Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials

Let R denote the reals, and let h: R^n --> R be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such h is representable in the form sup_i inf_j f_{ij}, for some finite collection of polynomials f_{ij} in R[x_1,...,x_n]. (A simple example is h(x_1) = |x_1| = sup{x_1, -x_1}.) In 1984, L. Mahe and, independently, G. Efroymson, proved this for n 2. In this paper we prove an analogous result for "generalized polynomials" (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers; in this version, we restrict to the positive orthant, where each x_i > 0. As before, our methods work only for n < 3.Comment: 16 pages, 4 figure

A quantitative method for maceration of hydra tissue

A method is described for the maceration (dissociation) of hydra tissue into single cells. The cells have characteristic morphology such that all basic types — epithelial, gland, mucous, interstitial, nematoblast, and nerve — can be distinguished. Criteria are given for identifying each cell type by phase contrast microscopy. It is shown that maceration quantitatively recovers cells from hydra tissue

The Principles of Exchange Rate Determination in an International Finance Experiment

This paper reports the first experiments designed to explore the behavior of economies with prominent features of international finance. Two “countries,” each with its own currency, were created. International trade could take place only through the operation of markets for currency. The law of one price and the flow of funds theory of exchange rate determination were used to produce general equilibrium models that captured much of the behavior of the economies. Prices of goods, as well as the exchange rate, evolve over time toward the predictions of the models. However, both the law of one price and purchasing power parity can be rejected for reasons that do not appear in the literature. Patterns of international trade were as predicted by the law of comparative advantage

Ferritin in the fungus Phycomyces

The iron-protein ferritin has been purified from mycelium, sporangiophores, and spores of the fungus Phycomyces blakesleeanus. It has a protein-to-iron ratio of 5, a sedimentation coefficient of 55S, a buoyant density in CsCl of 1.82 g/cm3, and the characteristic morphology of ferritin in the electron microscope. Apoferritin prepared from Phycomyces ferritin has a sedimentation coefficient of 18S and consists of subunits of molecular weight 25,000. In the cytoplasm of Phycomyces, ferritin is located on the surface of lipid droplets (0.5–2.0 µ in diameter) where it forms crystalline monolayers which are conspicuous in electron micrographs of sporangiophore thin-sections. Ferritin is found in all developmental stages of Phycomyces but is concentrated in spores. The level of ferritin iron is regulated by the iron level in the growth medium, a 50-fold increase occurring on iron-supplemented medium

LOSS OF DIFFERENTIATING NEMATOCYTES INDUCED BY REGENERATION AND WOUND HEALING IN HYDRA

Cell death was observed in the nematocyte differentiation pathway in Hydra during head and foot regeneration. This death occurs throughout the regenerating piece, is transient in nature and is selective for committed stenotele and desmoneme precursors. Proliferating nematoblasts are unaffected. Cell death appears to be caused by release of a toxic factor rather than loss of a hormone required for differentiation, since regenerating pieces released a factor that inactivated differentiating nematocytes, and injured animals that had intact head and foot tissue also lost differentiating nematocytes. The inactivated nematocytes are removed by phagocytosis by epitheliomuscular cells