http://www.rssboard.org/rss-specification 720 https://escholarship.org/uc/ucdavismath_undergraduate/rss Recent eScholarship items from Undergraduate Sun, 9 Aug 2026 05:04:50 +0000 https://escholarship.org/uc/item/9w16t6jg We present a change of basis that may allow more efficient calculation of the volumes of Birkhoff polytopes using a slicing method. We construct the basis from a special set of square matrices. We explain how to construct this basis easily for any Birkhoff polytope, and give examples of its use. We also discuss possible directions for future work. https://escholarship.org/uc/item/9w16t6jg Thu, 22 Feb 2018 00:00:00 +0000 Glynn, Trevor https://escholarship.org/uc/item/8x5458p6 We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves. https://escholarship.org/uc/item/8x5458p6 Tue, 20 Feb 2018 00:00:00 +0000 Chapman, Kevin M. Mulase, Motohico Safnuk, Brad https://escholarship.org/uc/item/5w08g4mg We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath. https://escholarship.org/uc/item/5w08g4mg Thu, 15 Feb 2018 00:00:00 +0000 Hong, Chun Yu Köppe, Matthias Zhou, Yuan https://escholarship.org/uc/item/5fs8k3gm We provide an upper bound on the number of ordered Reidemeister moves required to pass between two diagrams of the same link. This bound is in terms of the number of unordered Reidemeister moves required. https://escholarship.org/uc/item/5fs8k3gm Thu, 15 Feb 2018 00:00:00 +0000 Gold, Julian https://escholarship.org/uc/item/5142f2wk We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software implementation and provide benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory. https://escholarship.org/uc/item/5142f2wk Thu, 15 Feb 2018 00:00:00 +0000 De Loera, Jesus Dutra, Brandon Koeppe, Matthias Moreinis, Stanislav Pinto, Gregory Wu, Jianqiu https://escholarship.org/uc/item/45z9x9z6 We present software for investigations with cut-generating functions in the Gomory--Johnson model and extensions, implemented in the computer algebra system SageMath. https://escholarship.org/uc/item/45z9x9z6 Wed, 14 Feb 2018 00:00:00 +0000 Hong, Chun Yu Köppe, Matthias Zhou, Yuan https://escholarship.org/uc/item/41m27520 This paper presents an elementary proof of Lifschitz tail behavior for random discrete Schr\"{o}dinger operators with a Bernoulli-distributed potential. The proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value. This is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the gap were infinite. https://escholarship.org/uc/item/41m27520 Wed, 14 Feb 2018 00:00:00 +0000 Bishop, Michael Borovyk, Vita Wehr, Jan https://escholarship.org/uc/item/34c043z9 A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh. https://escholarship.org/uc/item/34c043z9 Thu, 1 Feb 2018 00:00:00 +0000 Mulase, Motohico Yu, Josephine T. https://escholarship.org/uc/item/152066g3 A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers. https://escholarship.org/uc/item/152066g3 Wed, 31 Jan 2018 00:00:00 +0000 Mulase, Motohico Yu, Josephine T. https://escholarship.org/uc/item/0ts4r1rd We encode the binomials belonging to the toric ideal $I_A$ associated with an integral $d \times n$ matrix $A$ using a short sum of rational functions as introduced by Barvinok \cite{bar,newbar}. Under the assumption that $d,n$ are fixed, this representation allows us to compute the Graver basis and the reduced Gr\"obner basis of the ideal $I_A$, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate further connections to integer programming and statistics. https://escholarship.org/uc/item/0ts4r1rd Fri, 26 Jan 2018 00:00:00 +0000 De Loera, Jesus Haws, David Hemmecke, Raymond Huggins, Peter Sturmfels, Bernd Yoshida, Ruriko https://escholarship.org/uc/item/0md8p6sm This paper focuses on determining the volumes of permutation polytopes associated to cyclic groups, dihedral groups, groups of automorphisms of tree graphs, and Frobenius groups. We do this through the use of triangulations and the calculation of Ehrhart polynomials. We also present results on the theta body hierarchy of various permutation polytopes. https://escholarship.org/uc/item/0md8p6sm Fri, 26 Jan 2018 00:00:00 +0000 Burggraf, Katherine De Loera, Jesús A. Omar, Mohamed