The Hecke Bicategory

We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid --- the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail.Comment: 27 pages, 11 .eps figures, Major revision of expositio

Comparison of an incline impact table to horizontal impact table as used in representing pallet marshaling

This study compared horizontal impact tables to incline impact tables . The hypothesis is that both tables could achieve the same resultant impact. This study proved that this is possible. However, the initial speed at which the table is set, and the type of programmers used on each table effected the resultant impact. A number of trial runs with setting adjustments were necessary before a three mile-per-hour resultant impact was achieved. Once this setting was achieved to create the proper impact, both tables were within a 5% error when thirty consecutive impacts were produced on each table. Reproducibility was achieved with minimum variation. A future study would be useful in breaking down the effects on impact based on varied duration and g force. This study only used the change in velocity which is a combination of the duration and g level. On any given product, the individual component or the duration or g force or any different combination of the two may have an impact on the product being tested

Higher-Dimensional Algebra VII: Groupoidification

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang-Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field with q elements. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify - or more precisely, groupoidify - the positive part of the quantum group associated to the quiver.Comment: 67 pages, 14 eps figures; uses undertilde.sty. This is an expanded version of arXiv:0812.486

A qualitative analysis of figural memory performance in persons with epilepsy

This study examined nonverbal memory in patients with intractable temporal lobe epilepsy (TLE) on a figural reproduction task, the Rey-Osterrieth Complex Figure (ROCF). The Boston Qualitative Scoring System (BQSS) was used to examine whether qualitative features of ROCF performance could discriminate between those with right and left TLE. As predicted, seizure groups did not differ on a standard quantitative scoring system for the ROCF. Contrary to prediction, the right TLE group did not perform more poorly on BQSS measures of quality or organization, and they did not have greater difficulty recalling the figure after a delay. There was a trend towards poorer performance by the right TLE group on 2 BQSS scales, those quantifying the presence or absence of elements of the figure. ROCF performance was more strongly correlated with measures of visuoperception than with additional measures of nonverbal memory. Thus, the BQSS does not appear to be assessing nonverbal memory, and the implications of the ROCF as a visuoperceptual task are discussed