Diependaal, Duifhuis, Hoogstraten and Viergever investigated three time-integration methods to solve a simplified one-dimensional model of the human cochlea. Two of these time-integration methods are dealt with in this paper, namely fourth-order Runge–Kutta and modified Sielecki. The stability of these two methods is examined, both theoretically and experimentally. This leads to the conclusion that in the case of the fourth-order Runge–Kutta method, a bigger time step can be used in comparison to the modified Sielecki method. This corresponds with the conclusion drawn in the article by Diependaal, Duifhuis, Hoogstraten and Viergever.

In this paper we extend the Kauffman bracket to singular links. Specifically, we define a polynomial invariant for singular links, and in doing this, we consider three approaches to our extended Kauffman bracket polynomial: (1) using skein relations involving singular link diagrams, (2) using representations of the singular braid monoid, (3) via a Yang–Baxter state model. We also study some properties of the extended Kauffman bracket.

McDuff and Schlenk recently determined exactly when a four-dimensional symplectic ellipsoid symplectically embeds into a symplectic ball. Similarly, Frenkel and Müller recently determined exactly when a symplectic ellipsoid symplectically embeds into a symplectic cube. Symplectic embeddings of more complicated sets, however, remain mostly unexplored. We study when a symplectic ellipsoid $E\left(a,b\right)$ symplectically embeds into a polydisc $P\left(c,d\right)$. We prove that there exists a constant $C$ depending only on $d∕c$ (here, $d$ is assumed greater than $c$) such that if $b∕a$ is greater than $C$, then the only obstruction to symplectically embedding $E\left(a,b\right)$ into $P\left(c,d\right)$ is the volume obstruction. We also conjecture exactly when an ellipsoid embeds into a scaling of $P\left(1,b\right)$ for $b\ge 6$, and conjecture about the set of $\left(a,b\right)$ such that the only obstruction to embedding $E\left(1,a\right)$ into a scaling of $P\left(1,b\right)$ is the volume. Finally, we verify our conjecture for $b=\frac{13}{2}$.

The question of whether a closed Riemannian manifold has infinitely many geometrically distinct closed geodesics has a long history. Though unsolved in general, it is well understood in the case of surfaces. For surfaces of revolution diffeomorphic to the sphere, a refinement of this problem was introduced by Borzellino, Jordan-Squire, Petrics, and Sullivan. In this article, we quantify their result by counting distinct geodesics of bounded length. In addition, we reframe these results to obtain a couple of characterizations of the round two-sphere.

For a topological space $X$, it is a natural undertaking to compare its topology with the weak topology generated by a family of real-valued continuous functions on $X$. We present a necessary and sufficient condition for the coincidence of these topologies for an arbitrary family $\mathcal{A}\subset C\left(X\right)$. As a corollary, we give a new proof of the fact that families of functions which separate points on a compact space induce topologies that coincide with the original topology.

We say a permutation $\pi ={\pi }_1{\pi }_2\cdots {\pi }_n$ in the symmetric group ${\mathfrak{S}}_n$ has a peak at index $i$ if ${\pi }_{i-1}<{\pi }_i>{\pi }_{i+1}$ and we let $P\left(\pi \right)=\left\{i\in \left\{1,2,\dots ,n\right\}\mid i\text{ is a peak of }\pi \right\}$. Given a set $S$ of positive integers, we let $P\left(S;n\right)$ denote the subset of ${\mathfrak{S}}_n$ consisting of all permutations $\pi$ where $P\left(\pi \right)=S$. In 2013, Billey, Burdzy, and Sagan proved $\left|P\left(S;n\right)\right|=p\left(n\right)2^{n-\left|S\right|-1}$, where $p\left(n\right)$ is a polynomial of degree $max\left(S\right)-1$. In 2014, Castro-Velez et al. considered the Coxeter group of type $B_n$ as the group of signed permutations on $n$ letters and showed that $\left|P_B\left(S;n\right)\right|=p\left(n\right)2^{2n-\left|S\right|-1}$, where $p\left(n\right)$ is the same polynomial of degree $max\left(S\right)-1$. In this paper we partition the sets $P\left(S;n\right)\subset {\mathfrak{S}}_n$ studied by Billey, Burdzy, and Sagan into subsets of permutations that end with an ascent to a fixed integer $k$ (or a descent to a fixed integer $k$) and provide polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie types $C_n$ and $D_n$ into ${\mathfrak{S}}_{2n}$, we partition these groups into bundles of permutations ${\pi }_1{\pi }_2\cdots {\pi }_n|{\pi }_{n+1}\cdots {\pi }_{2n}$ such that ${\pi }_1{\pi }_2\cdots {\pi }_n$ has the same relative order as some permutation ${\sigma }_1{\sigma }_2\cdots {\sigma }_n\in {\mathfrak{S}}_n$. This allows us to count the number of permutations in types $C_n$ and $D_n$ with a given peak set $S$ by reducing the enumeration to calculations in the symmetric group and sums across the rows of Pascal’s triangle.

We study Fox colorings of knots that are 13-colorable. We prove that any 13-colorable knot has a diagram that uses exactly five of the thirteen colors that are assigned to the arcs of the diagram. Due to an existing lower bound, this gives that the minimum number of colors of any 13-colorable knot is 5.

Let $X$ be the affine flag manifold of Lie type $A_1^1$. Its moment graph encodes the torus fixed points (which are elements of the infinite dihedral group $D_{\infty }$) and the torus stable curves in $X$. Given a fixed point $u\in D_{\infty }$ and a degree $d=\left(d_0,d_1\right)\in {\mathbb{Z}}_{\ge 0}^2$, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of $X$ which can be reached from $u$ using a chain of curves of total degree $\le d$. In this paper we give a formula for these elements, using combinatorics of the affine root system of type $A_1^1$.

In this paper, we introduce a new algorithm based on total variation for denoising speckle noise images. Total variation was introduced by Rudin, Osher, and Fatemi in 1992 for regularizing images. Chambolle proposed a faster algorithm based on the duality of convex functions for minimizing the total variation, but his algorithm was built for Gaussian noise removal. Unlike Gaussian noise, which is additive, speckle noise is multiplicative. We modify the original Chambolle algorithm for speckle noise images using the first noise equation for speckle denoising, proposed by Krissian, Kikinis, Westin and Vosburgh in 2005. We apply the Chambolle algorithm to the Krissian et al. speckle denoising model to develop a faster algorithm for speckle noise images.

We define an abstract Apollonian supergasket using the solution set of a certain Diophantine equation, showing that the solutions are in bijective correspondence with the circles of any concrete supergasket. Properties of the solution set translate directly to geometric and algebraic properties of Apollonian gaskets, facilitating their study. In particular, curvatures of individual circles are explored and geometric relationships among multiple circles are given simple algebraic expressions. All results can be applied to a concrete gasket using the curvature-center coordinates of its four defining circles. These techniques can also be applied to other types of circle packings and higher-dimensional analogs.