The functional-differential equation $f^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$,$$\underset{1,}{\underbrace{1,}}\underset{2,}{\underbrace{2,2,}}\underset{2,}{\underbrace{3,3,}}\underset{3,}{\underbrace{4,4,4}}\underset{3,}{\underbrace{5,5,5,}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$$We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number $p\ge 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact...

In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range $3332-4096$ which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range $128-3272$. We also construct some dense lattices of dimensions in the range $4098-8232$. Finally we also obtain some new lattices of moderate dimensions such as $68,84,85,86$, which are denser than the...

Some aspects of duality triads introduced recently are discussed. In particular, the general solution for the triad polynomials is given. Furthermore, a generalization of the notion of duality triad is proposed and some simple properties of these generalized duality triads are derived.

We prove that for every ε > 0 and every nonnegative integer w there exist primes $p_1,...,p_w$ such that for $n=p_1...p_w$ the height of the cyclotomic polynomial ${\Phi }_n$ is at least $(1-\epsilon )c_w{M}_n$, where $M_n={\prod }_{i=1}^{w-2}{p}_i^{2^{w-1-i}-1}$ and $c_w$ is a constant depending only on w; furthermore $li{m}_{w\to \infty }{c}_w^{2^{-w}}\approx 0.71$. In our construction we can have $p_i>h(p_1...p_{i-1})$ for all i = 1,...,w and any function h: ℝ₊ → ℝ₊.

The Pell sequence ${\left(P_n\right)}_{n=0}^{\infty }$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$. In this paper, we investigate a generalization of the Pell sequence called the $k$-generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced....

The set $𝒰{𝒟}_r$ of point sets of ${ℝ}^n,n\ge 1$, having the property that their minimal interpoint distance is greater than a given strictly positive constant $r\>0$ is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset $𝒰{𝒟}_{r,f}\subset 𝒰{𝒟}_r$ of the finite point sets is compatible with the restriction of this topology to $𝒰{𝒟}_{r,f}$. We show that its subsets of Delone sets of given constants in ${ℝ}^n,n\ge 1$, are compact. Three (classes of) metrics, whose one of crystallographic nature,...