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.

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.

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.

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.

The spectral operator was introduced for the first time by M. L. Lapidus and his collaborator M. van Frankenhuijsen in their theory of complex dimensions in fractal geometry cite,The corresponding inverse spectral problem was first considered by M. L. Lapidus and H. Maier in their work on a spectral reformulation of the Riemann hypothesis in connection with the question "Can One Hear The Shape of a Fractal String?". The spectral operator is defined on a suitable Hilbert space as the operator mapping the counting function of a generalized fractal string to the counting function of its associated spectral measure. It relates the spectrum of a fractal string with its geometry. The inverse spectral problem for vibrating fractal strings studied by M. L. Lapidus and H. Maier has a positive answer if and only if the Riemann zeta function has no zeros on Re(s)=D, where D is in (0,1) is the dimension of the fractal string. In this work, we provide a functional analytic framework allowing us to study the spectral operator. In particular, by determining the spectrum of the spectral operator, we give a necessary and sufficient condition providing its invertibility in the critical strip. We show that such a condition is related to the location of the critical zeroes of the Riemann zeta function or equivalently that the spectral operator is invertible if and only if the Riemann hypothesis is true. As a result, the spectral operator is invertible for any D in (0,1)-{1/2} if and only if the Riemann hypothesis is true.The latter results provides a spectral reformulation of the Riemann hypotesis in terms of a rigorously defined map (the spectral operator). Hence, it sheds new light to the earlier work obtained by M. L. Lapidus and H. Maier and later revisited by M. L. Lapidus and M. van Frankenhuijsen.

The study of fractals and their associated complex dimensions has led to the development of anew form of calculus. In [LvF13], Michel Lapidus and Machiel van Frankenhuijsen introduce two kinds of fractals, ordinary fractal strings and generalized fractal strings. Generalized fractal strings are locally compact measures taking place over the positive real line with mass near zero. In [LvF13], Lapidus and van Frankenhuijsen derive a way of recovering a generalized fractal string from it known complex dimensions via an explicit formula. In this thesis, we offer two key results which involve Taylor series expansions of fractals. The first result involves writing the explicit formula as a Taylor series, summing over the fractional derivatives of δ where the order is taken at the complex dimensions of the generalized fractal string. The second result is specific to more recent work done by Michel Lapidus and Claire David in [DL22a] and [DL22b]. This result involves the determination of coefficients of a fractal power series for the Weierstrass graph, mentioned in [DL22b]. Both results are important as they contribute to the development of fractal calculus. vi