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~~xmx S. Bhupatiraju, P. Etingof, D. Jordan, W. Kuszmaul, and J. Li, Lower central series of a free associative algebra over the integers and finite fields, J. Algebra 372(2012), 251-274. L. M. Braswell and T. Khovanova, On the Cookie Monster Problem, arXiv: 1309 .5985. --, Cookie Monster devours naccis, College Mathematics Journal 45 (2014), no. 2. Y. Chen, P. Etingof, D. Jordan, and M. Zhang, Poisson traces in positive characteristic, arXiv: 1112.6385. C. Chen, Maximizing volume ratios for shadow covering by tetrahedra, arXiv: 1201.2580. C. Chen, T. Khovanova,and D. A. Klain, Volume bounds for shadow covering, Trans. Amer. Math. Soc. 366 (2014), 1161-1177. S. Devadas and S. Sam, Representations of rational Cherednik algebras of G(m,r,n) in positive characteristic, J. Commut. Algebra (online). F. Ding and A. Tsymbaliuk, Representations of infinitesimal Cherednik algebras, Repre­sent Theory (electronic) 17 (2013), 557-583. C. Lian, Representations of Cherednik algebras associated to symmetric and dihedral groups in positive characteristic, arXiv:l207.0182. Yael Fregier and Isaac Xia, Lower Central Series Ideal Quotients Over fp and Z, arXiv: 1506.08469. N. Golowich, Resolving a Conjecture on degree of regularity of linear homogeneous equations, Electro. J. Combin. 21 (2014), no. 3. R. Jagadeesan, A new Gal (Q/Q)-invariant of dessins d' enfants, arXiv: 1403. 7690. Shashwat Kishore and Augustus Lonergan, Signatures of Multiplicity Spaces in tensor

products of s{z and Uq(slz) Representations, arXiv: 1506.02680. T. Khovanova and J. Xiong, Cookie Monster plays games, arXiv: 1407.1533. T.-Y. Jiang, Z. Scully, and Y. X. Zhang, Motors and impossible firing patterns in the parallel chipfiring game, SIAM J. Discrete Math. 29 (2015), 615-630. J. Zurier, Generalizations of the Joints Problem, math. mi t. edu/research/high­school/ primes/materials/2©14/Zurier.pdf.

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