Regeneration in rhizostoma pulmo

The several experiments, of which tlhs paper presets a resume, were conducted during the early summer of 1903, at the Naples Zoölogical Station, while occupying the table of the Smithsonian Institution, for the courtesy of which it is a pleasure to express my obligations. The primary object of the experiments was to test the regenerative capacity of the Scyphomedusae and to institute certain comparisons between these results and those obtained by similar experiments previously made upon the Hydromedusae. So far as I am aware no similar experiments have been made upon the Scyphomedusae with the definite purpose of testing this particular aspect: of their physiological constitution. Romanes in his experiments upon "Primitive Nervous Systems", 85, has recorded incidentally the fact that certain mutilations of medusae are promptly healed, but gave no details. Eimer, '78, has also carried on similar experiments and with the Same general purpose of testing the character and distribution of nervous centers, but makes no reference to the matter of regeneration. And quite recently Uexküll, has likewise reviewed these experiments of Romanes and Eimer and carried them sornewhat farther than they had done. But while arrivirlg at somewhat different conclusions, drawn from a series of experiments in some features coincident with those to be described now, he makes no reference to any regenerative processes, devoting attentioti almost exclusively to the movements, specially those of rhytmic character, and seeking physical explanations of them. The earlier references of Haeckel to the capacity of larvae of certain medusae to regenerate entire organisms are likewise indefinite. Morgan in referring to the subject in his recent book on "Regeneration", merely remarks that among Scyphozoa "the jelly-fishes belonging to this group have a limited amount of regenerative power". I very much regret that an unusual scarcity of material compels me to leave several points somewhat less fully considered than is desirable, but I trust they are not of suffiicient gravity to seriously mar the general value of the results as a whole. In one respect this scarcity of material, making necessary successive experiments on the same specimen in many cases, proved fortunate rather than otherwise, since facts of importance were thus brought to light which might otherwise have been overlooked. Some of these will be referred to specifically in another connection

Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to \infty$. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n \to \infty$. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.Comment: 12 pages, 2 figures, acroread recommended for figure displa

Apollonian structure in the Abelian sandpile

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.Comment: 27 Pages, 7 Figure

A Survey: What Elementary Teachers Recommend About Their Multi-Purpose Rooms

The present study was undertaken (1) to compile objective evidence of current uses of multi-purpose rooms; (2) to obtain recommendations of elementary teachers concerning the utilization of their multi-purpose rooms