We study stochastic convergence of random polygon sequences and establish several criteria for a sequence of random polygons to converge almost surely to a random limit point. We also explore a special case in which the limit point is prescribed. Existing literature on convergence of polygon sequences can be considered as a case study herein when all the participating random variables obey Dirac delta distributions.

Suppose $X$ is a smooth projective connected curve defined over an algebraically closed field of characteristic $p>0$ and $B\subset X$ is a finite, possibly empty, set of points. Booher and Cais determined a lower bound for the $a$-number of a $\mathbb{Z}∕p\mathbb{Z}$-cover of $X$ with branch locus $B$. For odd primes $p$, in most cases it is not known if this lower bound is realized. In this note, when $X$ is ordinary, we use formal patching to reduce that question to a computational question about $a$-numbers of $\mathbb{Z}∕p\mathbb{Z}$-covers of the affine line. As an application, when $p=3$ or $p=5$, for any ordinary curve $X$ and any choice of $B$, we prove that the lower bound is realized for Artin–Schreier covers of $X$ with branch locus $B$.

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\ge q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot \text{Cr}{(K)}^{1∕2}$. This provides an example of an upper bound on folded ribbonlength that is sublinear in crossing number.

We identify real numbers $c>1$, as large as possible, such that every large integer $N$ can be expressed as

$$N=[u^c]+[v^c]+[w^c],$$

where $u,v,w$ are positive integers. We also consider the cases with one or two of $u,v,w$ from a sparse subset of positive integers.

Repeatedly differentiating a polynomial with distinct real roots and then finding the roots of each derivative produces a sequence of real numbers. The classical Rolle’s theorem, typically studied in first-semester calculus, provides some constraints on the ordering of these roots. However, not all root sequences that are allowed by Rolle’s theorem occur for polynomials with all real roots. We use elementary methods to prove several Rolle’s-type theorems that further constrain the orderings of the roots of polynomials and their derivatives.

We consider the problem of fair pricing and hedging in the sense of Föllmer and Schweizer (1989) under small perturbations of the numéraire. We show that for replicable claims, the change of numéraire affects neither the fair price nor the hedging strategy. For nonreplicable claims, we demonstrate that is not the case. By reformulating the key stochastic control problem in a more tractable form, we show that both the fair price and optimal strategy are stable with respect to small perturbations of the numéraire. Further, our approach allows for explicit asymptotic formulas describing the fair price and hedging strategy’s leading order correction terms. Mathematically, our results constitute stability and asymptotic analysis of a stochastic control problem under certain perturbations of the integrator of the controlled process, where constraints make this problem hard to analyze.

The obstruction set for graphs with knotless embeddings is not known, but a recent paper of Goldberg, Mattman, and Naimi indicates that it is quite large. Almost all known obstructions fall into four triangle-Y families and they ask if there is an efficient way of finding or estimating the size of such graph families. Inspired by this question, we investigate the family size for complete multipartite graphs. Aside from three families that appear to grow exponentially, these families stabilize: after a certain point, increasing the number of vertices in a fixed part does not change family size.

We prove the scale-invariant elliptic Harnack inequality (EHI) for nonnegative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of the EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the local central limit theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical balayage formula then imply the EHI.

We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.