Mathematics Commons^™

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

Mathematics Faculty Works

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

Situating Sotl Within The Disciplines: Mathematics In The United States As A Case Study, Jacqueline Dewar, Curtis Bennett

Mathematics Faculty Works

After two decades of work, many in the SoTL community are pondering the future of the SoTL movement. Will it sustain its influence? Will it continue to attract new participants? What role should the disciplines play? From the perspective of mathematics, this paper examines efforts by the Carnegie Academy and individuals within the mathematical community to build disciplinary support for the scholarship of teaching and learning. The authors, both mathematicians and Carnegie scholars, restrict their observations to the efforts undertaken in the United States during the last decade and examine the situation in mathematics in greater depth than has heretofore …

Spark-Induced Sparks As A Mechanism Of Intracellular Calcium Alternans In Cardiac Myocytes, Robert J. Rovetti, Xiaohua Cui, Alan Garfinkel, James N. Weiss, Zhilin Qu

Mathematics Faculty Works

Rationale: Intracellular calcium (Ca) alternans has been widely studied in cardiac myocytes and tissue, yet the underlying mechanism remains controversial.

Objective: In this study, we used computational modeling and simulation to study how randomly occurring Ca sparks interact collectively to result in whole-cell Ca alternans.

Methods and Results: We developed a spatially-distributed intracellular Ca cycling model in which Ca release units (CRUs) are locally coupled by Ca diffusion throughout the myoplasm and sarcoplasmic reticulum (SR) network. Ca sparks occur randomly in the CRU network when periodically paced with a clamped voltage waveform, but Ca alternans develops as the pacing speeds …

Epimorphisms And Boundary Slopes Of 2–Bridge Knots, Jim Hoste, Patrick D. Shanahan

Mathematics Faculty Works

In this article we study a partial ordering on knots in S³ where K₁≥K₂ if there is an epimorphism from the knot group of K₁ onto the knot group of K₂ which preserves peripheral structure. If K₁ is a 2–bridge knot and K₁≥K₂, then it is known that K₂ must also be 2–bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2–bridge knot K_p∕q, produces infinitely many 2–bridge knots K_p′/q′ with K_p′∕q′≥K_p∕q. After characterizing all 2–bridge knots …

Derived Categories And The Analytic Approach To General Reciprocity Laws. Part Iii, Michael Berg

Mathematics Faculty Works

Building on the scaffolding constructed in the first two articles in this series, we now proceed to the geometric phase of our sheaf (-complex) theoretic quasidualization of Kubota's formalism for n-Hilbert reciprocity. Employing recent work by Bridgeland on stability conditions, we extend our yoga of t-structures situated above diagrams of specifically designed derived categories to arrangements of metric spaces or complex manifolds. This prepares the way for proving n-Hilbert reciprocity by means of singularity analysis.