Approximate well-supported Nash equilibria in symmetric bimatrix games

The $\varepsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\varepsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\varepsilon$ currently known for which there is a polynomial-time algorithm that computes an $\varepsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2/3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1/2+\delta)$-well-supported Nash equilibrium, for an arbitrarily small positive constant $\delta$

Energy and helicity budgets of solar quiet regions

We investigate the free magnetic energy and relative magnetic helicity budgets of solar quiet regions. Using a novel non-linear force-free method requiring single solar vector magnetograms we calculate the instantaneous free magnetic energy and relative magnetic helicity budgets in 55 quiet-Sun vector magnetograms. As in a previous work on active regions, we construct here for the first time the (free) energy-(relative) helicity diagram of quiet-Sun regions. We find that quiet-Sun regions have no dominant sense of helicity and show monotonic correlations a) between free magnetic energy/relative helicity and magnetic network area and, consequently, b) between free magnetic energy and helicity. Free magnetic energy budgets of quiet-Sun regions represent a rather continuous extension of respective active-region budgets towards lower values, but the corresponding helicity transition is discontinuous due to the incoherence of the helicity sense contrary to active regions. We further estimate the instantaneous free magnetic-energy and relative magnetic-helicity budgets of the entire quiet Sun, as well as the respective budgets over an entire solar cycle. Derived instantaneous free magnetic energy budgets and, to a lesser extent, relative magnetic helicity budgets over the entire quiet Sun are comparable to the respective budgets of a sizeable active region, while total budgets within a solar cycle are found higher than previously reported. Free-energy budgets are comparable to the energy needed to power fine-scale structures residing at the network, such as mottles and spicules

Polylogarithmic Supports are required for Approximate Well-Supported Nash Equilibria below 2/3

In an epsilon-approximate Nash equilibrium, a player can gain at most epsilon in expectation by unilateral deviation. An epsilon well-supported approximate Nash equilibrium has the stronger requirement that every pure strategy used with positive probability must have payoff within epsilon of the best response payoff. Daskalakis, Mehta and Papadimitriou conjectured that every win-lose bimatrix game has a 2/3-well-supported Nash equilibrium that uses supports of cardinality at most three. Indeed, they showed that such an equilibrium will exist subject to the correctness of a graph-theoretic conjecture. Regardless of the correctness of this conjecture, we show that the barrier of a 2/3 payoff guarantee cannot be broken with constant size supports; we construct win-lose games that require supports of cardinality at least Omega((log n)^(1/3)) in any epsilon-well supported equilibrium with epsilon < 2/3. The key tool in showing the validity of the construction is a proof of a bipartite digraph variant of the well-known Caccetta-Haggkvist conjecture. A probabilistic argument shows that there exist epsilon-well-supported equilibria with supports of cardinality O(log n/(epsilon^2)), for any epsilon> 0; thus, the polylogarithmic cardinality bound presented cannot be greatly improved. We also show that for any delta > 0, there exist win-lose games for which no pair of strategies with support sizes at most two is a (1-delta)-well-supported Nash equilibrium. In contrast, every bimatrix game with payoffs in [0,1] has a 1/2-approximate Nash equilibrium where the supports of the players have cardinality at most two.Comment: Added details on related work (footnote 7 expanded

A persistent quiet-Sun small-scale tornado. II. Oscillations

Recently, the characteristics, and dynamics of a persistent 1.7 h vortex flow, resembling a small-scale tornado, have been investigated with ground-base and space-based observations and for the first time in the Ha line centre. The vortex flow showed significant substructure in the form of several intermittent chromospheric swirls. We investigate the oscillatory behaviour of various physical parameters in the vortex area, with a 2D wavelet analysis performed within the vortex flow area and in a quiet-Sun region (for comparison), using the same high spatial and temporal resolution Ha and Ca II 8542 CRISP observations, as well as Doppler velocities and FWHM derived from the Ha line profiles. The vortex flow shows significant oscillatory power in the 3-5 min range that peaks around 4 min and behaves differently than the reference quiet-Sun region. Oscillations reflect the cumulative action of different components such as swaying motions, rotation, and waves. The derived swaying motion periods are in the range of 200-220 s, and the rotation periods are ~270 s for Ha and ~215 s for Ca II. Periods increase with atmospheric height and seem to decrease with radial distance from the vortex centre, suggesting a deviation from a rigid rotation. The behaviour of power within the vortex flow as a function of period and height implies the existence of evanescent waves and the excitation of different types of waves, such as magnetoacoustic (e.g. kink) or Alfven waves. The vortex flow seems to be dominated by two motions: a transverse (swaying) motion, and a rotational motion while oscillations point to the propagation of waves within it. Nearby fibril-like flows could play an important role in the rotational modulation of the vortex flow. Indirect evidence exists that the structure is magnetically supported while the central swirl seems to be acting as a "central engine" to the vortex flow

