Trace Identities for the Topological Vertex

The topological vertex is a universal series which can be regarded as an object in combinatorics, representation theory, geometry, or physics. It encodes the combinatorics of 3D partitions, the action of vertex operators on Fock space, the Donaldson-Thomas theory of toric Calabi-Yau threefolds, or the open string partition function of $\mathbb{C}^3$. We prove several identities in which a sum over terms involving the topological vertex is expressed as a closed formula, often a product of simple terms, closely related to Fourier expansions of Jacobi forms. We use purely combinatorial and representation theoretic methods to prove our formulas, but we discuss applications to the Donaldson-Thomas invariants of elliptically fibered Calabi-Yau threefolds at the end of the paper.Comment: 21 pages. Published versio

Mapping the Rainfall Event for Stormwater Quality Control

Stormwater runoff from transportation facilities and urban areas can contain significant concentrations of suspended solids, metals, and oil and grease. In some cases, best management practices (BMPs) are required for treatment of this contaminated runoff. Current Center for Watershed Protection (CWP) guidelines suggest that BMPs be designed to treat 90% of the annual runoff. A survey of state BMP design manuals shows that many states are adopting the 90% runoff guideline. The objective of this study was to determine the daily rainfall depth that should be used for sizing BMPs in Kansas. This report presents two methods for determining this rainfall depth: (a) the 90th percentile daily rainfall and (b) the 90% volume daily rainfall. Records for 623 raingages in and within 100 miles of Kansas were analyzed to determine the design rainfall event using these two methods. Results are presented as contour maps and maps showing the design depths for all Kansas counties

Sparing brain damage in severe sepsis: a beginning

Sepsis-associated encephalopathy (SAE) resembles metabolic encephalopathies but with a difference: there is the potential for enduring brain damage/dysfunction. The pathogenesis of SAE is likely multifactorial. However, the severity of SAE parallels the severity of the septic illness and the brain's microcirculation is probably affected in a similar manner to that of other organs. Mild cases of SAE are often completely reversible, but there is increasing evidence that severe cases have neurological sequelae. A better understanding of the mechanisms may lead to brain-sparing, protective strategies

Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds

We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.Comment: 38 pages, 10 figures. v2:fixed errors in defn 1.1, defn of e_n and h_n, the first displayed equation in the appendix (by Jim Bryan), and the sign for orbifold DT invariants of [C3/Z_3] in Remark A.5. Added example in section 3, updated bibliography, and made slight clarifications and changes in wording throughou

Getting back to basics : bimanual interaction on mobile touch screen devices

The availability, and popularity, of touch screen tablets is drastically increasing with over 30% of internet users now owning one. However the lack of bimanual interaction in touch screen tablets is presenting product designers with serious challenges. Several attempts have been made to facilitate bimanual interaction in such products but results are not comparable to that of their non-mobile cousins, e.g. laptops. This paper presents the finding of a group collaboration aimed at prototyping a mobile touch screen device which supports bimanual interaction during internet browser navigation through rear mounted inputs. The researchers found it problematic to add basic bimanual interactions for internet browser navigation to the rear of a prototype mobile touch screen device due to issues regarding grip type, finger movement and hand position. This paper concludes that in order to achieve bimanual interaction researchers need to return to basics and consider how to free the hand and fingers from current constraints

Non-functional biomimicry : utilising natural patterns to provoke attention responses

Natural reoccurring patterns arise from chaos and are prevalent throughout nature. The formation of these patterns is controlled by, or produces, underlying geometrical structures. Biomimicry is the study of nature’s structure, processes and systems, as models and solutions for design challenges and is being widely utilized in order to address many issues of contemporary engineering. Many academics now believe that aesthetics stem from pattern recognition, consequently, aesthetic preference may be a result of individuals recognising, and interacting with, natural patterns. The goal of this research was to investigate the impact of specific naturally occurring pattern types (spiral, branching, and fractal patterns) on user behaviour; investigating the potential of such patterns to control and influence how individuals interact with their surrounding environment. The results showed that the underlying geometry of natural patterns has the potential to induce attention responses to a statistically significant level