increasing the order of dissipativity (alpha) in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case alpha -> infinity U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of alpha greater than a crossover value alpha(crossover). We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows.

Strings vs. spins on the null orbifold

We study the null orbifold singularity in 2+1 d flat space higher spin theory as well as string theory. Using the Chern-Simons formulation of 2+1 d Einstein gravity, we first observe that despite the singular nature of this geometry, the eigenvalues of its Chern-Simons holonomy are trivial. Next, we construct a resolution of the singularity in higher spin theory: a Kundt spacetime with vanishing scalar curvature invariants. We also point out that the UV divergences previously observed in the 2-to-2 tachyon tree level string amplitude on the null orbifold do not arise in the at alpha' -> infinity limit. We find all the divergences of the amplitude and demonstrate that the ones remaining in the tensionless limit are physical IR-type divergences. We conclude with a discussion on the meaning and limitations of higher spin (cosmological) singularity resolution and its potential connection to string theory.

Belief propagation for minimum weight many-to-one matchings in the random complete graph

In a complete bipartite graph with vertex sets of cardinalities n and n', assign random weights from exponential distribution with mean 1, independently to each edge. We show that, as n -> infinity, with n' = n/alpha] for any fixed alpha > 1, the minimum weight of many-to-one matchings converges to a constant (depending on alpha). Many-to-one matching arises as an optimization step in an algorithm for genome sequencing and as a measure of distance between finite sets. We prove that a belief propagation (BP) algorithm converges asymptotically to the optimal solution. We use the objective method of Aldous to prove our results. We build on previous works on minimum weight matching and minimum weight edge cover problems to extend the objective method and to further the applicability of belief propagation to random combinatorial optimization problems.