We generate Fibonacci-type recursive polynomials through continued fractions. These polynomials generalize the well-known Fibonacci and Lucas polynomials. We present several results such as generating functions, finite sum representations and identities satisfied by these polynomials. Analytic results about the zeros of these polynomials are also presented.

We investigate when sets which are winning in the sense of Schmidt games have full Hausdorff dimension. The classical result by Schmidt asserts that winning sets for games played in Euclidean spaces have full dimension. We recover this type of result for games played on attractors of contracting iterated function systems: either on a complete metric space with semiconformal contractions or on ${ℝ}^n$ and an iterated function system of $C^1$ conformal maps with Hölder continuous derivative.

We consider a model of electrons in a finite topological insulator. We numerically study the propagation of electronic wave-packets localized near edges of the structure in the presence of defects and random disorder. We compare the propagation with computations of the spectral localizer index: a spatially local topological index. We find that without disorder, wave-packets propagate along boundaries between regions of differing spectral localizer index with minimal loss, even in the presence of strong defects. With disorder, wave-packets still propagate along boundaries between regions of differing localizer index, but lose significant mass as they propagate. We also find that with disorder, the localizer gap, a measure of the localizer index “strength”, is generally smaller away from the boundary than without disorder. Based on this result, we conjecture that wave-packets propagating along boundaries between regions of differing spectral localizer index do not lose significant mass whenever the localizer gap is sufficiently large on both sides of the boundary.

In 2000, Thomas Fink and Yong Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The ends of a neck tie can be joined together, which gives a physical model of a mathematical knot that we call a tie knot. In this paper we classify the knot type of each of Fink and Mao’s 85 tie knots. We describe how the unknot, left and right trefoil, twist knots and $\left(2,p\right)$ torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating.

We study the expected number of zeros of

$P_n\left(z\right)=\sum _{k=0}^n{\eta }_k{p}_k\left(z\right),$

where $\left\{{\eta }_k\right\}$ are complex-valued independent and identically distributed standard Gaussian random variables, and $\left\{p_k\left(z\right)\right\}$ are polynomials orthogonal on the unit disk. When $p_k\left(z\right)=\sqrt{\left(k+1\right)∕\pi }{z}^k$, $k\in \left\{0,1,\dots ,n\right\}$, we give an explicit formula for the expected number of zeros of $P_n\left(z\right)$ in a disk of radius $r\in \left(0,1\right)$ centered at the origin. From our formula we establish the limiting value of the expected number of zeros, the expected number of zeros in a radially expanding disk, and show that the expected number of zeros in the unit disk is $2n∕3$. Generalizing our basis functions $\left\{p_k\left(z\right)\right\}$ to be regular in the sense of Ullman–Stahl–Totik and that the measure of orthogonality associated to polynomials is absolutely continuous with respect to planar Lebesgue measure, we give the limiting value of the expected number of zeros in a disk of radius $r\in \left(0,1\right)$ centered at the origin, and show that asymptotically the expected number of zeros in the unit disk is $2n∕3$.

It is a surprising fact that the proportion of integer lattice points visible from the origin is exactly $6∕{\pi }^2$, or approximately 60%. Hence, approximately 40% of the integer lattice is hidden from the origin. Since 1971, many have studied a variety of problems involving lattice-point visibility, in particular, searching for patterns in that 40% of the lattice composed of invisible points. One such pattern is a square patch, an $n\times n$ grid of $n^2$ invisible points, which we call a hidden forest. It is known that there exist arbitrarily large hidden forests in the integer lattice. However, the methods up to now involve the Chinese remainder theorem (CRT) on the rows and columns of matrices with prime number entries, and they have only been able to locate hidden forests very far from the origin. For example, using this method the closest known $4\times 4$ hidden forest is over 3 quintillion, or $3\times 10^{18}$, units away from the origin. We introduce the concept of quasiprime matrices and utilize a variety of computational and theoretical techniques to find some of the closest known hidden forests to date. Using these new techniques, we find a $4\times 4$ hidden forest that is merely 184 million units away from the origin. We conjecture that every hidden forest can be found via the CRT-algorithm on a quasiprime matrix.

We find explicit formulas for the coefficients of a family of polynomials arising in higher-dimensional complex analysis.

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $\left(x_1,\dots ,x_n\right)\in {\left[0,1\right]}^n$ cyclic if there exist independent random variables $U_1,\dots ,U_n$ with $P\left(U_i=U_j\right)=0$ for $i\ne j$ such that for $i=1,\dots ,n-1$ and . We call the tuple $\left(x_1,\dots ,x_n\right)$ nontransitive if it is cyclic and in addition satisfies for all $i$.

Let $p_n$ (resp. $p_n^{\ast }$)denote the probability that a randomly chosen $n$-tuple $\left(x_1,\dots ,x_n\right)\in {\left[0,1\right]}^n$ is cyclic (resp. nontransitive). We determine $p_3$ and $p_3^{\ast }$ exactly, while for $n\ge 4$ we give upper and lower bounds for $p_n$ that show that $p_n$ converges to $1$ as $n\to \infty$. We also determine the distribution of the smallest, middle, and largest elements in a cyclic triple.

Let $H\left(ℂ\right)$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda ,b\in ℂ$, let $C_{\lambda ,b}:H\left(ℂ\right)\to H\left(ℂ\right)$ be the composition operator $C_{\lambda ,b}f\left(z\right)=f\left(\lambda z+b\right)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the nonconvolution operators $T_{\lambda ,b}=C_{\lambda ,b}\circ D$ by showing that whenever $\left|\lambda \right|\ge 1$, the collection of operators

$\left\{\psi \left(T_{\lambda ,b}\right):\psi \left(z\right)\in H\left(ℂ\right),\psi \left(0\right)=0\text{ and }\psi \left(T_{\lambda ,b}\right)\text{ is continuous}\right\}$

forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators

$\left\{C_{\lambda ,b}\circ \phi \left(D\right):\phi \left(z\right)\text{ is an entire function of exponential type with }\phi \left(0\right)=0\right\}$

consists entirely of hypercyclic operators.