We use Nathanson’s $g$-adic representation of integers to relate metric properties of Cayley graphs of the integers with respect to various infinite generating sets $S$ to problems in additive number theory. If $S$ consists of all powers of a fixed integer $g$, we find explicit formulas for the smallest positive integer of a given length. This is related to finding the smallest positive integer expressible as a fixed number of sums and differences of powers of $g$. We also consider $S$ to be the set of all powers of all primes and bound the diameter of this Cayley graph by relating it to Goldbach’s conjecture.

Given a graph $G$, the exponential distance matrix is defined entrywise by letting the $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then $q^{\text{dist}(u,v)}=0$. We will establish several properties of the characteristic polynomial (spectrum) for this matrix, give some families of graphs which are uniquely determined by their spectrum, and produce cospectral constructions.

We present three classes of abstract prearithmetics, ${\{A_M\}}_{M\ge 1}$, ${\{A_{-M,M}\}}_{M\ge 1}$, and . The first is weakly projective with respect to the nonnegative real Diophantine arithmetic $R_{+}=({ℝ}_{+},+,\times ,{\le }_{{ℝ}_{+}})$, the second is weakly projective with respect to the real Diophantine arithmetic $R=(ℝ,+,\times ,{\le }_{ℝ})$, while the third is exactly projective with respect to the extended real Diophantine arithmetic $\overline{R}=(\overline{ℝ},+,\times ,{\le }_{\overline{ℝ}})$. In addition, we have that every $A_M$ and every $B_M$ is a complete totally ordered semiring, while every $A_{-M,M}$ is not. We show that the projection of any series of elements of ${ℝ}_{+}$ converges in $A_M$, for any $M\ge 1$, and that the projection of any nonindeterminate series of elements of $ℝ$ converges in $A_{-M,M}$, for any $M\ge 1$, and in $B_M$, for all . We also prove that working in $A_M$ and in $A_{-M,M}$, for any $M\ge 1$, and in $B_M$, for all , allows us to overcome a version of the paradox of the heap.

The main contribution of this manuscript is a local normal form for Hamiltonian actions of Poisson–Lie groups $K$ on a symplectic manifold equipped with an $AN$-valued moment map, where $AN$ is the dual Poisson–Lie group of $K$. Our proof uses the delinearization theorem of Alekseev which relates a classical Hamiltonian action of $K$ with a ${\mathfrak{𝔨}}^{\ast }$-valued moment map to a Hamiltonian action with an $AN$-valued moment map, via a deformation of symplectic structures. We obtain our main result by proving a “delinearization commutes with symplectic quotients” theorem which is also of independent interest and then putting this together with the local normal form theorem for classical Hamiltonian actions with $\mathfrak{𝔨}\ast$-valued moment maps. A key ingredient for our main result is the delinearization $\mathcal{𝒟}({\omega }_{\text{can}})$ of the canonical symplectic structure on $T^{\ast }K$, so we additionally take some steps toward explicit computations of $\mathcal{𝒟}({\omega }_{\text{can}})$. In particular, in the case $K=\text{SU}(2)$, we obtain explicit formulas for the matrix coefficients of $\mathcal{𝒟}({\omega }_{\text{can}})$ with respect to a natural choice of coordinates on $T^{\ast }\text{SU}(2)$.

We construct, for any given positive integer $n$, Reeb flows on contact integral homology 3-spheres which do not admit global surfaces of section with fewer than $n$ boundary components. We use a connected sum operation for open books to construct such systems. Moreover, we prove that this property is stable with respect to $C^{4+𝜖}$-small perturbations of the Hamiltonian given on the symplectization.

One of the most important historical contributions to the inverse eigenvalue problem associated with trees is the celebrated Parter–Wiener theorem. We offer an alternate elementary proof of this seminal result, utilizing the basic matrix tool known as the Schur-complement of a matrix, in connection with analyzing the nullspace structure of matrices whose graph is a tree.

For integers $k,r\ge 2$, the diagonal Ramsey number $R_r(k)$ is the minimum $N\in \mathbb{N}$ such that every $r$-coloring of the edges of a complete graph on $N$ vertices yields a monochromatic subgraph on $k$ vertices. Here we make a careful effort of extracting explicit upper bounds for $R_r(k)$ from the pigeonhole principle alone. Our main term improves on previously documented explicit bounds for $r\ge 3$, and we also consider an often-ignored secondary term, which allows us to subtract a positive proportion of the main term that is uniformly bounded below. Asymptotically, we give a self-contained proof that

$$R_r(k)\le \left(\frac{3+e}{2}\right)\frac{(r(k-2))!}{{((k-2)!)}^r}(1+o_{r\to \infty }(1)),$$

and we conclude by noting that our methods combine with previous estimates on $R_r(3)$ to improve the constant $\frac{1}{2}(3+e)$ to $\frac{1}{2}(3+e)-\frac{1}{48}d$, where $d=66-R_4(3)\ge 4$. We also compare our formulas, and previously documented formulas, to some collected numerical data.

Let $\tau$ be an $n$-simplex and let $g$ be a metric on $\tau$ with constant curvature $\kappa$. The lengths that $g$ assigns to the edges of $\tau$, along with the value of $\kappa$, uniquely determine all of the geometry of $(\tau ,g)$. We focus on hyperbolic simplices ($\kappa =-1$) and develop geometric formulas which rely only on the edge lengths of $\tau$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $\kappa$.