In this paper, we investigate the dynamics of iterating the Weierstrass elliptic functions on vertical real rhombic lattices. The main result of this paper is to show that these functions can have at most one real attracting or parabolic cycle. If there is no real attracting or parabolic cycle, we prove that the real and imaginary axes, as well as translations of these lines by the lattice, lie in the Julia set. Further, we prove that if there exists a real attracting fixed point, then the intersection of the Julia set with the real axis is a Cantor set. Finally, we apply the theorem to find parameters in every real rhombic shape equivalence class for which the Julia set is the entire sphere.

We consider pattern avoidance in a subset of words on $\left\{1,1,2,2,\dots ,n,n\right\}$ called double lists. We enumerate double lists avoiding any permutation pattern of length at most 4 and completely determine the corresponding Wilf classes.

The focus of this note is to learn more about the Kolmogorov equation describing the dynamics of a randomly accelerated particle. We first explore some existing results of the Kolmogorov equation from the stochastic and differential equation points of view and discuss its solvability with and without boundary conditions. More specifically, we introduce stochastic processes and Brownian motion and we present a connection between a stochastic process and a differential equation. After looking at stochastic processes, we introduce generalized functions and derive the fundamental solution to the heat equation and to the Fokker–Planck equation. The problem with a reflecting boundary condition is also studied by using various methods such as separation of variables, self-similarity, and the reflection method.

The link of the $A_n$ singularity, $L_{A_n}\subset {ℂ}^3$ admits a natural contact structure ${\xi }_0$ coming from the set of complex tangencies. The canonical contact form ${\alpha }_0$ associated to ${\xi }_0$ is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to ${\xi }_0$ that has two isolated simple periodic Reeb orbits. We compute the Conley–Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive $S^1$-equivariant symplectic homology groups for $\left(L_{A_n},{\xi }_0\right)$. In addition, we prove that $\left(L_{A_n},{\xi }_0\right)$ is contactomorphic to the lens space $L\left(n+1,n\right)$, equipped with its canonical contact structure ${\xi }_{std}$.

The asymptotic estimation of the vibration spectrum for a system of two identical Euler–Bernoulli beams coupled via each of the four standard types of linear dissipative joint has been solved for the case when one beam is clamped and the other beam is free at the outer ends. Here, we generalize those results and solve the problem for all 40 combinations of energy-conserving end conditions. We provide both asymptotic and numerical results, and we compare the various systems with an eye toward determining which configurations lead to asymptotically equivalent vibration spectra.

A loxodrome is a curve that makes a constant angle with the meridians. We use conformal maps and the notion of parallel transport in differential geometry to investigate loxodromes on hypersurfaces of revolution and their spiral behavior near a pole.

Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast with high-order regularizations, the low-order regularizations are easier to implement numerically. Moreover, our methods give a theoretical foundation for this approach.

We study the dynamical behavior of functions on vertices of a graph that are contractions in the graph metric. We show that the fixed point set of such functions must be convex. If a function has no fixed points and the graph is a tree, we prove that every dynamical cycle must have an even period and the function behaves eventually like a symmetry.

In this paper, we study tilings of annular regions in the integer lattice by skew and T-tetrominoes. We demonstrate the tileability of most annular regions by the given tile set, enumerate the tilings of width-2 annuli, and determine the tile counting group associated to this tile set and the family of all width-2 annuli.

We present a $q$-analogue of the sum of cubes, give an interpretation in terms of overpartitions, and provide a combinatorial proof. In addition, we note a connection between a generating function for overpartitions and the $q$-Delannoy numbers.

Ulrich bundles play a central role in singularity theory, liaison theory and Boij–Söderberg theory. It was proved by the first author together with Costa, Huizenga, Miró-Roig and Woolf that Schur bundles on flag varieties of three or more steps are not Ulrich and conjectured a classification of Ulrich Schur bundles on two-step flag varieties. By the Borel–Weil–Bott theorem, the conjecture reduces to classifying integer sequences satisfying certain combinatorial properties. In this paper, we resolve the first instance of this conjecture and show that Schur bundles on $F\left(k,k+3;n\right)$ are not Ulrich if $n>6$ or $k>2$.