# Your search: "author:"Lapidus, A.""

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## Scholarly Works (302 results)

Brachybacterium faecium Collins et al. 1988 is the type species of the genus, and is of phylogenetic interest because of its location in the Dermabacteraceae, a rather isolated family within the actinobacterial suborder Micrococcineae. B. faecium is known for its rod-coccus growth cycle and the ability to degrade uric acid. It grows aerobically or weakly anaerobically. The strain described in this report is a free-living, nonmotile, Gram-positive bacterium, originally isolated from poultry deep litter. Here we describe the features of this organism, together with the complete genome sequence, and annotation. This is the first complete genome sequence of a member of the actinobacterial family Dermabacteraceae, and the 3,614,992 bp long single replicon genome with its 3129 protein-coding and 69 RNA genes is part of the Genomic Encyclopedia of Bacteria and Archaea project.

This dissertation demonstrates the contingent and contextual nuances of Islamic legal history by balancing precise legal case studies with broad-spectrum jurisprudential surveys. This work places Islamic legal history within diverse late antique (seventh to tenth centuries CE) and medieval (tenth to fifteenth centuries CE) contexts through specific comparisons with rabbinic legal traditions. By delineating intricate legal changes involving several generations of jurists, my research demonstrates the flexibility, expansiveness, and contingency of Islamic legal traditions within a meta-narrative about the transformations of law in the "Near East." I offer a historical understanding of the ambiguous and mutable nature of law and illustrate the complexity of legal pluralism and the struggle for legal-politcal authority that underlies the formation of orthodoxy. This research challenges common reifications of "Islamic law" as an inevitable outcome or a static, monolithic whole.

In crystallography, it was an axiom that any material with a diffraction pattern consisting of sharp spots must have an atomic structure that is a repetition of a unit cell consisting of finitely many atoms, i.e., having translational symmetry in three linearly independent spacial directions. However, the discovery of quasicrystals in 1982 by Dan Shechtman, which defied this law, started an investigation into the question of exactly which atomic structure is necessary to produce a pure point diffraction pattern. In this thesis, we give an extended version of the Poisson summation formula, and use it to address a question by Lapidus and van Frankenhuijsen on whether or not the complex dimensions of a nonlattice self-similar fractal string can be understood as a mathematical analog for a quasicrystal, in the context of a mathematical idealization of diffraction developed by A. Hof. Also, we provide an implementation of the LLL algorithm to give an exploration of the quasiperiodic structure in the nonlattice case.

While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of these spaces. In this dissertation we will focus on the geometry of Ahlfors regular spaces, Metric Measure spaces with an additional regularity condition. Historically, fractals have been studied using different ideas of dimension which have all proven to be unsatisfactory to some degree. The theory of complex dimensions, developed by M.L. Lapidus and a number of collaborators, was developed in part to better understand fractality in the Euclidean case and seeks to overcome these problems. Of particular interest is the recent theory of complex dimensions in higher-dimensional Euclidean spaces, as studied by M.L.Lapidus, G. Radunovic, and D. Zubrinic, who introduced and studied the properties of the distance zeta function $\ze_A$. We will show that this theory of complex dimensions naturally generalizes to the case of Ahlfors regular spaces, as the distance zeta function can be modified to these spaces and all of its main properties carry over. In particular, we will show that we can reconstruct information about the geometry of a subset from their associated distance zeta function through fractal tube formulas. We also provide a selection of examples in Ahlfors spaces, as well as hints that the theory can be expanded to a more general setting.

Fractal sets are sets that show self-similarity meaning that if one zooms in on some part of the fractal, the close up view exhibits the same patterns as the larger whole. Fractals are difficult to study using the usual tools of geometry and analysis; often classical notions from calculus cannot be meaningfully defined on fractals. The study of analysis on fractals seeks to develop analytic tools analogous to those used on ``nice" spaces but that can be used on fractal sets. One can then ask if these fractal tools give results analogous to the results in the classical setting. This text contributes to a new way of thinking about fractals by developing operator algebraic tools that can provide an alternative way of studying geometry and analysis on fractals.

Work in noncommutative fractal geometry involves an operator algebraic tool kit known as a spectral triple which is constructed based on the fractal being studied. Building upon previous works, we give the construction of a spectral triple for the fractal sets known as the stretched Sierpinski gasket and the Harmonic Sierpinski gasket. We show how these spectral triples can be used to describe fractal geometric properties: Hausdorff dimension, geodesic distance, and certain "fractal" measures. We then describe a spectral triple which can be used to describe the standard energy on the Sierpinski gasket.

This thesis investigates the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e., infinite) Sierpinski gasket and a self-similar Sturm--Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, C. Sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables, called the renormalization map. We obtain a factorization of the spectral zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on the complex plane (in the first case) or (in the second case) as a polynomial on the complex projective plane. Our first main result extends to the case of the fractal Laplacian on the unbounded Sierpinski gasket a factorization formula obtained by M. Lapidus for the spectral zeta function of a fractal string and later extended by A. Teplyaev to the bounded (i.e., finite) Sierpinski gasket and some other decimable fractals. Furthermore, our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm--Liouville operators. Moreover, as a corollary, in the very special case when the underlying self-similar measure is Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial on the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev.