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1. 𝐾-classes of Brill–Noether Loci and a Determinantal Formula

   https://doi.org/10.1093/imrn/rnab025  

   Anderson, Dave ; Chen, Linda ; Tarasca, Nicola ( April 2021 , International Mathematics Research Notices)

   Abstract We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $K$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux. 

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2. A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

   https://doi.org/10.37236/10602  

   Lee, Kyungyong ; Nasr, George D. ; Radcliffe, Jamie ( October 2021 , The Electronic Journal of Combinatorics)

   We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation. 

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3. Schubert Polynomials, the Inhomogeneous TASEP, and Evil-Avoiding Permutations

   https://doi.org/10.1093/imrn/rnac083  

   Kim, Donghyun ; Williams, Lauren K. ( April 2022 , International Mathematics Research Notices)

   Consider a lattice of n sites arranged around a ring, with the $n$ sites occupied by particles of weights $\{1,2,\ldots ,n\}$; the possible arrangements of particles in sites thus correspond to the $n!$ permutations in $S_n$. The inhomogeneous totally asymmetric simple exclusion process (or TASEP) is a Markov chain on $S_n$, in which two adjacent particles of weights $i<j$ swap places at rate $x_i - y_{n+1-j}$ if the particle of weight $j$ is to the right of the particle of weight $i$. (Otherwise, nothing happens.) When $y_i=0$ for all $i$, the stationary distribution was conjecturally linked to Schubert polynomials [18], and explicit formulas for steady state probabilities were subsequently given in terms of multiline queues [4, 5]. In the case of general $y_i$, Cantini [7] showed that $n$ of the $n!$ states have probabilities proportional to double Schubert polynomials. In this paper, we introduce the class of evil-avoiding permutations, which are the permutations avoiding the patterns $2413, 4132, 4213,$ and $3214$. We show that there are $\frac {(2+\sqrt {2})^{n-1}+(2-\sqrt {2})^{n-1}}{2}$ evil-avoiding permutations in $S_n$, and for each evil-avoiding permutation $w$, we give an explicit formula for the steady state probability $\psi _w$ as a product of double Schubert polynomials. (Conjecturally, all other probabilities are proportional to a positive sum of at least two Schubert polynomials.) When $y_i=0$ for all $i$, we give multiline queue formulas for the $\textbf {z}$-deformed steady state probabilities and use this to prove the monomial factor conjecture from [18]. Finally, we show that the Schubert polynomials arising in our formulas are flagged Schur functions, and we give a bijection in this case between multiline queues and semistandard Young tableaux.

    

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4. Fast Barycentric-Based Evaluation Over Spectral/hp Elements

   https://doi.org/10.1007/s10915-021-01750-2  

   Laughton, Edward ; Zala, Vidhi ; Narayan, Akil ; Kirby, Robert M. ; Moxey, David ( January 2022 , Journal of Scientific Computing)

   As the use of spectral/hpelement methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hpelement methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hpelement libraryNektar++by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst$$50\%$$$50\%$slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of$$7\times $$$7\times$. Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to$$30\%$$$30\%$speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around$$6\times $$$6\times$speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires$${\mathcal {O}}(k)$$$O\left(k\right)$storage compared to a best case space complexity of$${\mathcal {O}}(k^2)$$$O\left(k^2\right)$for the Lagrangian interpolation matrix method.

    

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5. Development of a Spatial Visualization Assessment Tool for Younger Students Using a Lego™ Assembly Task

   https://doi.org/10.18260/1-2--34452  

   Delson, Nathan ; Van Den Einde, Lelli ; Tuazon, Jessica ; Yang, Daniel ( June 2020 , 2020 ASEE Virtual Annual Conference Content Access)

   null (Ed.)

   In recent years, it has increasingly been recognized that spatial visualization skills are important in supporting student success in Science, Technology, Engineering, and Math (STEM) education and retention of these students in STEM careers. Many first year college engineering programs and high schools with pre-engineering curriculum have incorporated spatial visualization training into their courses. However, there is no reason why spatial visualization training could not occur at a much younger age, like elementary school. Often at the high school and college level, the Purdue Spatial Visualization Test: Rotations (PSVT:R), which is recognized as a gold standard assessment tool, is used to measure students’ spatial skills learning gains. The PSVT:R is a 20 minute timed test consisting of 30 three-dimensional rotations problems. While the PSVT:R test has been well validated, there are benefits to developing alternative methods of assessing spatial visualization skills suitable for elementary school grades. Researchers at XXX developed an assembly pre- and post- test based upon a timed Lego™ exercise. Students are timed to see how long it would take them to build small Lego shapes using only a picture of the final assembly, but no instructions. The test was implemented at the beginning and then at the end of the quarter/semester. The beauty of this assessment is that it lends itself well to K-12 students. The 20 minute, timed PSVT:R test is too challenging for elementary aged children and is not engaging. In order to validate the new instrument, the Lego™ Assembly test was implemented in a pilot study in 2 college freshman engineering courses using students who could do both the PSVT:R and the Lego Assembly™ assessments. At the beginning of the class all students took the PSVT:R test, and half the students performed a Lego™ assembly of one shape and the other half did the assembly test with another shape. During the class the students completed spatial visualization training which taught them how to sketch orthographic and isometric assignments using a mobile sketching app. At the end of the class the PSVT:R test was repeated for all students. The Lego Assembly™ test was also completed, but the students switched which shape they were building. This approach allowed us to normalize the difficulty of the assembly tasks based upon the average time it took to build the shapes in the pre-test. The Lego Assembly™ test was first implemented in winter and spring of 2018. The data showed a correlation of the R-Squared of 0.11 between the assembly times and the PSVT:R scores in pre-test and 0.14 in post-test. However, analysis of the assembly times indicated that the difficulty of the 2 Lego shapes were significantly different, which could skew the normalization of the assembly times. Accordingly, the test was repeated in winter and spring 2019 with Lego shapes of similar difficulty. This paper describes the results of the assembly tess in all 4 classes, and its suitability for a spatial visualization assessment for elementary school age students. 

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