The Prism tableau model for Schubert polynomials

The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1,x2,...]. We suggest the "prism tableau model" for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Groebner geometry of matrix Schubert varieties.Comment: 23 page

Study of passive optical techniques for detecting clear air turbulence

Passive optical techniques evaluated for detecting clear air turbulence

The influence of reviewers\u27 characteristics on their evaluations of instructional technology integration

The purpose of this study was to determine what effect the personal characteristics of reviewers might have on their reviews. The participants in this study were nine instructional leaders who used a rubric to evaluate P-12 teachers\u27 web-based instructional units, but still scored them differently.;The characteristics examined in this study included the reviewers\u27 styles of teaching, personal computer use, concerns about instructional technology integration, professional positions, perceived roles, and personal criteria for what makes a good web-based instructional unit.;The study employed both quantitative and qualitative research methodologies. The quantitative instruments used were the Stages of Concern Questionnaire (SoCQ), the Principles of Adult learning Scale (PALS), and the Survey of Computer Use (SCU). The qualitative methods used were interviews of the instructional leaders and coding and compiling of comments made by the instructional leaders when they were evaluating the units.;The results of this study indicate that the reviewers who had teaching styles that were more learner-centered were inclined to pass more units than their teacher-centered counterparts. The reviewers who had the lowest concerns overall about technology passed less units than those with higher technology concerns. Reviewers who were not in P-12 teaching positions passed more units, while the reviewer whose job it was to teach technology in the public school system passed the least. In addition, the group identified as advanced appeared to be judged more stringently, and the group identified as beginners appeared to be evaluated more generously. Other factors were less clear; however, when interviewed, the reviewers mentioned various factors that may have influenced their scoring, including their perceptions of their roles and their personal feelings about the participants.;The results of this study imply that reviewers, even when using a rubric, will still incorporate their own opinions and expertise into their reviews. When choosing and assigning reviewers, characteristics that might have an effect should be kept in mind and controlled as much as possible by having at least three reviewers for each product being evaluated and by balancing groups of reviewers by assigning reviewers with similar characteristics to different groups

Bumpless pipe dreams encode Gr\"obner geometry of Schubert polynomials

In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the TxT-equivariant cohomology classes of matrix Schubert varieties and of their flat degenerations. We give diagonal term orders with respect to which bumpless pipe dreams index the irreducible components of diagonal Gr\"obner degenerations of matrix Schubert varieties, counted with scheme-theoretic multiplicity. This indexing was conjectured by Hamaker, Pechenik, and Weigandt (2022). We also give a generalization to equidimensional unions of matrix Schubert varieties. This result establishes that bumpless pipe dreams are dual to and as geometrically natural as classical pipe dreams, for which an analogous anti-diagonal theory was developed by Knutson and Miller (2005).Comment: In the original, we made an error in a claimed reduction from arbitrary diagonal term orders. We now only address term orders compatible with droop moves. The case of an arbitrary diagonal term order remains open. We are very grateful to Kuei-Nuan Lin and Yi-Huang Shen for alerting us to a flipped inequality in Lemma 5.10. The proof has been amended. Other more minor improvements throughou