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Articles 1 - 8 of 8
Full-Text Articles in Physical Sciences and Mathematics
Classification Of Book Representations Of K6, Dana Rowland
Classification Of Book Representations Of K6, Dana Rowland
Mathematics Faculty Publications
A book representation of a graph is a particular way of embedding a graph in three dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to K6, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for K6, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of K6 contain between one and seven links, and …
Candy Crush Combinatorics, Dana Rowland
Candy Crush Combinatorics, Dana Rowland
Mathematics Faculty Publications
In the popular game Candy Crush, differently colored candies are arranged in a grid and a player swaps adjacent candies in order to crush them by lining up three or more of the same color. At the beginning of each game, the grid cannot have three consecutive candies of the same color in a row or column, but it must be possible to swap two adjacent candies in order to get at least three consecutive candies of the same color. How many starting configurations are there? We derive recurrence relations to answer this question for a single line of candy, …
Knots In The Canonical Book Representation Of Complete Graphs, Dana Rowland, Andrea Politano
Knots In The Canonical Book Representation Of Complete Graphs, Dana Rowland, Andrea Politano
Mathematics Faculty Publications
We describe which knots can be obtained as cycles in the canonical book representation of the complete graph Kn, and we conjecture that the canonical book representation of Kn attains the least possible number of knotted cycles for any embedding of Kn. The canonical book representation of Kn contains a Hamiltonian cycle that is a composite knot if and only if n ≥12. When p and q are relatively prime, the (p, q) torus knot is a Hamiltonian cycle in the canonical book representation of K2p+q. …
Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson
Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson
Mathematics Faculty Publications
Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/`Z) b o Z/pZ where b is a positive integer and ` is a prime distinct from p. In this paper, we study Galois covers ψ : Z → P 1 k ramified only over ∞ with Galois group G. We find the minimal genus of a curve Z which admits a covering map of this form and we give an explicit formula for this genus in terms of ` and p. The minimal genus occurs when b equals the …
Calculus Students’ Difficulties In Using Variables As Changing Quantities, Susan S. Gray, Barbara J. Loud, Carole Sokolowski
Calculus Students’ Difficulties In Using Variables As Changing Quantities, Susan S. Gray, Barbara J. Loud, Carole Sokolowski
Mathematics Faculty Publications
The study of calculus requires an ability to understand algebraic variables as generalized numbers and as functionally-related quantities. These more advanced uses of variables are indicative of algebraic thinking as opposed to arithmetic thinking. This study reports on entering Calculus I students’ responses to a selection of test questions that required the use of variables in these advanced ways. On average, students’ success rates on these questions were less than 50%. An analysis of errors revealed students’ tendencies toward arithmetic thinking when they attempted to answer questions that required an ability to think of variables as changing quantities, a characteristic …
Mathematics Placement Test: Helping Students Succeed, Norma Rueda, Carole Sokolowski
Mathematics Placement Test: Helping Students Succeed, Norma Rueda, Carole Sokolowski
Mathematics Faculty Publications
A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult …
Classroom Capsules: Additivity ⊕ Homogeneity, Michael J. Bradley, Michael St. Vincent, David L. Finn
Classroom Capsules: Additivity ⊕ Homogeneity, Michael J. Bradley, Michael St. Vincent, David L. Finn
Mathematics Faculty Publications
A Classroom Capsule is a short article that contains a new insight on a topic taught in the earlier years of undergraduate mathematics.
Building Home Plate: Field Of Dreams Or Reality?, Michael J. Bradley
Building Home Plate: Field Of Dreams Or Reality?, Michael J. Bradley
Mathematics Faculty Publications
No abstract provided.