Chromospheric swirls I. Automated detection in Hα\alpha observations and their statistical properties

Chromospheric swirls are considered to play a significant role in the dynamics and heating of the upper solar atmosphere. It is important to automatically detect and track them in chromospheric observations and determine their properties. We applied a recently developed automated chromospheric swirl detection method to time-series observations of a quiet region of the solar chromosphere obtained in the H$\alpha$-0.2 \r{A} wavelength of the H$\alpha$ spectral line by the CRISP instrument at the Swedish 1-m Solar Telescope. The algorithm exploits the morphological characteristics of swirling events in high contrast chromospheric observations and results in the detection of these structures in each frame of the time series and their tracking over time. We conducted a statistical analysis to determine their various properties, including a survival analysis for deriving the mean lifetime. A mean number of 146 $\pm$ 9 swirls was detected within the FOV at any given time. The mean surface density is found equal to $\sim$0.08 swirls$$Mm$^{-2}$ and the occurrence rate is $\sim$10$^{-2}$ swirls$$Mm$^{-2}$ min$^{-1}$. These values are much higher than those previously reported from chromospheric observations. The radii of the detected swirls range between 0.5 and 2.5 Mm, with a mean value equal to 1.3 $\pm$ 0.3 Mm, which is slightly higher than previous reports. The lifetimes range between 1.5 min and 33.7 min with an arithmetic mean value of $\sim$8.5 min. A survival analysis of the lifetimes, however, using the Kaplan-Meier estimator in combination with a parametric model results in a mean lifetime of 10.3 $\pm$ 0.6 min. An automated method sheds more light on their abundance than visual inspection, while higher cadence, higher resolution observations will most probably result in the detection of a higher number of such features on smaller scales and with shorter lifetimes

State–of–the–art report on nonlinear representation of sources and channels

This report consists of two complementary parts, related to the modeling of two important sources of nonlinearities in a communications system. In the first part, an overview of important past work related to the estimation, compression and processing of sparse data through the use of nonlinear models is provided. In the second part, the current state of the art on the representation of wireless channels in the presence of nonlinearities is summarized. In addition to the characteristics of the nonlinear wireless fading channel, some information is also provided on recent approaches to the sparse representation of such channels

Emergence of small-scale magnetic flux in the quiet Sun

We study the evolution of a small-scale emerging flux region (EFR) in the quiet Sun, from its emergence to its decay. We track processes and phenomena across all atmospheric layers, explore their interrelations and compare our findings with recent numerical modelling studies. We used imaging, spectral and spectropolarimetric observations from space-borne and ground-based instruments. The EFR appears next to the chromospheric network and shows all characteristics predicted by numerical simulations. The total magnetic flux of the EFR exhibits distinct evolutionary phases, namely an initial subtle increase, a fast increase and expansion of the region area, a more gradual increase, and a slow decay. During the initial stages, bright points coalesce, forming clusters of positive- and negative-polarity in a largely bipolar configuration. During the fast expansion, flux tubes make their way to the chromosphere, producing pressure-driven absorption fronts, visible as blueshifted chromospheric features. The connectivity of the quiet-Sun network gradually changes and part of the existing network forms new connections with the EFR. A few minutes after the bipole has reached its maximum magnetic flux, it brightens in soft X-rays forming a coronal bright point, exhibiting episodic brightenings on top of a long smooth increase. These coronal brightenings are also associated with surge-like chromospheric features, which can be attributed to reconnection with adjacent small-scale magnetic fields and the ambient magnetic field. The emergence of magnetic flux even at the smallest scales can be the driver of a series of energetic phenomena visible at various atmospheric heights and temperature regimes. Multi-wavelength observations reveal a wealth of mechanisms which produce diverse observable effects during the different evolutionary stages of these small-scale structures.Comment: Accepted for publication in Astronomy & Astrophysics 14 pages, 14 figure

A Direct Reduction from k-Player to 2-Player Approximate Nash Equilibrium

We present a direct reduction from k-player games to 2-player games that preserves approximate Nash equilibrium. Previously, the computational equivalence of computing approximate Nash equilibrium in k-player and 2-player games was established via an indirect reduction. This included a sequence of works defining the complexity class PPAD, identifying complete problems for this class, showing that computing approximate Nash equilibrium for k-player games is in PPAD, and reducing a PPAD-complete problem to computing approximate Nash equilibrium for 2-player games. Our direct reduction makes no use of the concept of PPAD, thus eliminating some of the difficulties involved in following the known indirect reduction.Comment: 21 page