Let G = G(p, r) be the Grassmann variety of r-dimensional quotients of k p . Let E be the quotient vector bundle of rank r on G. Note that G is the homogeneous space SL(p)/P, where P is a parabolic subgroup of SL(p), and E corresponds to a representation of P. Any element of G can be written as (y), where y ek rXp is a matrix of rank r. For any areGL(r), we have (y) = (ay). Any element of the fibre of E over (y) can be written as (y, v) where v e k r , so A.M.S. (1980) subject classification: 14F05, 14D20.

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For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder-Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification.In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder-Narasimhan filtration with a given Harder-Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder-Narasimhan type.The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder-Narasimhan type form an algebraic stack in the sense of Artin.

A monolayer of granular spheres in a cylindrical vial, driven continuously by an orbital shaker and subjected to a symmetric confining centrifugal potential, self-organizes to form a distinctively asymmetric structure which occupies only the rear half-space. It is marked by a sharp leading edge at the potential minimum and a curved rear. The area of the structure obeys a power-law scaling with the number of spheres. Imaging shows that the regulation of motion of individual spheres occurs via toggling between two types of motion, namely, rolling and sliding. A low density of weakly frictional rollers congregates near the sharp leading edge whereas a denser rear comprises highly frictional sliders. Experiments further suggest that because the rolling and sliding friction coefficients differ substantially, the spheres acquire a local time-averaged coefficient of friction within a large range of intermediate values in the system. The various sets of spatial and temporal configurations of the rollers and sliders constitute the internal states of the system. Experiments demonstrate and simulations confirm that the global features of the structure are maintained robustly by autotuning of friction through these internal states, providing a previously unidentified route to self-organization of a many-body system.riven, dissipative, and nonlinear many-body systems are capable of self-organizing (1). Examples include avalanches in sand piles (2), earthquakes (3), pinned or depinned disordered systems (4), and animals in collective locomotion patterns (5). The selforganized state in these systems is usually characterized by a macroscopically observable or measurable quantity, for example, the slope of a sand pile or the appearance of large-scale coherent motion of motile objects. The robustness of these selforganized states against external perturbation is achieved through feedback processes which suitably change the internal states of the system. It is thus important to identify the relevant internal states of the system and the feedback mechanism at play to understand the self-organization process itself. Such knowledge is useful, for example, in deciphering the connection between spatial structures and a frequently encountered complex rheology of flowing particulate systems (6), designing adaptive locomotion strategies and decentralized-decision modular robots (7) and swarm robotics (8), and understanding behavioral patterns of active matter (9). For a wide variety of such systems, the basic ingredient of the feedback processes should be, but seldom is, achievable by a set of simple rules via which the individual component responds to a changing environment.Over the last decade, flow properties of moving granular systems, as archetypes of driven dissipative systems, have been extensively studied (10, 11). The emphasis in these studies has been on understanding their flow behavior (10), occasional loss of translational mobility caused by jamming, which leads to self-organization of stress patterns (12, 13), and differenti...