Symmetric polynomials are polynomials that are invariant under the action of the symmetric group, and they play an integral role in mathematics. The space of quasiin- variant polynomials, polynomials that are invariant under the action of the symmetric group to a certain order, were introduced by Feigin and Veselov. These spaces are modules over the ring of symmetric polynomials, and their Hilbert series in elds of characteristic 0 were also computed by Feigin and Veselov. In this paper, we study the Hilbert series of these spaces in elds of positive char- acteristic. Braverman, Etingof, and Finkelberg recently introduced spaces of twisted quasiinvariant polynomials, a generalization of quasiinvariant polynomials in which the space is twisted by a monomial. We extend some of their results to spaces twisted by a product of smooth functions and compute the Hilbert series of the space in certain cases.

Because of its importance in number theory and singularity theory, the problem of nding a polynomial sharp upper estimate of the number of positive integral points in an n- dimensional (n  3) polyhedron has received attention by a lot of mathematicians. S. S.-T. Yau proposed the upper estimate, so-called the Yau Number Theoretic Conjecture. The previous results on the Yau Number Theoretic Conjecture in low dimension cases (n  6) have been proved by using the sharp GLY conjecture. Unfortunately, it is only valid in low dimension. The Yau Number Theoretic Conjecture for n = 7 has been shown with a completely new method in [19]. In this paper, the similar method has been applied to prove the Yau Number Theoretic Conjecture for n = 8, but with more meticulous analyses. The main method of proof is summing existing sharp upper bounds for the number of points in 7-dimensional simplexes over the cross sections of eight-dimensional simplex. This reasearch project paves the way for the proof of a fully general sharp upper bound for the number of lattice points in a simplex. It also moves the mathematical community one step closer towards proving the Yau Number Theoretic Conjecture in full generality. As an application, we give a sharper estimate of the Dickman-De Bruijn function (x; y) for 5  y < 23, compared with the result obtained by Ennola.

Polyhedral combinatorics has been a topic of interest in modern day’s computational geometry. The founding of Steinitz’s Theorem in 1922 revealed consequential relations between graph theory and polyhedral combinatorics. It allows us to better investigate on the topology of convex polyhedrons. In this paper, we proposed an algorithm that generates a unique sequence of points, using the vertices of a triangulated polyhedron, pre-determined by the selection of the starting 3 vertices in the sequence. Following that, we discover an interesting relation between the sequence and the volume of the polyhedron itself, in which we presented in the form of a sufficient condition. To further investigate which polyhedrons generate sequences that satisfy the sufficient condition, we study the problem in the context of graph theory, that is, the explorer walk (corresponding to the sequence of vertices) in maximal planar graphs (skeletons of triangulated convex polyhedrons). With that, we uncovered a family of maximal planar graphs, called the explorer graphs, which exhibits volumetric properties in the polyhedrons constructed from them, in regard to the explorer walk. In this paper, we also introduce generalized methods of constructing explorer graphs of higher order from explorer graphs of lower order, demonstrating the prevalence of explorer graphs. As the edges of a maximal planar graph is of great importance in tracing an explorer walk, we investigate on the line graph of maximal planar graphs, and re-establish a better definition of explorer graphs. Lastly, our paper covers the edge contraction of explorer graphs, which allows us to solve the volume of polyhedrons constructed from non-explorer graphs. For this, we presented a possible bound for the minimum number of edge contractions a non-explorer graph requires from an explorer graph. This will generalize the proposed method of finding volumes to any triangulated convex polyhedron.

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Xingdu is an emerging mathematical game about searching for a corresponding polyline for a given polyline in a 2-dimensional or 3-dimensional mesh grid, meanwhile the searched polyline should have the same start and end points as those of the given polyline and the corresponding line segments on the searched and given polylines are perpendicular each other respectively. Studying from mathematical perspective the problem solving and designing of Xingdu is helpful of finding a general method for Xingdu problem solving, but still not attract much more attentions due to the relatively short history of Xingdu. Referring to the achievements in mathematical methods for Sudoku, we investigated preliminarily the possibility of using optimization theory to solve the problem of Xingdu. According to the definition of Xingdu, the points on the given polyline and the solution polyline have to be the grid nodes in the mesh grid so that their coordinates are integer. In addition, there should not be any points appearing repeatedly and any three sequential points on the solution polyline cannot be located on a same line. Due to these constraints, the optimization problem related to Xingdu belongs to integer programming with nonlinear constraints, which is very difficult and usually can only be solved by enumerating. For this reason, we solve the related problem first by neglecting the nonlinear constraints so that it can be converted as an integer programming with linear constraints only. Afterwards, we can check whether it fulfill the related nonlinear constraints if a feasible solution is obtained. Based on that, we proposed two integer programming models for Xingdu problem, which are the model using only line segments perpendicular property as constraints and the binary integer programming model using line segments perpendicular property together with no repeated points as constraints. The test results show that the proposed binary integer programming model has a very good performance in finding the solution of Xingdu problem. We also discussed the possibility of using the proposed binary integer programming model together with random simulation to design a Xingdu problem. Due to the challenge of discriminating the uniqueness of Xingdu solution, we finally presented an approach, inspired from the branch and bound method for integer programming, which solves and designs the Xingdu problem by constructing and pruning a search tree.