The Indefinite Logarithm, Logarithmic Units, and the Nature of Entropy

We define the indefinite logarithm [log x] of a real number x>0 to be a mathematical object representing the abstract concept of the logarithm of x with an indeterminate base (i.e., not specifically e, 10, 2, or any fixed number). The resulting indefinite logarithmic quantities naturally play a mathematical role that is closely analogous to that of dimensional physical quantities (such as length) in that, although these quantities have no definite interpretation as ordinary numbers, nevertheless the ratio of two of these entities is naturally well-defined as a specific, ordinary number, just like the ratio of two lengths. As a result, indefinite logarithm objects can serve as the basis for logarithmic spaces, which are natural systems of logarithmic units suitable for measuring any quantity defined on a logarithmic scale. We illustrate how logarithmic units provide a convenient language for explaining the complete conceptual unification of the disparate systems of units that are presently used for a variety of quantities that are conventionally considered distinct, such as, in particular, physical entropy and information-theoretic entropy.Comment: Manuscript of a 15 pp. review article. Suggestions for additional appropriate references to relevant prior work are solicited from the communit

Can technology help solve the Arab-Israeli conflict in Palestine?

This repository item contains a single issue of Which Way?, a series of occasional papers published by The Frederick S. Pardee Center for the Study of the Longer-Range Future at Boston University. Which Way? pamphlets highlight emerging controversies at the crossroads where decisions must be made about choices that will affect the future of humankind through the twenty-first century and into the next. They are intended to illuminate, inform, arouse interest, and inspire debate among opinion-molders, decisionmakers, and an informed and thoughtful public.This paper looks at the lack of land in Palestine as one part of the problem that might have a low technology solution if the right pressures were applied. If the Gaza settlements were extended and Israel itself was built out into the eastern Mediterranean, then if a time came when peace was in reach, the struggle for land might not remain quite so desperate an issue. This is modeled after the “Dutch Solution,” in the hopes that their success could likewise be achieved using this obvious yet overlooked idea. Dr. Davidson encourages more practical collaboration between the academic sphere and those in positions to make change. Calling it “the quite unnecessary human tragedy in the Middle East,” he focuses on instances of past cooperation and exchange between the cultures of East and West. The paper also notes how, with technological vision, Death Valley was transformed into one of the most agriculturally productive regions in the United States. Dr. Davidson calls for greater real international support, pointing out the high tariffs of the U.S. and France on exports from North Africa which discourage economic expansion

Inflation Driven by Unification Energy

We examine the hypothesis that inflation is primarily driven by vacuum energy at a scale indicated by gauge coupling unification. Concretely, we consider a class of hybrid inflation models wherein the vacuum energy associated with a grand unified theory condensate provides the dominant energy during inflation, while a second "inflaton" scalar slow-rolls. We show that it is possible to obtain significant tensor-to-scalar ratios while fitting the observed spectral index.Comment: 5 double column pages, 1 figure. V2: Updated to resemble version published in PR

Isometric Tensor Network States in Two Dimensions

Tensor network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm (TEBD$^2$) for approximating the ground state of a Hamiltonian as an isometric TNS, which we demonstrate for the 2D transverse field Ising model.Comment: 5 pages, 4 figure

The Formation of Globules in Planetary Nebulae

We discuss the formation of globules in planetary nebulae, typified by those observed in the Helix Nebula. We show that the properties of the globules, their number, mass, separation, and overall geometry strongly support a scenario in which globules are formed by the fragmentation of a swept-up shell as opposed to models in which the knots form in the AGB wind. We show that the RT or other instabilities which lead to the break-up of shells formed in the nebulae by fast winds or ionization fronts can produce arrays of globules with the overall geometry and within the mass range observed. We also show that the presence of a magnetic field in the circumstellar gas may play an important role in controlling the fragmentation process. Using field strengths measured in the precursor AGB envelopes, we find that close to the central star where the fields are relatively strong, the wavelengths of unstable MRT modes are larger than the shell dimensions, and the fragmentation of the shell is suppressed. The wavelength of the most unstable MRT mode decreases with increasing distance from the star, and when it becomes comparable to the shell thickness, it can lead to the sudden, rapid break-up of an accelerating shell. For typical nebula parameters, the model results in numerous fragments with a mass scale and a separation scale similar to those observed. Our results provide a link between global models of PN shaping in which shells form via winds and ionization fronts, and the formation of small scale structures in the nebulae.Comment: 4 pages, 2 figures, to appear in IAU Symp. 234, Planetary Nebulae in Our Galaxy and Beyond, eds. M. J. Barlow, R. H. Mende