Quantum Holonomies in (2+1)-Dimensional Gravity

We describe an approach to the quantisation of (2+1)-dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q-commutation relation. Solutions of diagonal and upper-triangular form are constructed, which in the latter case exhibit additional, non-trivial internal relations for each holonomy matrix. Representations are constructed and a group of transformations - a quasi-modular group - which preserves this structure, is presented.Comment: 10 pages Latex no figure

Quantum geometry from 2+1 AdS quantum gravity on the torus

Wilson observables for 2+1 quantum gravity with negative cosmological constant, when the spatial manifold is a torus, exhibit several novel features: signed area phases relate the observables assigned to homotopic loops, and their commutators describe loop intersections, with properties that are not yet fully understood. We describe progress in our study of this bracket, which can be interpreted as a q-deformed Goldman bracket, and provide a geometrical interpretation in terms of a quantum version of Pick's formula for the area of a polygon with integer vertices.Comment: 19 pages, 11 figures, revised with more explanations, improved figures and extra figures. To appear GER

ACL and God’s Call to Librarianship

Many people have asked me how, as a librarian in a secular institution, I became so involved in ACL. I have been at Auburn University Libraries since 1978, but before that was at a Christian liberal arts college, now defunct. ACL has been part of my life since the early 70’s and, of all the professional organizations to which I have belonged, it is the one that has had the greatest effect on me personally

The Christian Periodical Index, ACL’s Longest Service Project

When the Christian Librarians’ Fellowship (CLF) was first formed in 1956, those early librarians were not just focused on their own needs, but they were also concerned with the information needs of students at Christian institutions. Part of their earliest vision was the establishment of the Christian Periodical Index (CPI).And so began a project which continues to the present day. It has far exceeded anyone’s vision at the time if its creation. Who knew about the coming explosion of publishing – to say nothing about the Internet! Yet the mission of CPI – to provide access to English language articles and reviews from an evangelical perspective – has not changed in 50 years, though it has broadened its target group from the original CLF membership

Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

Automatic transversality in contact homology I: Regularity

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, therby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder index under iteration, typically up to large action; see Definition 1.15 These contact forms arise naturally as perturbations of Morse-Bott contact forms such as those associated to $S^1$-bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples.Comment: 68 pages, added more information about bad Reeb orbits, added a proof of a beloved folk theorem concerning the factorization of multiply covered curves, contains expository revisions helpfully suggested by the refere

Localization and Toeplitz Operators on Polyanalytic Fock Spaces

The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $\psi$ and symbol ${\bf a}(x,\omega)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $\Psi$ and symbol $\sigma(z,\bar{z})$ coincides with a {\it Toeplitz} operator with symbol $D\sigma(z,\bar{z})$ for some polynomial differential operator $D$. Using the Berezin quantization approach, we will extend the proof for polyanalytic Fock spaces. While the generation is almost mimetic for two-windowed localization operators, the Gabor analysis framework for vector-valued windows will provide a meaningful generalization of this conjecture for {\it true polyanalytic} Fock spaces and moreover for polyanalytic Fock spaces. Further extensions of this conjecture to certain classes of Gel'fand-Shilov spaces will also be considered {\it a-posteriori}.Comment: 23 page

Autonomous Boat Control Software Design Using Model-Based Systems Engineering

While there is considerable buzz about self-driving cars, self-driving boats are actually more fully developed. The Boat Hardware Control Platform Team was tasked with developing a fleet of small autonomous boats that travel to a destination while avoiding obstacles and staying in formation. The author’s specific task was to develop software used by the boats to detect obstacles and plan a route to a destination. This was done using a method inspired by self-driving cars, which shows promise, but is still being tested at the time of writing. The entire project incorporated model-based systems engineering, which proved to be useful