Descent c-Wilf Equivalence

Let $S_n$ denote the symmetric group. For any $\sigma \in S_n$, we let $\mathrm{des}(\sigma)$ denote the number of descents of $\sigma$, $\mathrm{inv}(\sigma)$ denote the number of inversions of $\sigma$, and $\mathrm{LRmin}(\sigma)$ denote the number of left-to-right minima of $\sigma$. For any sequence of statistics $\mathrm{stat}_1, \ldots \mathrm{stat}_k$ on permutations, we say two permutations $\alpha$ and $\beta$ in $S_j$ are $(\mathrm{stat}_1, \ldots \mathrm{stat}_k)$-c-Wilf equivalent if the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\alpha$ equals the generating function of $\prod_{i=1}^k x_i^{\mathrm{stat}_i}$ over all permutations which have no consecutive occurrences of $\beta$. We give many examples of pairs of permutations $\alpha$ and $\beta$ in $S_j$ which are $\mathrm{des}$-c-Wilf equivalent, $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent, and $(\mathrm{des},\mathrm{inv},\mathrm{LRmin})$-c-Wilf equivalent. For example, we will show that if $\alpha$ and $\beta$ are minimally overlapping permutations in $S_j$ which start with 1 and end with the same element and $\mathrm{des}(\alpha) = \mathrm{des}(\beta)$ and $\mathrm{inv}(\alpha) = \mathrm{inv}(\beta)$, then $\alpha$ and $\beta$ are $(\mathrm{des},\mathrm{inv})$-c-Wilf equivalent.Comment: arXiv admin note: text overlap with arXiv:1510.0431

Neural activity associated with the passive prediction of ambiguity and risk for aversive events

In economic decision making, outcomes are described in terms of risk (uncertain outcomes with certain probabilities) and ambiguity (uncertain outcomes with uncertain probabilities). Humans are more averse to ambiguity than to risk, with a distinct neural system suggested as mediating this effect. However, there has been no clear disambiguation of activity related to decisions themselves from perceptual processing of ambiguity. In a functional magnetic resonance imaging (fMRI) experiment, we contrasted ambiguity, defined as a lack of information about outcome probabilities, to risk, where outcome probabilities are known, or ignorance, where outcomes are completely unknown and unknowable.Wemodified previously learned pavlovian CSstimuli such that they became an ambiguous cue and contrasted evoked brain activity both with an unmodified predictive CS(risky cue), and a cue that conveyed no information about outcome probabilities (ignorance cue). Compared with risk, ambiguous cues elicited activity in posterior inferior frontal gyrus and posterior parietal cortex during outcome anticipation. Furthermore, a similar set of regions was activated when ambiguous cues were compared with ignorance cues. Thus, regions previously shown to be engaged by decisions about ambiguous rewarding outcomes are also engaged by ambiguous outcome prediction in the context of aversive outcomes. Moreover, activation in these regions was seen even when no actual decision is made. Our findings suggest that these regions subserve a general function of contextual analysis when search for hidden information during outcome anticipation is both necessary and meaningful

Threshold games and cooperation on multiplayer graphs

Objective: The study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured - i.e. when a given individual is only competing with a small subset of the entire population. Method: To optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators - this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named "threshold games", and are defined by a threshold, $M > 0$, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks. Results: When comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model - it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for.Comment: in PLOS ONE, 4th Feb 201

A flight-test methodology for identification of an aerodynamic model for a V/STOL aircraft

Described is a flight test methodology for developing a data base to be used to identify an aerodynamic model of a vertical and short takeoff and landing (V/STOL) fighter aircraft. The aircraft serves as a test bed at Ames for ongoing research in advanced V/STOL control and display concepts. The flight envelope to be modeled includes hover, transition to conventional flight, and back to hover, STOL operation, and normaL cruise. Although the aerodynamic model is highly nonlinear, it has been formulated to be linear in the parameters to be identified. Motivation for the flight test methodology advocated in this paper is based on the choice of a linear least-squares method for model identification. The paper covers elements of the methodology from maneuver design to the completed data base. Major emphasis is placed on the use of state estimation with tracking data to ensure consistency among maneuver variables prior to their entry into the data base. The design and processing of a typical maneuver is illustrated

Flight testing a V/STOL aircraft to identify a full-envelope aerodynamic model

Flight-test techniques are being used to generate a data base for identification of a full-envelope aerodynamic model of a V/STOL fighter aircraft, the YAV-8B Harrier. The flight envelope to be modeled includes hover, transition to conventional flight and back to hover, STOL operation, and normal cruise. Standard V/STOL procedures such as vertical takeoff and landings, and short takeoff and landings are used to gather data in the powered-lift flight regime. Long (3 to 5 min) maneuvers which include a variety of input types are used to obtain large-amplitude control and response excitations. The aircraft is under continuous radar tracking; a laser tracker is used for V/STOL operations near the ground. Tracking data are used with state-estimation techniques to check data consistency and to derive unmeasured variables, for example, angular accelerations. A propulsion model of the YAV-8B's engine and reaction control system is used to isolate aerodynamic forces and moments for model identification. Representative V/STOL flight data are presented. The processing of a typical short takeoff and slow landing maneuver is illustrated

An O(M(n) log n) algorithm for the Jacobi symbol

The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Sch\"onhage's fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) log n) algorithm based on the binary recursive gcd algorithm of Stehl\'e and Zimmermann. Our implementation - which to our knowledge is the first to run in time O(M(n) log n) - is faster than GMP's quadratic implementation for inputs larger than about 10000 decimal digits.Comment: Submitted to ANTS IX (Nancy, July 2010

Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant lambda. We show that the ground state energy is an analytic function of lambda and that the corresponding ground state can also be chosen to be an analytic function of lambda. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground state energy can be calculated using regular analytic perturbation theory

Unbiased flux calibration methods for spectral-line radio observations

Position and frequency switching techniques used for the removal of the bandpass dependence of radio astronomical spectra are presented and discussed in detail. Both methods are widely used, although the frequency dependence of the system temperature and/or noise diode is often neglected. This leads to systematic errors in the calibration that potentially have a significant impact on scientific results, especially when using large-bandwidth receivers or performing statistical analyses. We present methods to derive an unbiased calibration using a noise diode, which is part of many heterodyne receivers. We compare the proposed methods and describe the advantages and bottlenecks of the various approaches. Monte Carlo simulations are used to qualitatively investigate both systematics and the error distribution of the reconstructed flux estimates about the correct flux values for the new methods but also the 'classical' case. Finally, the determination of the frequency-dependent noise temperature of the calibration diode using hot-cold measurements or observations of well-known continuum sources is also briefly discussed.Comment: 25 pages, 30 figures. Accepted for publication in A&