Generalized Irreducible Divisor Graphs

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $\tau$-factorization. The goal of this paper is to synthesize the notions of $\tau$-factorization and irreducible divisor graphs in domains. We will define a $\tau$-irreducible divisor graph for non-zero non-unit elements of a domain. We show that by studying $\tau$-irreducible divisor graphs, we find equivalent characterizations of several finite $\tau$-factorization properties.Comment: 17 pages, 2 figures, to appear in Communications in Algebr

Generalized U-factorization in Commutative Rings with Zero-Divisors

Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular $\tau$-factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing $\tau$-factorization to rings with zero-divisors by using the notion of U-factorization.Comment: 16 pages, to appear in Rocky Mountain Journal of Mathematic

Tracing technological development trajectories: A genetic knowledge persistence-based main path approach

The aim of this paper is to propose a new method to identify main paths in a technological domain using patent citations. Previous approaches for using main path analysis have greatly improved our understanding of actual technological trajectories but nonetheless have some limitations. They have high potential to miss some dominant patents from the identified main paths; nonetheless, the high network complexity of their main paths makes qualitative tracing of trajectories problematic. The proposed method searches backward and forward paths from the high-persistence patents which are identified based on a standard genetic knowledge persistence algorithm. We tested the new method by applying it to the desalination and the solar photovoltaic domains and compared the results to output from the same domains using a prior method. The empirical results show that the proposed method overcomes the aforementioned drawbacks defining main paths that are almost 10x less complex while containing more of the relevant important knowledge than the main path networks defined by the existing method.Comment: 20 pages, 7 figure

Model-Independent Production of a Top-Philic Resonance at the LHC

We investigate the collider phenomenology of a color-singlet vector resonance, which couples to the heaviest quarks, the top quarks, but very weakly to the rest of the fermions in the Standard Model. We find that the dominant production of such a resonance does not appear at the tree level -- it rather occurs at the one-loop level in association with an extra jet. Signatures like t anti-t plus jets readily emerge as a result of the subsequent decay of the resonance into a pair of top quarks. Without the additional jet, the resonance can still be produced off-shell, which gives a sizeable contribution at low masses. The lower top quark multiplicity of the loop induced resonance production facilitates its reconstruction as compared to the tree level production that gives rise to more exotic signatures involving three or even four top quarks in the final state. For all these cases, we discuss the constraints on the resonance production stemming from recent experimental measurements in the top quark sector. We find that the top-philic vector resonance remains largely unconstrained for the majority of the parameter space, although this will be scrutinized closely in the Run 2 phase of the LHC.Comment: 32 pages, 16 figure

Concordance of knots in S1×S2S^1\times S^2

We establish a number of results about smooth and topological concordance of knots in $S^1\times S^2$. The winding number of a knot in $S^1\times S^2$ is defined to be its class in $H_1(S^1\times S^2;\mathbb{Z})\cong \mathbb{Z}$. We show that there is a unique smooth concordance class of knots with winding number one. This improves the corresponding result of Friedl-Nagel-Orson-Powell in the topological category. We say a knot in $S^1\times S^2$ is slice (resp. topologically slice) if it bounds a smooth (resp. locally flat) disk in $D^2\times S^2$. We show that there are infinitely many topological concordance classes of non-slice knots, and moreover, for any winding number other than $\pm 1$, there are infinitely many topological concordance classes even within the collection of slice knots. Additionally we demonstrate the distinction between the smooth and topological categories by constructing infinite families of slice knots that are topologically but not smoothly concordant, as well as non-slice knots that are topologically slice and topologically concordant, but not smoothly concordant.Comment: 25 pages, 19 figures, final version, to appear in Journal of London Mathematical Societ

Computed tomographic imaging characteristics of the normal canine lacrimal glands.

BackgroundThe canine lacrimal gland (LG) and accessory lacrimal gland of the third eyelid (TEG) are responsible for production of the aqueous portion of the precorneal tear film. Immune-mediated, toxic, neoplastic, or infectious processes can affect the glands directly or can involve adjacent tissues, with secondary gland involvement. Disease affecting these glands can cause keratoconjunctivitis sicca, corneal ulcers, and loss of vision. Due to their location in the orbit, these small structures are difficult to evaluate and measure, making cross-sectional imaging an important diagnostic tool. The detailed cross-sectional imaging appearance of the LG and TEG in dogs using computed tomography (CT) has not been reported to date.ResultsForty-two dogs were imaged, and the length, width, and height were measured and the volume calculated for the LGs & TEGs. The glands were best visualized in contrast-enhanced CT images. The mean volume of the LG was 0.14 cm3 and the TEG was 0.1 cm3. The mean height, width, and length of the LG were, 9.36 mm, 4.29 mm, and 9.35 mm, respectively; the corresponding values for the TEG was 2.02 mm, 9.34 mm, and 7.90 mm. LG and TEG volume were positively correlated with body weight (p < 0.05).ConclusionsContrast-enhanced CT is a valuable tool for noninvasive assessment of canine lacrimal glands