Fractal Strings and Multifractal Zeta Functions

For a Borel measure on the unit interval and a sequence of scales that tend to zero, we define a one-parameter family of zeta functions called multifractal zeta functions. These functions are a first attempt to associate a zeta function to certain multifractal measures. However, we primarily show that they associate a new zeta function, the topological zeta function, to a fractal string in order to take into account the topology of its fractal boundary. This expands upon the geometric information garnered by the traditional geometric zeta function of a fractal string in the theory of complex dimensions. In particular, one can distinguish between a fractal string whose boundary is the classical Cantor set, and one whose boundary has a single limit point but has the same sequence of lengths as the complement of the Cantor set. Later work will address related, but somewhat different, approaches to multifractals themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and figures illustrating the main results of this paper and recent results from others. Sections 0, 2, and 6 have been significantly rewritte

Deployment and testing of a second prototype expandable surgical chamber in microgravity

During microgravity exposure, two separate expandable surgical chambers were tested. Both chambers had been modified to fit the microgravity work station without extending over the sides of the table. Both chambers were attached to a portable laminar flow generator which served two purposes: to keep the chambers expanded during use; and to provide an operative area environment free of contamination. During the tests, the chambers were placed on various parts of a total body moulage to simulate management of several types of trauma. The tests consisted of cleansing contusions, debridement of burns, and suturing of lacerations. Also, indigo carmine dye was deliberately injected into the chamber during the tests to determine the ease of cleansing the chamber walls after contamination by escaping fluids. Upon completion of the tests, the expandable surgical chambers were deflated, folded, and placed in a flattened state back into their original containers for storage and later disposal. Results are briefly discussed

Multifractal analysis via scaling zeta functions and recursive structure of lattice strings

The multifractal structure underlying a self-similar measure stems directly from the weighted self-similar system (or weighted iterated function system) which is used to construct the measure. This follows much in the way that the dimension of a self-similar set, be it the Hausdorff, Minkowski, or similarity dimension, is determined by the scaling ratios of the corresponding self-similar system via Moran's theorem. The multifractal structure allows for our definition of scaling regularity and scaling zeta functions motivated by geometric zeta functions and, in particular, partition zeta functions. Some of the results of this paper consolidate and partially extend the results regarding a multifractal analysis for certain self-similar measures supported on compact subsets of a Euclidean space based on partition zeta functions. Specifically, scaling zeta functions generalize partition zeta functions when the choice of the family of partitions is given by the natural family of partitions determined by the self-similar system in question. Moreover, in certain cases, self-similar measures can be shown to exhibit lattice or nonlattice structure with respect to specified scaling regularity values. Additionally, in the context provided by generalized fractal strings viewed as measures, we define generalized self-similar strings, allowing for the examination of many of the results presented here in a specific overarching context and for a connection to the results regarding the corresponding complex dimensions as roots of Dirichlet polynomials. Furthermore, generalized lattice strings and recursive strings are defined and shown to be very closely related.Comment: 33 pages, no figures, in pres

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of R which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.National Science Foundation (U.S.) (Grant DMS–1247679