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We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue $lambda$ of this matrix, the following inequality holds: $frac{|Im lambda |}{|Re lambda |} leq cotfrac{pi}{n}$. This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower bound $sigma_c$ for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: $sigma_c=frac{1}{1-cosleft( 2pi /nright)} $. Numerical analysis revealed that, depending on the network size $n$, multiple dynamic regimes co-exist in the state space of the system. In addition to the fully synchronous state a rotating wave solution occurs. The effect is observed in networks exceeding a certain critical size. The emergence of a rotating wave highlights the importance of long chains and loops in networks of oscillators: the larger the size of chains and loops, the more sensitive the network dynamics becomes to removal or addition of a single connection.
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Fragmentation of filaments into dense cores is thought to be an important step in forming stars. The bar-mode instability of spherically collapsing cores found in previous linear analysis invokes a possibility of re-fragmentation of the cores due to their ellipsoidal (prolate or oblate) deformation. To investigate this possibility, here we perform three-dimensional self-gravitational hydrodynamics simulations that follow all the way from filament fragmentation to subsequent core collapse. We assume the gas is polytropic with index gamma, which determines the stability of the bar-mode. For the case that the fragmentation of isolated hydrostatic filaments is triggered by the most unstable fragmentation mode, we find the bar mode grows as collapse proceeds if gamma < 1.1, in agreement with the linear analysis. However, it takes more than ten orders-of-magnitude increase in the central density for the distortion to become non-linear. In addition to this fiducial case, we also study non-fiducial ones such as the fragmentation is triggered by a fragmentation mode with a longer wavelength and it occurs during radial collapse of filaments and find the distortion rapidly grows. In most of astrophysical applications, the effective polytropic index of collapsing gas exceeds 1.1 before ten orders-of-magnitude increase in the central density. Thus, supposing the fiducial case of filament fragmentation, re-fragmentation of dense cores would not be likely and their final mass would be determined when the filaments fragment.
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Lampros Flokas
, Emmanouil-Vasileios Vlatakis-Gkaragkounis andn Thanasis Lianeas
, Panayotis Mertikopoulos
2020
Understanding the behavior of no-regret dynamics in general $N$-player games is a fundamental question in online learning and game theory. A folk result in the field states that, in finite games, the empirical frequency of play under no-regret learning converges to the games set of coarse correlated equilibria. By contrast, our understanding of how the day-to-day behavior of the dynamics correlates to the games Nash equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this paper, we study the dynamics of follow-the-regularized-leader (FTRL), arguably the most well-studied class of no-regret dynamics, and we establish a sweeping negative result showing that the notion of mixed Nash equilibrium is antithetical to no-regret learning. Specifically, we show that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTRL. This result has significant implications for predicting the outcome of a learning process as it shows unequivocally that only strict (and hence, pure) Nash equilibria can emerge as stable limit points thereof.
Classical turning surfaces of Kohn-Sham potentials, separating classically-allowed regions (CARs) from classically-forbidden regions (CFRs), provide a useful and rigorous approach to understanding many chemical properties of molecules. Here we calculate such surfaces for several paradigmatic solids. Our study of perfect crystals at equilibrium geometries suggests that CFRs are absent in metals, rare in covalent semiconductors, but common in ionic and molecular crystals. A CFR can appear at a monovacancy in a metal. In all materials, CFRs appear or grow as the internuclear distances are uniformly expanded. Calculations with several approximate density functionals and codes confirm these behaviors. A classical picture of conduction suggests that CARs should be connected in metals, and disconnected in wide-gap insulators. This classical picture is confirmed in the limits of extreme uniform compression of the internuclear distances, where all materials become metals without CFRs, and extreme expansion, where all materials become insulators with disconnected and widely-separated CARs around the atoms.
We apply a simple, one-equation, galaxy formation model on top of the halos and subhalos of a high-resolution dark matter cosmological simulation to study how dwarf galaxies acquire their mass and, for better mass resolution, on over 10^5 halo merger trees, to predict when they form their stars. With the first approach, we show that the large majority of galaxies within group- and cluster-mass halos have acquired the bulk of their stellar mass through gas accretion and not via galaxy mergers. We deduce that most dwarf ellipticals are not built up by galaxy mergers. With the second approach, we constrain the star formation histories of dwarfs by requiring that star formation must occur within halos of a minimum circular velocity set by the evolution of the temperature of the IGM, starting before the epoch of reionization. We qualitatively reproduce the downsizing trend of greater ages at greater masses and predict an upsizing trend of greater ages as one proceeds to masses lower than m_crit. We find that the fraction of galaxies with very young stellar populations (more than half the mass formed within the last 1.5 Gyr) is a function of present-day mass in stars and cold gas, which peaks at 0.5% at m_crit=10^6-8 M_Sun, corresponding to blue compact dwarfs such as I Zw 18. We predict that the baryonic mass function of galaxies should not show a maximum at masses above 10^5.5, M_Sun, and we speculate on the nature of the lowest mass galaxies.
The large class of moving boundary processes in the plane modeled by the so-called Laplacian growth, which describes, e.g., solidification, electrodeposition, viscous fingering, bacterial growth, etc., is known to be integrable and to exhibit a large number of exact solutions. In this work, the boundaries are assumed to be in the class of lemniscates with all zeros inside the bounded component of the complex plane. We prove that for any initial boundary taken from this class, the evolving boundary instantly stops being in the class, or else Laplacian growth destroys lemniscates instantly.
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